Big O notation - Wikipedia

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</li> <li id="toc-Multiple_variables" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Multiple_variables"> <div class="vector-toc-text"> 5 Multiple variables </div> </a> <ul id="toc-Multiple_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matters_of_notation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Matters_of_notation"> <div class="vector-toc-text"> 6 Matters of notation </div> </a> <button aria-controls="toc-Matters_of_notation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Matters of notation subsection </button> <ul id="toc-Matters_of_notation-sublist" class="vector-toc-list"> <li id="toc-Equals_sign" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equals_sign"> <div class="vector-toc-text"> 6.1 Equals sign </div> </a> <ul id="toc-Equals_sign-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_arithmetic_operators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_arithmetic_operators"> <div class="vector-toc-text"> 6.2 Other arithmetic operators </div> </a> <ul id="toc-Other_arithmetic_operators-sublist" class="vector-toc-list"> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> 6.2.1 Example </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Multiple_uses" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_uses"> <div class="vector-toc-text"> 6.3 Multiple uses </div> </a> <ul id="toc-Multiple_uses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Typesetting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Typesetting"> <div class="vector-toc-text"> 6.4 Typesetting </div> </a> <ul id="toc-Typesetting-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orders_of_common_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Orders_of_common_functions"> <div class="vector-toc-text"> 7 Orders of common functions </div> </a> <ul id="toc-Orders_of_common_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_asymptotic_notations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_asymptotic_notations"> <div class="vector-toc-text"> 8 Related asymptotic notations </div> </a> <button aria-controls="toc-Related_asymptotic_notations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Related asymptotic notations subsection </button> <ul id="toc-Related_asymptotic_notations-sublist" class="vector-toc-list"> <li id="toc-Little-o_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Little-o_notation"> <div class="vector-toc-text"> 8.1 Little-o notation </div> </a> <ul id="toc-Little-o_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Big_Omega_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Big_Omega_notation"> <div class="vector-toc-text"> 8.2 Big Omega notation </div> </a> <ul id="toc-Big_Omega_notation-sublist" class="vector-toc-list"> <li id="toc-The_Hardy–Littlewood_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_Hardy–Littlewood_definition"> <div class="vector-toc-text"> 8.2.1 The Hardy–Littlewood definition </div> </a> <ul id="toc-The_Hardy–Littlewood_definition-sublist" class="vector-toc-list"> <li id="toc-Simple_examples" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Simple_examples"> <div class="vector-toc-text"> 8.2.1.1 Simple examples </div> </a> <ul id="toc-Simple_examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_Knuth_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_Knuth_definition"> <div class="vector-toc-text"> 8.2.2 The Knuth definition </div> </a> <ul id="toc-The_Knuth_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Family_of_Bachmann–Landau_notations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Family_of_Bachmann–Landau_notations"> <div class="vector-toc-text"> 8.3 Family of Bachmann–Landau notations </div> </a> <ul id="toc-Family_of_Bachmann–Landau_notations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Use_in_computer_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Use_in_computer_science"> <div class="vector-toc-text"> 8.4 Use in computer science </div> </a> <ul id="toc-Use_in_computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_notation"> <div class="vector-toc-text"> 8.5 Other notation </div> </a> <ul id="toc-Other_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extensions_to_the_Bachmann–Landau_notations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extensions_to_the_Bachmann–Landau_notations"> <div class="vector-toc-text"> 8.6 Extensions to the Bachmann–Landau notations </div> </a> <ul id="toc-Extensions_to_the_Bachmann–Landau_notations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations_and_related_usages" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations_and_related_usages"> <div class="vector-toc-text"> 9 Generalizations and related usages </div> </a> <ul id="toc-Generalizations_and_related_usages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History_(Bachmann–Landau,_Hardy,_and_Vinogradov_notations)" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History_(Bachmann–Landau,_Hardy,_and_Vinogradov_notations)"> <div class="vector-toc-text"> 10 History (Bachmann–Landau, Hardy, and Vinogradov notations) </div> </a> <ul id="toc-History_(Bachmann–Landau,_Hardy,_and_Vinogradov_notations)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 11 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References_and_notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References_and_notes"> <div class="vector-toc-text"> 12 References and notes </div> </a> <ul id="toc-References_and_notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 13 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Big O notation</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 36 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-36" aria-hidden="true" > 36 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%85%D8%AB%D9%8A%D9%84_O_%D8%A7%D9%84%D9%83%D8%A8%D8%B1%D9%89" title="تمثيل O الكبرى – Arabic" lang="ar" hreflang="ar" data-title="تمثيل O الكبرى" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/B%C3%B6y%C3%BCk_O_i%C5%9Far%C9%99l%C9%99r_sistemi" title="Böyük O işarələr sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Böyük O işarələr sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%A1%E0%A6%BC_O_%E0%A6%B2%E0%A6%BF%E0%A6%96%E0%A6%A8%E0%A6%AA%E0%A6%A6%E0%A7%8D%E0%A6%A7%E0%A6%A4%E0%A6%BF" title="বড় O লিখনপদ্ধতি – Bangla" lang="bn" hreflang="bn" data-title="বড় O লিখনপদ্ধতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9E-%D0%BD%D0%B0%D1%82%D0%B0%D1%86%D1%8B%D1%8F" title="О-натацыя – Belarusian" lang="be" hreflang="be" data-title="О-натацыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Notaci%C3%B3_de_Landau" title="Notació de Landau – Catalan" lang="ca" hreflang="ca" data-title="Notació de Landau" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Landauova_notace" title="Landauova notace – Czech" lang="cs" hreflang="cs" data-title="Landauova notace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Landau-Symbole" title="Landau-Symbole – German" lang="de" hreflang="de" data-title="Landau-Symbole" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cota_superior_asint%C3%B3tica" title="Cota superior asintótica – Spanish" lang="es" hreflang="es" data-title="Cota superior asintótica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Granda_O" title="Granda O – Esperanto" lang="eo" hreflang="eo" data-title="Granda O" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Goi_borne_asintotiko" title="Goi borne asintotiko – Basque" lang="eu" hreflang="eu" data-title="Goi borne asintotiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%85%D8%A7%D8%AF_O_%D8%A8%D8%B2%D8%B1%DA%AF" title="نماد O بزرگ – Persian" lang="fa" hreflang="fa" data-title="نماد O بزرگ" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Comparaison_asymptotique" title="Comparaison asymptotique – French" lang="fr" hreflang="fr" data-title="Comparaison asymptotique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%90%EA%B7%BC_%ED%91%9C%EA%B8%B0%EB%B2%95" title="점근 표기법 – Korean" lang="ko" hreflang="ko" data-title="점근 표기법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%A1%E0%A4%BC%E0%A4%BE_%E0%A4%93_%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A5%87%E0%A4%A4%E0%A4%A8" title="बड़ा ओ संकेतन – Hindi" lang="hi" hreflang="hi" data-title="बड़ा ओ संकेतन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Notasi_O_besar" title="Notasi O besar – Indonesian" lang="id" hreflang="id" data-title="Notasi O besar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/O-grande" title="O-grande – Italian" lang="it" hreflang="it" data-title="O-grande" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99" title="סימון אסימפטוטי – Hebrew" lang="he" hreflang="he" data-title="סימון אסימפטוטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%A1%E1%83%98%E1%83%9B%E1%83%9E%E1%83%A2%E1%83%9D%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%90%E1%83%A6%E1%83%9C%E1%83%98%E1%83%A8%E1%83%95%E1%83%9C%E1%83%90_O-%E1%83%93%E1%83%98%E1%83%93%E1%83%98" title="ასიმპტოტური აღნიშვნა O-დიდი – Georgian" lang="ka" hreflang="ka" data-title="ასიმპტოტური აღნიშვნა O-დიდი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/O_jel%C3%B6l%C3%A9s" title="O jelölés – Hungarian" lang="hu" hreflang="hu" data-title="O jelölés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Grote-O-notatie" title="Grote-O-notatie – Dutch" lang="nl" hreflang="nl" data-title="Grote-O-notatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%B3%E3%83%80%E3%82%A6%E3%81%AE%E8%A8%98%E5%8F%B7" title="ランダウの記号 – Japanese" lang="ja" hreflang="ja" data-title="ランダウの記号" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Stor_O-notasjon" title="Stor O-notasjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Stor O-notasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Asymptotyczne_tempo_wzrostu" title="Asymptotyczne tempo wzrostu – Polish" lang="pl" hreflang="pl" data-title="Asymptotyczne tempo wzrostu" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grande-O" title="Grande-O – Portuguese" lang="pt" hreflang="pt" data-title="Grande-O" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Nota%C8%9Bia_Big_O" title="Notația Big O – Romanian" lang="ro" hreflang="ro" data-title="Notația Big O" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%C2%ABO%C2%BB_%D0%B1%D0%BE%D0%BB%D1%8C%D1%88%D0%BE%D0%B5_%D0%B8_%C2%ABo%C2%BB_%D0%BC%D0%B0%D0%BB%D0%BE%D0%B5" title="«O» большое и «o» малое – Russian" lang="ru" hreflang="ru" data-title="«O» большое и «o» малое" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Big_O_notation" title="Big O notation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Big O notation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/O_notacija" title="O notacija – Slovenian" lang="sl" hreflang="sl" data-title="O notacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%92%D0%B5%D0%BB%D0%B8%D0%BA%D0%BE_%D0%9E" title="Велико О – Serbian" lang="sr" hreflang="sr" data-title="Велико О" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ordo" title="Ordo – Swedish" lang="sv" hreflang="sv" data-title="Ordo" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%B1%E0%B8%8D%E0%B8%81%E0%B8%A3%E0%B8%93%E0%B9%8C%E0%B9%82%E0%B8%AD%E0%B9%83%E0%B8%AB%E0%B8%8D%E0%B9%88" title="สัญกรณ์โอใหญ่ – Thai" lang="th" hreflang="th" data-title="สัญกรณ์โอใหญ่" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/B%C3%BCy%C3%BCk_O_g%C3%B6sterimi" title="Büyük O gösterimi – Turkish" lang="tr" hreflang="tr" data-title="Büyük O gösterimi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%82%D0%B0%D1%86%D1%96%D1%8F_%D0%9B%D0%B0%D0%BD%D0%B4%D0%B0%D1%83" title="Нотація Ландау – Ukrainian" 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class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Big-O-notation.png/300px-Big-O-notation.png" decoding="async" width="300" height="282" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Big-O-notation.png/450px-Big-O-notation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Big-O-notation.png/600px-Big-O-notation.png 2x" data-file-width="661" data-file-height="622" /></a><figcaption>Example of Big O notation: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {red}f(x)}=O{\color {blue}(g(x))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <mo>=</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {red}f(x)}=O{\color {blue}(g(x))}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5ddc938d43066aaaaffe5b12a4462ae9594ae5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.354ex; height:2.843ex;" alt="{\displaystyle {\color {red}f(x)}=O{\color {blue}(g(x))}}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda2caf97ec29f30d5f0c0cd7135393361efc020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\to \infty }"> since there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f423ab77b3411ec2803520a07c0dfae6ceb826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.703ex; height:2.176ex;" alt="{\displaystyle M>0}"> (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca00f82608101fe96e18b56e5a0626193feff2c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.703ex; height:2.176ex;" alt="{\displaystyle M=1}">) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> (e.g.,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb111f85ebf1abbc8b78932a621d47dab364bda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{0}=5}">) such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq {\color {red}f(x)}\leq M{\color {blue}g(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <mo>≤</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq {\color {red}f(x)}\leq M{\color {blue}g(x)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f601e4f2f09c6dd249ce7762a8facefef534fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.474ex; height:2.843ex;" alt="{\displaystyle 0\leq {\color {red}f(x)}\leq M{\color {blue}g(x)}}"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06eb265a902eeabdb1149c4e0682a9a52ec3a4a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.343ex;" alt="{\displaystyle x\geq x_{0}}">.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline 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.mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks" style="width:18em; text-align:center; font-size:95%;"><tbody><tr><th class="sidebar-title" style="font-size:125%; font-weight:bold;">Fit approximation</th></tr><tr><td class="sidebar-image"><a href="/wiki/File:BigOnotationfunctionapprox_popular.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/BigOnotationfunctionapprox_popular.svg/80px-BigOnotationfunctionapprox_popular.svg.png" decoding="async" width="80" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/BigOnotationfunctionapprox_popular.svg/120px-BigOnotationfunctionapprox_popular.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/BigOnotationfunctionapprox_popular.svg/160px-BigOnotationfunctionapprox_popular.svg.png 2x" data-file-width="35" data-file-height="13" /></a></td></tr><tr><th class="sidebar-heading"> Concepts</th></tr><tr><td class="sidebar-content hlist" style="line-height:1.4em;"> <ul><li><a href="/wiki/Order_of_approximation" title="Order of approximation">Orders of approximation</a></li> <li><a href="/wiki/Scale_analysis_(mathematics)" title="Scale analysis (mathematics)">Scale analysis</a></li> <li><a class="mw-selflink selflink">Big O notation</a></li> <li><a href="/wiki/Curve_fitting" title="Curve fitting">Curve fitting</a></li> <li><a href="/wiki/False_precision" title="False precision">False precision</a></li> <li><a href="/wiki/Significant_figures" title="Significant figures">Significant figures</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Other fundamentals</th></tr><tr><td class="sidebar-content hlist" style="line-height:1.4em;"> <ul><li><a href="/wiki/Approximation" title="Approximation">Approximation</a></li> <li><a href="/wiki/Generalization_error" title="Generalization error">Generalization error</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor polynomial</a></li> <li><a href="/wiki/Scientific_modelling" title="Scientific modelling">Scientific modelling</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Order-of-approx" title="Template:Order-of-approx"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Order-of-approx" title="Template talk:Order-of-approx"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Order-of-approx" title="Special:EditPage/Template:Order-of-approx"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Big O notation is a mathematical notation that describes the <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">limiting behavior</a> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> when the <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a> tends towards a particular value or infinity. Big O is a member of a <a href="#Related_asymptotic_notations">family of notations</a> invented by German mathematicians <a href="/wiki/Paul_Gustav_Heinrich_Bachmann" title="Paul Gustav Heinrich Bachmann">Paul Bachmann</a>,<a href="#cite_note-Bachmann-1">[1]</a> <a href="/wiki/Edmund_Landau" title="Edmund Landau">Edmund Landau</a>,<a href="#cite_note-Landau-2">[2]</a> and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for <a href="https://en.wiktionary.org/wiki/Ordnung#German" class="extiw" title="wikt:Ordnung">Ordnung</a>, meaning the <a href="/wiki/Order_of_approximation" title="Order of approximation">order of approximation</a>. In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, big O notation is used to <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">classify algorithms</a> according to how their run time or space requirements grow as the input size grows.<a href="#cite_note-:0-3">[3]</a> In <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>, big O notation is often used to express a bound on the difference between an <a href="/wiki/Arithmetic_function" title="Arithmetic function">arithmetical function</a> and a better understood approximation; a famous example of such a difference is the remainder term in the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: different functions with the same asymptotic growth rate may be represented using the same O notation. The letter O is used because the growth rate of a function is also referred to as the order of the function. A description of a function in terms of big O notation usually only provides an <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.<a href="#cite_note-:0-3">[3]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=1" title="Edit section: Formal definition">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9687ea22c0f310582e97ee5f6c6a5fca28203d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f,}"> the function to be estimated, be a <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> valued function, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>g</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4f953d0045fe6c7829caf82d345f4545c84ca71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.084ex; height:2.009ex;" alt="{\displaystyle \ g\,}"> the comparison function, be a real valued function. Let both functions be defined on some <a href="/wiki/Bounded_set" title="Bounded set">unbounded</a> <a href="/wiki/Subset" title="Subset">subset</a> of the positive <a href="/wiki/Real_number" title="Real number">real numbers</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"> be non-zero (often, but not necessarily, strictly positive) for all large enough values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"><a href="#cite_note-LandauO-4">[4]</a> One writes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45ededce98fd7f1d8866cff71d6311e6510e850" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.504ex; height:3.176ex;" alt="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to \infty }"> and it is read "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93e672050841d667b55e1ff33d567a3ed879d883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.579ex; height:2.843ex;" alt="{\displaystyle \ f(x)\ }"> is big O of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}">" or more often "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is of the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}">" if the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is at most a positive constant multiple of the absolute value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"> for all sufficiently large values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"> That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b171ef5a4bf9979e88b862b43b44196c70a2cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.674ex; height:3.176ex;" alt="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}"> if there exists a positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> and a real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)|\leq M\ |g(x)|\quad {\text{ for all }}x\geq x_{0}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>M</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>x</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq M\ |g(x)|\quad {\text{ for all }}x\geq x_{0}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c616aa2ad74f1d52a60648eedf973b7f06b96733" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.727ex; height:2.843ex;" alt="{\displaystyle |f(x)|\leq M\ |g(x)|\quad {\text{ for all }}x\geq x_{0}~.}"> In many contexts, the assumption that we are interested in the growth rate as the variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8870480257369121bf218c4814f88d4f5c43108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.491ex; height:1.676ex;" alt="{\displaystyle \ x\ }"> goes to infinity or to zero is left unstated, and one writes more simply that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd5c1bc330a1a7a4a1784b75d08d386a426434c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.321ex; height:3.176ex;" alt="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}.}"> The notation can also be used to describe the behavior of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> near some real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> (often, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}">): we say <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d86a6ccab54b0c0969bcda1c5021ced57fcc155" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.991ex; height:3.176ex;" alt="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a}"> if there exist positive numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> such that for all defined <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<|x-a|<\delta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<|x-a|<\delta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c09500e7e9381b238aa30d66c37a707ecc5781da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.748ex; height:2.843ex;" alt="{\displaystyle 0<|x-a|<\delta ,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)|\leq M|g(x)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq M|g(x)|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3b1907dc88a74ce67749e0ecb3ed7c4a48f0aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.447ex; height:2.843ex;" alt="{\displaystyle |f(x)|\leq M|g(x)|.}"> As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"> is non-zero for adequately large (or small) values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"> both of these definitions can be unified using the <a href="/wiki/Limit_superior" class="mw-redirect" title="Limit superior">limit superior</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d86a6ccab54b0c0969bcda1c5021ced57fcc155" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.991ex; height:3.176ex;" alt="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a}"> if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \limsup _{x\to a}{\frac {\left|f(x)\right|}{\left|g(x)\right|}}<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \limsup _{x\to a}{\frac {\left|f(x)\right|}{\left|g(x)\right|}}<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176cfe091b474a99b1a9e6c2c86b6a2cfd5d80aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.122ex; height:6.509ex;" alt="{\displaystyle \limsup _{x\to a}{\frac {\left|f(x)\right|}{\left|g(x)\right|}}<\infty .}"> And in both of these definitions the <a href="/wiki/Limit_point" class="mw-redirect" title="Limit point">limit point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> (whether <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"> or not) is a <a href="/wiki/Cluster_point" class="mw-redirect" title="Cluster point">cluster point</a> of the domains of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f2986cd965e404a1ee33ec84baee5c43da47fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g,}"> i. e., in every neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> there have to be infinitely many points in common. Moreover, as pointed out in the article about the <a href="/wiki/Limit_inferior_and_limit_superior#Real-valued_functions" title="Limit inferior and limit superior">limit inferior and limit superior</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \limsup _{x\to a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mrow> </munder> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \limsup _{x\to a}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/053814e8519054bbf8e10109acd77ad38e1052c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.804ex; height:2.676ex;" alt="{\displaystyle \textstyle \limsup _{x\to a}}"> (at least on the <a href="/wiki/Extended_real_number_line" title="Extended real number line">extended real number line</a>) always exists. In computer science, a slightly more restrictive definition is common: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> are both required to be functions from some unbounded subset of the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">positive integers</a> to the nonnegative real numbers; then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b171ef5a4bf9979e88b862b43b44196c70a2cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.674ex; height:3.176ex;" alt="{\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}"> if there exist positive integer numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63584d203ecb012a7bcb90f422408bbfe4018956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{0}}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(n)|\leq M|g(n)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(n)|\leq M|g(n)|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3346019a35f57a8347a1af4bc0aa46060e1cc73c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.931ex; height:2.843ex;" alt="{\displaystyle |f(n)|\leq M|g(n)|}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq n_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq n_{0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d69154ee97b1c6af1b7fc6b0240891a549c3666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.589ex; height:2.343ex;" alt="{\displaystyle n\geq n_{0}.}"><a href="#cite_note-5">[5]</a> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=2" title="Edit section: Example">edit</a>]</div> In typical usage the O notation is asymptotical, that is, it refers to very large x. In this setting, the contribution of the terms that grow "most quickly" will eventually make the other ones irrelevant. As a result, the following simplification rules can be applied: <ul><li>If f(x) is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted.</li> <li>If f(x) is a product of several factors, any constants (factors in the product that do not depend on x) can be omitted.</li></ul> For example, let f(x) = 6x4 − 2x3 + 5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x4, −2x3, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the second rule: 6x4 is a product of 6 and x4 in which the first factor does not depend on x. Omitting this factor results in the simplified form x4. Thus, we say that f(x) is a "big O" of x4. Mathematically, we can write f(x) = O(x4). One may confirm this calculation using the formal definition: let f(x) = 6x4 − 2x3 + 5 and g(x) = x4. Applying the <a href="#Formal_definition">formal definition</a> from above, the statement that f(x) = O(x4) is equivalent to its expansion, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)|\leq Mx^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>M</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq Mx^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05e585265f72da0079873c3bc4a8711ebd115d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.636ex; height:3.176ex;" alt="{\displaystyle |f(x)|\leq Mx^{4}}"> for some suitable choice of a real number x0 and a positive real number M and for all x > x0. To prove this, let x0 = 1 and M = 13. Then, for all x > x0: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}|6x^{4}-2x^{3}+5|&\leq 6x^{4}+|2x^{3}|+5\\&\leq 6x^{4}+2x^{4}+5x^{4}\\&=13x^{4}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≤</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≤</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>13</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|6x^{4}-2x^{3}+5|&\leq 6x^{4}+|2x^{3}|+5\\&\leq 6x^{4}+2x^{4}+5x^{4}\\&=13x^{4}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc926b093c721dee9f23185efef90e122c0acdf4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:35.4ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}|6x^{4}-2x^{3}+5|&\leq 6x^{4}+|2x^{3}|+5\\&\leq 6x^{4}+2x^{4}+5x^{4}\\&=13x^{4}\end{aligned}}}"> so <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |6x^{4}-2x^{3}+5|\leq 13x^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>13</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |6x^{4}-2x^{3}+5|\leq 13x^{4}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb23b946983d89787b6ee97bc4cab8ebd39aca96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.684ex; height:3.176ex;" alt="{\displaystyle |6x^{4}-2x^{3}+5|\leq 13x^{4}.}"> <div class="mw-heading mw-heading2"><h2 id="Use">Use</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=3" title="Edit section: Use">edit</a>]</div> Big O notation has two main areas of application: <ul><li>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, it is commonly used to describe <a href="#Infinitesimal_asymptotics">how closely a finite series approximates a given function</a>, especially in the case of a truncated <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> or <a href="/wiki/Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a>.</li> <li>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, it is useful in the <a href="#Infinite_asymptotics">analysis of algorithms</a>.<a href="#cite_note-:0-3">[3]</a></li></ul> In both applications, the function g(x) appearing within the O(·) is typically chosen to be as simple as possible, omitting constant factors and lower order terms. There are two formally close, but noticeably different, usages of this notation:[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <ul><li><a href="/wiki/Infinity" title="Infinity">infinite</a> asymptotics</li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> asymptotics.</li></ul> This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.[<a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research">original research?</a>] <div class="mw-heading mw-heading3"><h3 id="Infinite_asymptotics">Infinite asymptotics</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=4" title="Edit section: Infinite asymptotics">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Comparison_computational_complexity.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Comparison_computational_complexity.svg/220px-Comparison_computational_complexity.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Comparison_computational_complexity.svg/330px-Comparison_computational_complexity.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Comparison_computational_complexity.svg/440px-Comparison_computational_complexity.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Graphs of functions commonly used in the analysis of algorithms, showing the number of operations N versus input size n for each function</figcaption></figure> Big O notation is useful when <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">analyzing algorithms</a> for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size n might be found to be T(n) = 4n2 − 2n + 2. As n grows large, the n2 <a href="/wiki/Summand" class="mw-redirect" title="Summand">term</a> will come to dominate, so that all other terms can be neglected—for instance when n = 500, the term 4n2 is 1000 times as large as the 2n term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> become irrelevant if we compare to any other <a href="/wiki/Orders_of_approximation" class="mw-redirect" title="Orders of approximation">order</a> of expression, such as an expression containing a term n3 or n4. Even if T(n) = 1,000,000n2, if U(n) = n3, the latter will always exceed the former once n grows larger than 1,000,000, viz. T(1,000,000) = 1,000,0003 = U(1,000,000). Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(n)=O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)=O(n^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8526253f8646d3899c4b3f99b6b1c1d3d911a33c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.97ex; height:3.176ex;" alt="{\displaystyle T(n)=O(n^{2})}"></dd></dl> or <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(n)\in O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)\in O(n^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc66b3cf4747029fad4ae4d3a3c565bcf22ed237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.712ex; height:3.176ex;" alt="{\displaystyle T(n)\in O(n^{2})}"></dd></dl> and say that the algorithm has order of n2 time complexity. The sign "=" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is sometimes considered more accurate (see the "<a href="#Equals_sign">Equals sign</a>" discussion below) while the first is considered by some as an <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a>.<a href="#cite_note-clrs3-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="Infinitesimal_asymptotics">Infinitesimal asymptotics</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=5" title="Edit section: Infinitesimal asymptotics">edit</a>]</div> Big O can also be used to describe the <a href="/wiki/Taylor_series#Approximation_error_and_convergence" title="Taylor series">error term</a> in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. Consider, for example, the <a href="/wiki/Exponential_function#Formal_definition" title="Exponential function">exponential series</a> and two expressions of it that are valid when x is small: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dotsb &{\text{for all finite }}x\\[4pt]&=1+x+{\frac {x^{2}}{2}}+O(x^{3})&{\text{as }}x\to 0\\[4pt]&=1+x+O(x^{2})&{\text{as }}x\to 0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all finite </mtext> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dotsb &{\text{for all finite }}x\\[4pt]&=1+x+{\frac {x^{2}}{2}}+O(x^{3})&{\text{as }}x\to 0\\[4pt]&=1+x+O(x^{2})&{\text{as }}x\to 0\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3a5d996be04cc29fdf69a93368abe98e94fd45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:53.377ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dotsb &{\text{for all finite }}x\\[4pt]&=1+x+{\frac {x^{2}}{2}}+O(x^{3})&{\text{as }}x\to 0\\[4pt]&=1+x+O(x^{2})&{\text{as }}x\to 0\end{aligned}}}"></dd></dl> The middle expression (the one with O(x3)) means the absolute-value of the error ex − (1 + x + x2/2) is at most some constant times |x3| when x is close enough to 0. <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=6" title="Edit section: Properties">edit</a>]</div> If the function f can be written as a finite sum of other functions, then the fastest growing one determines the order of f(n). For example, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=9\log n+5(\log n)^{4}+3n^{2}+2n^{3}=O(n^{3})\qquad {\text{as }}n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>9</mn> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=9\log n+5(\log n)^{4}+3n^{2}+2n^{3}=O(n^{3})\qquad {\text{as }}n\to \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89af85ea4fc64d706852594e7dea55f5c64f3e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.822ex; height:3.176ex;" alt="{\displaystyle f(n)=9\log n+5(\log n)^{4}+3n^{2}+2n^{3}=O(n^{3})\qquad {\text{as }}n\to \infty .}"></dd></dl> In particular, if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial. The sets O(nc) and O(cn) are very different. If c is greater than one, then the latter grows much faster. A function that grows faster than nc for any c is called superpolynomial. One that grows more slowly than any exponential function of the form cn is called subexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a> and the function nlog n. We may ignore any powers of n inside of the logarithms. The set O(log n) is exactly the same as O(log(nc)). The logarithms differ only by a constant factor (since log(nc) = c log n) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. On the other hand, exponentials with different bases are not of the same order. For example, 2n and 3n are not of the same order. Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of n2, replacing n by cn means the algorithm runs in the order of c2n2, and the big O notation ignores the constant c2. This can be written as c2n2 = O(n2). If, however, an algorithm runs in the order of 2n, replacing n with cn gives 2cn = (2c)n. This is not equivalent to 2n in general. Changing variables may also affect the order of the resulting algorithm. For example, if an algorithm's run time is O(n) when measured in terms of the number n of digits of an input number x, then its run time is O(log x) when measured as a function of the input number x itself, because n = O(log x). <div class="mw-heading mw-heading3"><h3 id="Product">Product</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=7" title="Edit section: Product">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}=O(g_{1}){\text{ and }}f_{2}=O(g_{2})\Rightarrow f_{1}f_{2}=O(g_{1}g_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}=O(g_{1}){\text{ and }}f_{2}=O(g_{2})\Rightarrow f_{1}f_{2}=O(g_{1}g_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/819a5b9bb47ab027ff7cb4240935e18487391599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.993ex; height:2.843ex;" alt="{\displaystyle f_{1}=O(g_{1}){\text{ and }}f_{2}=O(g_{2})\Rightarrow f_{1}f_{2}=O(g_{1}g_{2})}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\cdot O(g)=O(fg)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>⋅</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\cdot O(g)=O(fg)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b82dc34c0c85245ebf04bb9b7202bd30931d7a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.732ex; height:2.843ex;" alt="{\displaystyle f\cdot O(g)=O(fg)}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Sum">Sum</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=8" title="Edit section: Sum">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}=O(g_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}=O(g_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10bbbef07d6840089dac53d3844220538055f114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.038ex; height:2.843ex;" alt="{\displaystyle f_{1}=O(g_{1})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}=O(g_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}=O(g_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c9c91c308dc88f1fba95d5b9307bc731b45936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.038ex; height:2.843ex;" alt="{\displaystyle f_{2}=O(g_{2})}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}+f_{2}=O(\max(|g_{1}|,|g_{2}|))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}+f_{2}=O(\max(|g_{1}|,|g_{2}|))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46bbf69733fada9e826e73f184e770f18f818e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.991ex; height:2.843ex;" alt="{\displaystyle f_{1}+f_{2}=O(\max(|g_{1}|,|g_{2}|))}">. It follows that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}=O(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}=O(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d51f82b5ddc09c23f6676096a19367cd136a366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.991ex; height:2.843ex;" alt="{\displaystyle f_{1}=O(g)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}=O(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}=O(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb9c640aca1f49030a10cef0b81fc17be012a2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.991ex; height:2.843ex;" alt="{\displaystyle f_{2}=O(g)}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}+f_{2}\in O(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}+f_{2}\in O(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6423bf44c7a2d4a0a299441c610ddecb63314fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.767ex; height:2.843ex;" alt="{\displaystyle f_{1}+f_{2}\in O(g)}">. In other words, this second statement says that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b2d25119aee12d89cada9a75c50c1dfb9e13384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.699ex; height:2.843ex;" alt="{\displaystyle O(g)}"> is a <a href="/wiki/Convex_cone" title="Convex cone">convex cone</a>. <div class="mw-heading mw-heading3"><h3 id="Multiplication_by_a_constant">Multiplication by a constant</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=9" title="Edit section: Multiplication by a constant">edit</a>]</div> Let k be a nonzero constant. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|k|\cdot g)=O(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(|k|\cdot g)=O(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985920563f4f78550f8a1fd39b05087419893328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.68ex; height:2.843ex;" alt="{\displaystyle O(|k|\cdot g)=O(g)}">. In other words, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=O(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=O(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beed823cad3e8b52dcfa798e8097d94b3c3c7541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.076ex; height:2.843ex;" alt="{\displaystyle f=O(g)}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot f=O(g).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <mi>f</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot f=O(g).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd52d7bf53187189e7f1703754a985e20b5f7d74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.613ex; height:2.843ex;" alt="{\displaystyle k\cdot f=O(g).}"> <div class="mw-heading mw-heading2"><h2 id="Multiple_variables">Multiple variables</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=10" title="Edit section: Multiple variables">edit</a>]</div> Big O (and little o, Ω, etc.) can also be used with multiple variables. To define big O formally for multiple variables, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> are two functions defined on some subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">. We say <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {x} ){\text{ is }}O(g(\mathbf {x} ))\quad {\text{ as }}\mathbf {x} \to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> is </mtext> </mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {x} ){\text{ is }}O(g(\mathbf {x} ))\quad {\text{ as }}\mathbf {x} \to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7795cfe1aba791ee68445acb0972e714e74822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.054ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} ){\text{ is }}O(g(\mathbf {x} ))\quad {\text{ as }}\mathbf {x} \to \infty }"></dd></dl> if and only if there exist constants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84d4126c6df243734f9355927c026df6b0d3859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C>0}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(\mathbf {x} )|\leq C|g(\mathbf {x} )|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(\mathbf {x} )|\leq C|g(\mathbf {x} )|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfaf93ff01d1ecce5ae7e5663e944e8bf031fc16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.287ex; height:2.843ex;" alt="{\displaystyle |f(\mathbf {x} )|\leq C|g(\mathbf {x} )|}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\geq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\geq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f38c48bb379042d7d38c69a508a179c8340479a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.67ex; height:2.509ex;" alt="{\displaystyle x_{i}\geq M}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffcf9ad7ad44f04fa43c5b604b4801e089981cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.449ex; height:2.176ex;" alt="{\displaystyle i.}"><a href="#cite_note-7">[7]</a> Equivalently, the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\geq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\geq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f38c48bb379042d7d38c69a508a179c8340479a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.67ex; height:2.509ex;" alt="{\displaystyle x_{i}\geq M}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> can be written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {x} \|_{\infty }\geq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>≥</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|_{\infty }\geq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/227752f0aea2730aa50e062e7f4d049acec3c5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.152ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {x} \|_{\infty }\geq M}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {x} \|_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|_{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9444d248dbfe32092e00fea179a754dab64812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.611ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {x} \|_{\infty }}"> denotes the <a href="/wiki/Chebyshev_norm" class="mw-redirect" title="Chebyshev norm">Chebyshev norm</a>. For example, the statement <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n,m)=n^{2}+m^{3}+O(n+m)\quad {\text{ as }}n,m\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n,m)=n^{2}+m^{3}+O(n+m)\quad {\text{ as }}n,m\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56fe59d65b70e4e5d736265ebaaf1e298112e7df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.707ex; height:3.176ex;" alt="{\displaystyle f(n,m)=n^{2}+m^{3}+O(n+m)\quad {\text{ as }}n,m\to \infty }"></dd></dl> asserts that there exist constants C and M such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(n,m)-(n^{2}+m^{3})|\leq C|n+m|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(n,m)-(n^{2}+m^{3})|\leq C|n+m|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d7b9ba195e7a6e61e9ee41b4db198cb8338398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.318ex; height:3.176ex;" alt="{\displaystyle |f(n,m)-(n^{2}+m^{3})|\leq C|n+m|}"></dd></dl> whenever either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7279b1c0ec22610b56ffbca3e53f7959cfeb53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.581ex; height:2.343ex;" alt="{\displaystyle m\geq M}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39da113317452e220c9cf19066d930c7c13886d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.935ex; height:2.343ex;" alt="{\displaystyle n\geq M}"> holds. This definition allows all of the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> to increase to infinity. In particular, the statement <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n,m)=O(n^{m})\quad {\text{ as }}n,m\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n,m)=O(n^{m})\quad {\text{ as }}n,m\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/647fb88e21638c3dae333c139e3dd7ea7c2c24a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.277ex; height:2.843ex;" alt="{\displaystyle f(n,m)=O(n^{m})\quad {\text{ as }}n,m\to \infty }"></dd></dl> (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists C\,\exists M\,\forall n\,\forall m\,\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>C</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>M</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>m</mi> <mspace width="thinmathspace" /> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists C\,\exists M\,\forall n\,\forall m\,\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b89f97936c678bd23ee08dcd0c61775893332436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.472ex; height:2.176ex;" alt="{\displaystyle \exists C\,\exists M\,\forall n\,\forall m\,\cdots }">) is quite different from <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall m\colon ~f(n,m)=O(n^{m})\quad {\text{ as }}n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>m</mi> <mo>:</mo> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall m\colon ~f(n,m)=O(n^{m})\quad {\text{ as }}n\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66054aa34578efeb81fc0cde681fb86f45073038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.15ex; height:2.843ex;" alt="{\displaystyle \forall m\colon ~f(n,m)=O(n^{m})\quad {\text{ as }}n\to \infty }"></dd></dl> (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall m\,\exists C\,\exists M\,\forall n\,\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>m</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>C</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>M</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mspace width="thinmathspace" /> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall m\,\exists C\,\exists M\,\forall n\,\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca18642648021a9f83a74e74d7baaf2b64d4229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.472ex; height:2.176ex;" alt="{\displaystyle \forall m\,\exists C\,\exists M\,\forall n\,\cdots }">). Under this definition, the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n,m)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n,m)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4d90d0dcfb967ec07bae7226112b9e9b000b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.818ex; height:2.843ex;" alt="{\displaystyle f(n,m)=1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n,m)=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n,m)=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad27ab0d17f4a109ed5405edaa12ff433e18e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.888ex; height:2.843ex;" alt="{\displaystyle g(n,m)=n}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n,m)=O(g(n,m))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n,m)=O(g(n,m))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d05254fc1d2f56769dfad2fa32f78148a297c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.632ex; height:2.843ex;" alt="{\displaystyle f(n,m)=O(g(n,m))}"> if we restrict <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,\infty )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,\infty )^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a753904fe6e79c3dc47a3625dec4f8cda69f3565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.126ex; height:3.176ex;" alt="{\displaystyle [1,\infty )^{2}}">, but not if they are defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty )^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9344632df798a9fb7d480004a0c78474c6fc7e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.126ex; height:3.176ex;" alt="{\displaystyle [0,\infty )^{2}}">. This is not the only generalization of big O to multivariate functions, and in practice, there is some inconsistency in the choice of definition.<a href="#cite_note-8">[8]</a> <div class="mw-heading mw-heading2"><h2 id="Matters_of_notation">Matters of notation</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=11" title="Edit section: Matters of notation">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Equals_sign">Equals sign</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=12" title="Edit section: Equals sign">edit</a>]</div> The statement "  f(x)   is   O[ g(x) ]  " as defined above is usually written as   f(x) = O[ g(x) ]  . Some consider this to be an <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a>, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As <a href="/wiki/Nicolaas_Govert_de_Bruijn" title="Nicolaas Govert de Bruijn">de Bruijn</a> says,   O[ x ] = O[ x2 ]   is true but   O[ x2 ] = O[ x ]   is not.<a href="#cite_note-deBruijn-9">[9]</a> <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth</a> describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like   n = n2   from the identities   n = O[ n2 ]   and   n2 = O[ n2 ]  ".<a href="#cite_note-Concrete_Mathematics-10">[10]</a> In another letter, Knuth also pointed out that <dl><dd>"the equality sign is not symmetric with respect to such notations", [as, in this notation,] "mathematicians customarily use the '=' sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle".<a href="#cite_note-11">[11]</a></dd></dl> For these reasons, it would be more precise to use <a href="/wiki/Set_notation" class="mw-redirect" title="Set notation">set notation</a> and write   f(x) ∈ O[ g(x) ]   (read as: "  f(x)   <a href="/wiki/Element_(mathematics)#Notation_and_terminology" title="Element (mathematics)">is an element of</a>   O[ g(x) ]  ", or "  f(x)   is in the set   O[ g(x) ]  "), thinking of   O[ g(x) ]   as the class of all functions   h(x)   such that   |h(x)| ≤ C |g(x)|   for some positive real number C.<a href="#cite_note-Concrete_Mathematics-10">[10]</a> However, the use of the equals sign is customary.<a href="#cite_note-deBruijn-9">[9]</a><a href="#cite_note-Concrete_Mathematics-10">[10]</a> <div class="mw-heading mw-heading3"><h3 id="Other_arithmetic_operators">Other arithmetic operators</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=13" title="Edit section: Other arithmetic operators">edit</a>]</div> Big O notation can also be used in conjunction with other arithmetic operators in more complicated equations. For example, h(x) + O(f(x)) denotes the collection of functions having the growth of h(x) plus a part whose growth is limited to that of f(x). Thus, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=h(x)+O(f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=h(x)+O(f(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73e8f66f701116890456d29ce8d01b807272d91d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.672ex; height:2.843ex;" alt="{\displaystyle g(x)=h(x)+O(f(x))}"></dd></dl> expresses the same as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)-h(x)=O(f(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)-h(x)=O(f(x)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3477cd71688b6a7a1769aadc28764efa090688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.319ex; height:2.843ex;" alt="{\displaystyle g(x)-h(x)=O(f(x)).}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Example_2">Example </h4>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=14" title="Edit section: Example">edit</a>]</div> Suppose an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> is being developed to operate on a set of n elements. Its developers are interested in finding a function T(n) that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set. The algorithm works by first calling a subroutine to sort the elements in the set and then perform its own operations. The sort has a known time complexity of O(n2), and after the subroutine runs the algorithm must take an additional 55n3 + 2n + 10 steps before it terminates. Thus the overall time complexity of the algorithm can be expressed as T(n) = 55n3 + O(n2). Here the terms 2n + 10 are subsumed within the faster-growing O(n2). Again, this usage disregards some of the formal meaning of the "=" symbol, but it does allow one to use the big O notation as a kind of convenient placeholder. <div class="mw-heading mw-heading3"><h3 id="Multiple_uses">Multiple uses</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=15" title="Edit section: Multiple uses">edit</a>]</div> In more complicated usage, O(·) can appear in different places in an equation, even several times on each side. For example, the following are true for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42312b6e624001cba2f95952067b93278ea868bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.21ex; margin-bottom: -0.294ex; width:48.295ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}}"> The meaning of such statements is as follows: for any functions which satisfy each O(·) on the left side, there are some functions satisfying each O(·) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function f(n) = O(1), there is some function g(n) = O(en) such that nf(n) = g(n)." In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side. In this use the "=" is a formal symbol that unlike the usual use of "=" is not a <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric relation</a>. Thus for example nO(1) = O(en) does not imply the false statement O(en) = nO(1). <div class="mw-heading mw-heading3"><h3 id="Typesetting">Typesetting</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=16" title="Edit section: Typesetting">edit</a>]</div> Big O is typeset as an italicized uppercase "O", as in the following example: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd9594a16cb898b8f2a2dff9227a385ec183392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{2})}">.<a href="#cite_note-KnuthArt-12">[12]</a><a href="#cite_note-ConcreteMath-13">[13]</a> In <a href="/wiki/TeX" title="TeX">TeX</a>, it is produced by simply typing O inside math mode. Unlike Greek-named Bachmann–Landau notations, it needs no special symbol. However, some authors use the calligraphic variant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}"> instead.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a> <div class="mw-heading mw-heading2"><h2 id="Orders_of_common_functions">Orders of common functions</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=17" title="Edit section: Orders of common functions">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Time_complexity#Table_of_common_time_complexities" title="Time complexity">Time complexity § Table of common time complexities</a></div> Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm. In each case, c is a positive constant and n increases without bound. The slower-growing functions are generally listed first. <table class="wikitable"> <tbody><tr> <th>Notation</th> <th>Name</th> <th>Example </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></td> <td><a href="/wiki/Constant_time" class="mw-redirect" title="Constant time">constant</a></td> <td>Finding the median value for a sorted array of numbers; Calculating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c490525b94310eb9d66c0282f8d28f652af9f40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.998ex; height:2.843ex;" alt="{\displaystyle (-1)^{n}}">; Using a constant-size <a href="/wiki/Lookup_table" title="Lookup table">lookup table</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\alpha (n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\alpha (n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f2f9c5bf5571b12dd0907f0a1ef917e1c082201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.274ex; height:2.843ex;" alt="{\displaystyle O(\alpha (n))}"></td> <td><a href="/wiki/Inverse_Ackermann_function" class="mw-redirect" title="Inverse Ackermann function">inverse Ackermann function</a></td> <td>Amortized complexity per operation for the <a href="/wiki/Disjoint-set_data_structure" title="Disjoint-set data structure">Disjoint-set data structure</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log \log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log \log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f3ded7e013a35c8039ff3276bee6e183d7f1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.695ex; height:2.843ex;" alt="{\displaystyle O(\log \log n)}"></td> <td>double logarithmic</td> <td>Average number of comparisons spent finding an item using <a href="/wiki/Interpolation_search" title="Interpolation search">interpolation search</a> in a sorted array of uniformly distributed values </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}"></td> <td><a href="/wiki/Logarithmic_time" class="mw-redirect" title="Logarithmic time">logarithmic</a></td> <td>Finding an item in a sorted array with a <a href="/wiki/Binary_search_algorithm" class="mw-redirect" title="Binary search algorithm">binary search</a> or a balanced search <a href="/wiki/Tree_data_structure" class="mw-redirect" title="Tree data structure">tree</a> as well as all operations in a <a href="/wiki/Binomial_heap" title="Binomial heap">binomial heap</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O((\log n)^{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O((\log n)^{c})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180638b34fb9d9a72d4448fa4136615b0487d52a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.09ex; height:2.843ex;" alt="{\displaystyle O((\log n)^{c})}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle c>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle c>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9057ec1c55def34aa105650bfe9491240553d160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\textstyle c>1}"></td> <td><a href="/wiki/Polylogarithmic_time" class="mw-redirect" title="Polylogarithmic time">polylogarithmic</a></td> <td>Matrix chain ordering can be solved in polylogarithmic time on a <a href="/wiki/Parallel_random-access_machine" class="mw-redirect" title="Parallel random-access machine">parallel random-access machine</a>. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{c})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949acd060374f98c6dc5f76ad962c7fe2d6636d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.922ex; height:2.843ex;" alt="{\displaystyle O(n^{c})}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0<c<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>c</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0<c<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e7a58e5049a6f4b9f57b980a1e4fc6d7481532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.529ex; height:2.176ex;" alt="{\textstyle 0<c<1}"></td> <td>fractional power</td> <td>Searching in a <a href="/wiki/K-d_tree" title="K-d tree">k-d tree</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></td> <td><a href="/wiki/Linear_time" class="mw-redirect" title="Linear time">linear</a></td> <td>Finding an item in an unsorted list or in an unsorted array; adding two n-bit integers by <a href="/wiki/Ripple_carry_adder" class="mw-redirect" title="Ripple carry adder">ripple carry</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log ^{*}n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n\log ^{*}n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6a679f4e6f08c50bfbdf85386ff72f320fc384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.172ex; height:2.843ex;" alt="{\displaystyle O(n\log ^{*}n)}"></td> <td>n <a href="/wiki/Log-star" class="mw-redirect" title="Log-star">log-star</a> n</td> <td>Performing <a href="/wiki/Polygon_triangulation" title="Polygon triangulation">triangulation</a> of a simple polygon using <a href="/wiki/Kirkpatrick%E2%80%93Seidel_algorithm" title="Kirkpatrick–Seidel algorithm">Seidel's algorithm</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log ^{*}(n)={\begin{cases}0,&{\text{if }}n\leq 1\\1+\log ^{*}(\log n),&{\text{if }}n>1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <msup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log ^{*}(n)={\begin{cases}0,&{\text{if }}n\leq 1\\1+\log ^{*}(\log n),&{\text{if }}n>1\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8ce0b95ab4bdc244b0ad464677f1709df7c562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.979ex; height:6.176ex;" alt="{\displaystyle \log ^{*}(n)={\begin{cases}0,&{\text{if }}n\leq 1\\1+\log ^{*}(\log n),&{\text{if }}n>1\end{cases}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log n)=O(\log n!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n\log n)=O(\log n!)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb5e834d1a8273752727953b92a514047d22484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.199ex; height:2.843ex;" alt="{\displaystyle O(n\log n)=O(\log n!)}"></td> <td><a href="/wiki/Linearithmic_time" class="mw-redirect" title="Linearithmic time">linearithmic</a>, loglinear, quasilinear, or "n log n"</td> <td>Performing a <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a>; fastest possible <a href="/wiki/Comparison_sort" title="Comparison sort">comparison sort</a>; <a href="/wiki/Heapsort" title="Heapsort">heapsort</a> and <a href="/wiki/Merge_sort" title="Merge sort">merge sort</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd9594a16cb898b8f2a2dff9227a385ec183392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{2})}"></td> <td><a href="/wiki/Quadratic_time" class="mw-redirect" title="Quadratic time">quadratic</a></td> <td>Multiplying two n-digit numbers by <a href="/wiki/Multiplication_algorithm#Long_multiplication" title="Multiplication algorithm">schoolbook multiplication</a>; simple sorting algorithms, such as <a href="/wiki/Bubble_sort" title="Bubble sort">bubble sort</a>, <a href="/wiki/Selection_sort" title="Selection sort">selection sort</a> and <a href="/wiki/Insertion_sort" title="Insertion sort">insertion sort</a>; (worst-case) bound on some usually faster sorting algorithms such as <a href="/wiki/Quicksort" title="Quicksort">quicksort</a>, <a href="/wiki/Shellsort" title="Shellsort">Shellsort</a>, and <a href="/wiki/Tree_sort" title="Tree sort">tree sort</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{c})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949acd060374f98c6dc5f76ad962c7fe2d6636d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.922ex; height:2.843ex;" alt="{\displaystyle O(n^{c})}"></td> <td><a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial</a> or algebraic</td> <td><a href="/wiki/Tree-adjoining_grammar" title="Tree-adjoining grammar">Tree-adjoining grammar</a> parsing; maximum <a href="/wiki/Matching_(graph_theory)" title="Matching (graph theory)">matching</a> for <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite graphs</a>; finding the <a href="/wiki/Determinant" title="Determinant">determinant</a> with <a href="/wiki/LU_decomposition" title="LU decomposition">LU decomposition</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>α</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40715162d91a34daf9bcfed5f41d8e336417f803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.754ex; height:3.676ex;" alt="{\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0<\alpha <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0<\alpha <1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2d230edd927b1759c4b4093fa4caaa8a7944cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.009ex; height:2.176ex;" alt="{\textstyle 0<\alpha <1}"></td> <td><a href="/wiki/L-notation" title="L-notation">L-notation</a> or <a href="/wiki/Sub-exponential_time" class="mw-redirect" title="Sub-exponential time">sub-exponential</a></td> <td>Factoring a number using the <a href="/wiki/Quadratic_sieve" title="Quadratic sieve">quadratic sieve</a> or <a href="/wiki/General_number_field_sieve" title="General number field sieve">number field sieve</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(c^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(c^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab28ca44589e77de77ab8cc55cdbd4a63f7ef8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.808ex; height:2.843ex;" alt="{\displaystyle O(c^{n})}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle c>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle c>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9057ec1c55def34aa105650bfe9491240553d160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\textstyle c>1}"></td> <td><a href="/wiki/Exponential_time" class="mw-redirect" title="Exponential time">exponential</a></td> <td>Finding the (exact) solution to the <a href="/wiki/Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a> using <a href="/wiki/Dynamic_programming" title="Dynamic programming">dynamic programming</a>; determining if two logical statements are equivalent using <a href="/wiki/Brute-force_search" title="Brute-force search">brute-force search</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n!)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12921c489714d475a454bd39ef644d4334d97113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.624ex; height:2.843ex;" alt="{\displaystyle O(n!)}"></td> <td><a href="/wiki/Factorial" title="Factorial">factorial</a></td> <td>Solving the <a href="/wiki/Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a> via brute-force search; generating all unrestricted permutations of a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">poset</a>; finding the <a href="/wiki/Determinant" title="Determinant">determinant</a> with <a href="/wiki/Laplace_expansion" title="Laplace expansion">Laplace expansion</a>; enumerating <a href="/wiki/Bell_number" title="Bell number">all partitions of a set</a> </td></tr></tbody></table> The statement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=O(n!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=O(n!)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2112c37f521132251407043c59aec72dbad126ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.205ex; height:2.843ex;" alt="{\displaystyle f(n)=O(n!)}"> is sometimes weakened to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=O\left(n^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=O\left(n^{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e476f151f8eebb41935f16a78e7d78b8bb8a75f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.164ex; height:2.843ex;" alt="{\displaystyle f(n)=O\left(n^{n}\right)}"> to derive simpler formulas for asymptotic complexity. For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba126f626d61752f62eaacaf11761a54de4dc84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{c}(\log n)^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{c}(\log n)^{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b83df58cf2ec028a8c249ec28c770e3bb0bf347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.573ex; height:3.176ex;" alt="{\displaystyle O(n^{c}(\log n)^{k})}"> is a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{c+\varepsilon })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{c+\varepsilon })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10906d93e6dc825b378d3642c1079925d87492bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.966ex; height:3.009ex;" alt="{\displaystyle O(n^{c+\varepsilon })}"> for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}">, so may be considered as a polynomial with some bigger order. <div class="mw-heading mw-heading2"><h2 id="Related_asymptotic_notations">Related asymptotic notations</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=18" title="Edit section: Related asymptotic notations">edit</a>]</div> Big O is widely used in computer science. Together with some other related notations, it forms the family of Bachmann–Landau notations.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Little-o_notation">Little-o notation</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=19" title="Edit section: Little-o notation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Little o" redirects here. For the baseball player, see <a href="/wiki/Omar_Vizquel" title="Omar Vizquel">Omar Vizquel</a>.</div> Intuitively, the assertion "f(x) is o(g(x))" (read "f(x) is little-o of g(x)" or "f(x) is of inferior order to g(x)") means that g(x) grows much faster than f(x), or equivalently f(x) grows much slower than g(x). As before, let f be a real or complex valued function and g a real valued function, both defined on some unbounded subset of the positive <a href="/wiki/Real_number" title="Real number">real numbers</a>, such that g(x) is strictly positive for all large enough values of x. One writes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/095a25369f3dec90b5123575eca7eb35c2bb2c63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.538ex; height:2.843ex;" alt="{\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to \infty }"></dd></dl> if for every positive constant ε there exists a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)|\leq \varepsilon g(x)\quad {\text{ for all }}x\geq x_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>ε</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>x</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq \varepsilon g(x)\quad {\text{ for all }}x\geq x_{0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b0e6df26708f200e053ff5fa852452c01a11d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.914ex; height:2.843ex;" alt="{\displaystyle |f(x)|\leq \varepsilon g(x)\quad {\text{ for all }}x\geq x_{0}.}"><a href="#cite_note-Landausmallo-16">[16]</a></dd></dl> For example, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x=o(x^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x=o(x^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5154f15b7298f3b33c2341117142eaf1890322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.911ex; height:3.176ex;" alt="{\displaystyle 2x=o(x^{2})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/x=o(1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/x=o(1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a546858e20b4fdffc68734738e18b4d30371c25c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.499ex; height:2.843ex;" alt="{\displaystyle 1/x=o(1),}">     both as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077f3645400fedfac226358e8dc3ead4949b1fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.914ex; height:1.843ex;" alt="{\displaystyle x\to \infty .}"></dd></dl> The difference between the <a href="#Formal_definition">definition of the big-O notation</a> and the definition of little-o is that while the former has to be true for at least one constant M, the latter must hold for every positive constant ε, however small.<a href="#cite_note-Introduction_to_Algorithms-17">[17]</a> In this way, little-o notation makes a stronger statement than the corresponding big-O notation: every function that is little-o of g is also big-O of g, but not every function that is big-O of g is little-o of g. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{2}=O(x^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}=O(x^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120bc341cdd3295546a6da949edc72d4ef04b053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.611ex; height:3.176ex;" alt="{\displaystyle 2x^{2}=O(x^{2})}"> but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{2}\neq o(x^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≠</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}\neq o(x^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3a0e26e2f0130c79181844f0bcf3be77b531de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.966ex; height:3.176ex;" alt="{\displaystyle 2x^{2}\neq o(x^{2})}">. If g(x) is nonzero, or at least becomes nonzero beyond a certain point, the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=o(g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=o(g(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66122afa55c2aa7af3f23d3caa66a5a6ca4a494c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.708ex; height:2.843ex;" alt="{\displaystyle f(x)=o(g(x))}"> is equivalent to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b124ab9f38e11ecaf524de2f501f4956ec7b0bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.128ex; height:6.509ex;" alt="{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0}"> (and this is in fact how Landau<a href="#cite_note-Landausmallo-16">[16]</a> originally defined the little-o notation).</dd></dl> Little-o respects a number of arithmetic operations. For example, <dl><dd>if c is a nonzero constant and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=o(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=o(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0023489f1bbadf8b2ee4b0257a0a7f6fb600784e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.43ex; height:2.843ex;" alt="{\displaystyle f=o(g)}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\cdot f=o(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>⋅</mo> <mi>f</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\cdot f=o(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07bfe7f3dfa6248159f2437946fb3623b7da292a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.116ex; height:2.843ex;" alt="{\displaystyle c\cdot f=o(g)}">, and</dd> <dd>if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=o(F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=o(F)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a702e6bf0cfdb968289ca99ca0ff50df5eac80f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.055ex; height:2.843ex;" alt="{\displaystyle f=o(F)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=o(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=o(G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc932d37954c06dd90db3fdc2d86df728c15b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.978ex; height:2.843ex;" alt="{\displaystyle g=o(G)}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\cdot g=o(F\cdot G).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>⋅</mo> <mi>g</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo>⋅</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\cdot g=o(F\cdot G).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec05071b3538b7537bceb558e8770698fe3d460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.003ex; height:2.843ex;" alt="{\displaystyle f\cdot g=o(F\cdot G).}"></dd></dl> It also satisfies a <a href="/wiki/Transitive_relation" title="Transitive relation">transitivity</a> relation: <dl><dd>if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=o(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=o(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0023489f1bbadf8b2ee4b0257a0a7f6fb600784e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.43ex; height:2.843ex;" alt="{\displaystyle f=o(g)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=o(h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=o(h)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30f22123250b0ea58e7d52f498ba1e9820168b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.49ex; height:2.843ex;" alt="{\displaystyle g=o(h)}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=o(h).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=o(h).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a25bacdf64773597665893fa46262103bf632f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.3ex; height:2.843ex;" alt="{\displaystyle f=o(h).}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Big_Omega_notation">Big Omega notation</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=20" title="Edit section: Big Omega notation">edit</a>]</div> Another asymptotic notation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }">, read "big omega".<a href="#cite_note-18">[18]</a> There are two widespread and incompatible definitions of the statement <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\Omega (g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\Omega (g(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83679ae2e7d7e9f0d0c0e7e750ee75ef824f74a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.258ex; height:2.843ex;" alt="{\displaystyle f(x)=\Omega (g(x))}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d566e0ecdd92745b25ad40bae39614468c01f6e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.174ex; height:1.843ex;" alt="{\displaystyle x\to a}">,</dd></dl> where a is some real number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }">, where f and g are real functions defined in a neighbourhood of a, and where g is positive in this neighbourhood. The Hardy–Littlewood definition is used mainly in <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>, and the Knuth definition mainly in <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>; the definitions are not equivalent. <div class="mw-heading mw-heading4"><h4 id="The_Hardy–Littlewood_definition">The Hardy–Littlewood definition</h4>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=21" title="Edit section: The Hardy–Littlewood definition">edit</a>]</div> In 1914 <a href="/wiki/Godfrey_Harold_Hardy" class="mw-redirect" title="Godfrey Harold Hardy">G.H. Hardy</a> and <a href="/wiki/John_Edensor_Littlewood" title="John Edensor Littlewood">J.E. Littlewood</a> introduced the new symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Omega \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Ω</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Omega \ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c5e43c8d2b22b4c0191ebd8bcc4c3f5d715179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.486ex; height:2.509ex;" alt="{\displaystyle \ \Omega \ ,}"><a href="#cite_note-HL-19">[19]</a> which is defined as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719b776f8a8486047d46d1c0b995fe92c8595df7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.063ex; height:3.176ex;" alt="{\displaystyle f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}\quad }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x\to \infty \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x\to \infty \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef653404ab1b7d3bdd8c266c8efc83dd9ad009c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.913ex; height:1.843ex;" alt="{\displaystyle \quad x\to \infty \quad }"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \limsup _{x\to \infty }\ \left|{\frac {\ f(x)\ }{g(x)}}\right|>0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mtext> </mtext> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \limsup _{x\to \infty }\ \left|{\frac {\ f(x)\ }{g(x)}}\right|>0~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1389f773c5e34441a282bcd9ebbbc0f8d1cd4237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.993ex; height:6.509ex;" alt="{\displaystyle \quad \limsup _{x\to \infty }\ \left|{\frac {\ f(x)\ }{g(x)}}\right|>0~.}"></dd></dl> Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76762849097e9094691cda705eb35448c9f470a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.901ex; height:3.176ex;" alt="{\displaystyle ~f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}~}"> is the negation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~f(x)=o{\bigl (}\ g(x)\ {\bigr )}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~f(x)=o{\bigl (}\ g(x)\ {\bigr )}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e1e79f0eee321331e7240bffa9b9547031bca1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.998ex; height:3.176ex;" alt="{\displaystyle ~f(x)=o{\bigl (}\ g(x)\ {\bigr )}~.}"> In 1916 the same authors introduced the two new symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Omega _{R}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Omega _{R}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1108f25bb73cbee35a299e688e3b5a85b57b261d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.319ex; height:2.509ex;" alt="{\displaystyle \ \Omega _{R}\ }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Omega _{L}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Omega _{L}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec9180b2c2ef15696ca5b770dd246ffa3a67380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.838ex; height:2.509ex;" alt="{\displaystyle \ \Omega _{L}\ ,}"> defined as:<a href="#cite_note-HL2-20">[20]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\Omega _{R}{\bigl (}\ g(x)\ {\bigr )}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\Omega _{R}{\bigl (}\ g(x)\ {\bigr )}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d742079135b22ce3732b229da3cba5a989d1b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.542ex; height:3.176ex;" alt="{\displaystyle f(x)=\Omega _{R}{\bigl (}\ g(x)\ {\bigr )}\quad }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x\to \infty \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x\to \infty \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef653404ab1b7d3bdd8c266c8efc83dd9ad009c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.913ex; height:1.843ex;" alt="{\displaystyle \quad x\to \infty \quad }"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \limsup _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}>0\ ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \limsup _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}>0\ ;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d794effaab534866d2da1f70098493b98e0c044a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.312ex; height:6.509ex;" alt="{\displaystyle \quad \limsup _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}>0\ ;}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\Omega _{L}{\bigl (}\ g(x)\ {\bigr )}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\Omega _{L}{\bigl (}\ g(x)\ {\bigr )}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21bb968446a8211a84485db0c46f58d8a56f8674" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.414ex; height:3.176ex;" alt="{\displaystyle f(x)=\Omega _{L}{\bigl (}\ g(x)\ {\bigr )}\quad }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x\to \infty \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x\to \infty \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef653404ab1b7d3bdd8c266c8efc83dd9ad009c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.913ex; height:1.843ex;" alt="{\displaystyle \quad x\to \infty \quad }"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad ~\liminf _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}<0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo><</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad ~\liminf _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}<0~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1e4c105c6213ddeb9ef8c068125840a6292d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.43ex; height:6.509ex;" alt="{\displaystyle \quad ~\liminf _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}<0~.}"></dd></dl> These symbols were used by <a href="/wiki/Edmund_Landau" title="Edmund Landau">E. Landau</a>, with the same meanings, in 1924.<a href="#cite_note-landau-21">[21]</a> Authors that followed Landau, however, use a different notation for the same definitions:[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Omega _{R}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Omega _{R}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1108f25bb73cbee35a299e688e3b5a85b57b261d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.319ex; height:2.509ex;" alt="{\displaystyle \ \Omega _{R}\ }"> has been replaced by the current notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Omega _{+}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Omega _{+}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed6eda01147bbaa4c155358a4baf6ab7f0672903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.35ex; height:2.509ex;" alt="{\displaystyle \ \Omega _{+}\ }"> with the same definition, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Omega _{L}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Omega _{L}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9c2f1022ac10a744c73930ec1998ad0d7dfb68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.191ex; height:2.509ex;" alt="{\displaystyle \ \Omega _{L}\ }"> became <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Omega _{-}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Omega _{-}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aac8b139956069742925d74e0d555f44c0fcb32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.997ex; height:2.509ex;" alt="{\displaystyle \ \Omega _{-}~.}"> These three symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Omega \ ,\Omega _{+}\ ,\Omega _{-}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Ω</mi> <mtext> </mtext> <mo>,</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mtext> </mtext> <mo>,</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Omega \ ,\Omega _{+}\ ,\Omega _{-}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4328f8e296d03a3a1ea9e5357d3c80a56429ab60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.093ex; height:2.509ex;" alt="{\displaystyle \ \Omega \ ,\Omega _{+}\ ,\Omega _{-}\ ,}"> as well as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=\Omega _{\pm }{\bigl (}\ g(x)\ {\bigr )}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)=\Omega _{\pm }{\bigl (}\ g(x)\ {\bigr )}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9980b9c62ae16aa7f7ceaa0d8ff8fa8ac2e62ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.412ex; height:3.176ex;" alt="{\displaystyle \ f(x)=\Omega _{\pm }{\bigl (}\ g(x)\ {\bigr )}\ }"> (meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=\Omega _{+}{\bigl (}\ g(x)\ {\bigr )}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)=\Omega _{+}{\bigl (}\ g(x)\ {\bigr )}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbb0203ff8768ab57fa2b604be332532c07f9b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.412ex; height:3.176ex;" alt="{\displaystyle \ f(x)=\Omega _{+}{\bigl (}\ g(x)\ {\bigr )}\ }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=\Omega _{-}{\bigl (}\ g(x)\ {\bigr )}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)=\Omega _{-}{\bigl (}\ g(x)\ {\bigr )}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5f878ae2b06d1ad5876e51cd45acc753cd7db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.412ex; height:3.176ex;" alt="{\displaystyle \ f(x)=\Omega _{-}{\bigl (}\ g(x)\ {\bigr )}\ }"> are both satisfied), are now currently used in <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>.<a href="#cite_note-Ivic-22">[22]</a><a href="#cite_note-23">[23]</a> <div class="mw-heading mw-heading5"><h5 id="Simple_examples">Simple examples</h5>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=22" title="Edit section: Simple examples">edit</a>]</div> We have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x=\Omega (1)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x=\Omega (1)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e844e31a8b6f2be13b24d7327ce25f26825f510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.643ex; height:2.843ex;" alt="{\displaystyle \sin x=\Omega (1)\quad }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x\to \infty \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x\to \infty \ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896ee8b534f618a0d1fd725a44e2a6db307cb48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.818ex; height:2.176ex;" alt="{\displaystyle \quad x\to \infty \ ,}"></dd></dl> and more precisely <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x=\Omega _{\pm }(1)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x=\Omega _{\pm }(1)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a513df63b2ec1f4c15d81fbcf22b5e8a999eb63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.154ex; height:2.843ex;" alt="{\displaystyle \sin x=\Omega _{\pm }(1)\quad }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x\to \infty ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x\to \infty ~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4284c8e3e549f79df084395b2199b0a4f1f20f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.818ex; height:1.843ex;" alt="{\displaystyle \quad x\to \infty ~.}"></dd></dl> We have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\sin x=\Omega (1)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\sin x=\Omega (1)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c825cc194b38f1fdd54b8d274628e4909c295e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.646ex; height:2.843ex;" alt="{\displaystyle 1+\sin x=\Omega (1)\quad }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x\to \infty \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x\to \infty \ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896ee8b534f618a0d1fd725a44e2a6db307cb48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.818ex; height:2.176ex;" alt="{\displaystyle \quad x\to \infty \ ,}"></dd></dl> and more precisely <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\sin x=\Omega _{+}(1)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\sin x=\Omega _{+}(1)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a662473108818abbcef72fcd9e562b5a634f4eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.157ex; height:2.843ex;" alt="{\displaystyle 1+\sin x=\Omega _{+}(1)\quad }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x\to \infty \ ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mtext> </mtext> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x\to \infty \ ;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03da1882cd535de5b695390951b4b96912de9a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.818ex; height:2.176ex;" alt="{\displaystyle \quad x\to \infty \ ;}"></dd></dl> however <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\sin x\neq \Omega _{-}(1)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>≠</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\sin x\neq \Omega _{-}(1)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f224454dd133f145d3fd6d2618decc113105ca66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.157ex; height:2.843ex;" alt="{\displaystyle 1+\sin x\neq \Omega _{-}(1)\quad }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x\to \infty ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x\to \infty ~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4284c8e3e549f79df084395b2199b0a4f1f20f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.818ex; height:1.843ex;" alt="{\displaystyle \quad x\to \infty ~.}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="The_Knuth_definition">The Knuth definition</h4>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=23" title="Edit section: The Knuth definition">edit</a>]</div> In 1976 <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> published a paper to justify his use of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }">-symbol to describe a stronger property.<a href="#cite_note-knuth-24">[24]</a> Knuth wrote: "For all the applications I have seen so far in computer science, a stronger requirement ... is much more appropriate". He defined <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\Omega (g(x))\Leftrightarrow g(x)=O(f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\Omega (g(x))\Leftrightarrow g(x)=O(f(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3df3f696c222490af2885947ea346e56f1288504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.226ex; height:2.843ex;" alt="{\displaystyle f(x)=\Omega (g(x))\Leftrightarrow g(x)=O(f(x))}"></dd></dl> with the comment: "Although I have changed Hardy and Littlewood's definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }">, I feel justified in doing so because their definition is by no means in wide use, and because there are other ways to say what they want to say in the comparatively rare cases when their definition applies."<a href="#cite_note-knuth-24">[24]</a> <div class="mw-heading mw-heading3"><h3 id="Family_of_Bachmann–Landau_notations">Family of Bachmann–Landau notations</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=24" title="Edit section: Family of Bachmann–Landau notations">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th>Notation </th> <th>Name<a href="#cite_note-knuth-24">[24]</a> </th> <th>Description </th> <th>Formal definition </th> <th>Limit definition<a href="#cite_note-Balcázar-25">[25]</a><a href="#cite_note-Cucker-26">[26]</a><a href="#cite_note-Wild-27">[27]</a><a href="#cite_note-knuth-24">[24]</a><a href="#cite_note-HL-19">[19]</a> </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=o(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=o(g(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/319350e8903fea3a1f83f0a91ec740d9694eef4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.838ex; height:2.843ex;" alt="{\displaystyle f(n)=o(g(n))}"> </td> <td>Small O; Small Oh; Little O; Little Oh </td> <td>f is dominated by g asymptotically (for any constant factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>k</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a4f61359500829ea3f53970881059e641a8cfdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.342ex; height:2.843ex;" alt="{\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/492b43a40ba041a1f958d4e3923eed6fd947f08c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.239ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=O(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=O(g(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4ecd455bbf94b9eeab71366619cb18e040dce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.484ex; height:2.843ex;" alt="{\displaystyle f(n)=O(g(n))}"> </td> <td>Big O; Big Oh; Big Omicron </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940e58aa437f429f47fd743a819e41fedaa1ff9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle |f|}"> is asymptotically bounded above by g (up to constant factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>k</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16115602291a322030d1f83eaa012a3e8f11e10d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.342ex; height:2.843ex;" alt="{\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \limsup _{n\to \infty }{\frac {f(n)}{g(n)}}<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \limsup _{n\to \infty }{\frac {f(n)}{g(n)}}<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d841dd81b7f9d43b4b2df003dbd15984ed846af3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.247ex; height:6.509ex;" alt="{\displaystyle \limsup _{n\to \infty }{\frac {f(n)}{g(n)}}<\infty }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\asymp g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≍</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\asymp g(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd1056a38d423d01ca060a8c50a760632b6d3b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.901ex; height:2.843ex;" alt="{\displaystyle f(n)\asymp g(n)}"> (Hardy's notation) or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=\Theta (g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=\Theta (g(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8445424def6bf9c759eff91dc041585caf4abd81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.519ex; height:2.843ex;" alt="{\displaystyle f(n)=\Theta (g(n))}"> (Knuth notation) </td> <td>Of the same order as (Hardy); Big Theta (Knuth) </td> <td>f is asymptotically bounded by g both above (with constant factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}">) and below (with constant factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}">) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists k_{1}>0\,\exists k_{2}>0\,\exists n_{0}\,\forall n>n_{0}\colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists k_{1}>0\,\exists k_{2}>0\,\exists n_{0}\,\forall n>n_{0}\colon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be379e6bea41362b562f8783c99e54cc4b716ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.422ex; height:2.509ex;" alt="{\displaystyle \exists k_{1}>0\,\exists k_{2}>0\,\exists n_{0}\,\forall n>n_{0}\colon }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}\,g(n)\leq f(n)\leq k_{2}\,g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{1}\,g(n)\leq f(n)\leq k_{2}\,g(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b53161e524a1b5489b581143dfa5f4d5f845673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.625ex; height:2.843ex;" alt="{\displaystyle k_{1}\,g(n)\leq f(n)\leq k_{2}\,g(n)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=O(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=O(g(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4ecd455bbf94b9eeab71366619cb18e040dce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.484ex; height:2.843ex;" alt="{\displaystyle f(n)=O(g(n))}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)=O(f(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)=O(f(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a666fe69b6e49bd7a1cabe96d5ee4b49fb0115c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.484ex; height:2.843ex;" alt="{\displaystyle g(n)=O(f(n))}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\sim g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\sim g(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21481c5c7ce37ba3b4a27c198ca59f4cfc92f3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.901ex; height:2.843ex;" alt="{\displaystyle f(n)\sim g(n)}"> </td> <td>Asymptotic equivalence </td> <td>f is equal to g <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotically</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \varepsilon >0\,\exists n_{0}\,\forall n>n_{0}\colon \left|{\frac {f(n)}{g(n)}}-1\right|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\,\exists n_{0}\,\forall n>n_{0}\colon \left|{\frac {f(n)}{g(n)}}-1\right|<\varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd9b982208142f0fa7be6058570298f37ec0d39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.218ex; height:6.509ex;" alt="{\displaystyle \forall \varepsilon >0\,\exists n_{0}\,\forall n>n_{0}\colon \left|{\frac {f(n)}{g(n)}}-1\right|<\varepsilon }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e40d97db198d24361a2a35fa8af2128e0523f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.239ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=\Omega (g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=\Omega (g(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df8a463b6e106cce5d68174c1df5768a1682ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.388ex; height:2.843ex;" alt="{\displaystyle f(n)=\Omega (g(n))}"> </td> <td>Big Omega in complexity theory (Knuth) </td> <td>f is bounded below by g asymptotically </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)\geq k\,g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>k</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)\geq k\,g(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5154c98624eb114e71485abb7ec1396d157e6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.048ex; height:2.843ex;" alt="{\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)\geq k\,g(n)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{g(n)}}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{g(n)}}>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9e629f09dbf872b91187297980c2e624d58a54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.235ex; height:6.509ex;" alt="{\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{g(n)}}>0}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=\omega (g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=\omega (g(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488ceb81e87597589fc9cd0ba7f54345f6569929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.156ex; height:2.843ex;" alt="{\displaystyle f(n)=\omega (g(n))}"> </td> <td>Small Omega; Little Omega </td> <td>f dominates g asymptotically </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)>k\,g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>k</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)>k\,g(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b016c45b28b07815ff7f1e86ce09064f61057faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.048ex; height:2.843ex;" alt="{\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)>k\,g(n)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94f1a706b1f85b13d89bb38ef7a1486a755caa3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.401ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=\Omega (g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=\Omega (g(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df8a463b6e106cce5d68174c1df5768a1682ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.388ex; height:2.843ex;" alt="{\displaystyle f(n)=\Omega (g(n))}"> </td> <td>Big Omega in number theory (Hardy–Littlewood) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940e58aa437f429f47fd743a819e41fedaa1ff9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle |f|}"> is not dominated by g asymptotically </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists k>0\,\forall n_{0}\,\exists n>n_{0}\colon |f(n)|\geq k\,g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>k</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists k>0\,\forall n_{0}\,\exists n>n_{0}\colon |f(n)|\geq k\,g(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1de6bf9135e3729da954f9a7e35225a1689c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.342ex; height:2.843ex;" alt="{\displaystyle \exists k>0\,\forall n_{0}\,\exists n>n_{0}\colon |f(n)|\geq k\,g(n)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \limsup _{n\to \infty }{\frac {\left|f(n)\right|}{g(n)}}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \limsup _{n\to \infty }{\frac {\left|f(n)\right|}{g(n)}}>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd52d75e1283fbf16cee8ade84c95a94b39e776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.379ex; height:6.509ex;" alt="{\displaystyle \limsup _{n\to \infty }{\frac {\left|f(n)\right|}{g(n)}}>0}"> </td></tr></tbody></table> The limit definitions assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f58b017ed2f12e925640328d716644bced11d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.581ex; height:2.843ex;" alt="{\displaystyle g(n)>0}"> for sufficiently large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. The table is (partly) sorted from smallest to largest, in the sense that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o,O,\Theta ,\sim ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo>,</mo> <mi>O</mi> <mo>,</mo> <mi mathvariant="normal">Θ</mi> <mo>,</mo> <mo>∼</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o,O,\Theta ,\sim ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e390d0aeb498be71e63699533c19791c7479950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.266ex; height:2.509ex;" alt="{\displaystyle o,O,\Theta ,\sim ,}"> (Knuth's version of) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ,\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ,\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae305c1f11e74128425067caff8f3c749ff61fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.158ex; height:2.509ex;" alt="{\displaystyle \Omega ,\omega }"> on functions correspond to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <,\leq ,\approx ,=,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> <mo>,</mo> <mo>≤</mo> <mo>,</mo> <mo>≈</mo> <mo>,</mo> <mo>=</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <,\leq ,\approx ,=,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb15153a0a17dfe2878ee7c7590230ad4c9d3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.981ex; height:2.343ex;" alt="{\displaystyle <,\leq ,\approx ,=,}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq ,>}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≥</mo> <mo>,</mo> <mo>></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \geq ,>}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0adc8772efbceb6ee25889368c7e0b08c1abf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.65ex; height:2.343ex;" alt="{\displaystyle \geq ,>}"> on the real line<a href="#cite_note-Wild-27">[27]</a> (the Hardy–Littlewood version of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }">, however, doesn't correspond to any such description). Computer science uses the big <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}">, big Theta <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }">, little <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}">, little omega <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> and Knuth's big Omega <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> notations.<a href="#cite_note-28">[28]</a> Analytic number theory often uses the big <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}">, small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}">, Hardy's <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \asymp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≍</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \asymp }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57855824799fd82a250ecfe10596a751bb1b091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \asymp }">,<a href="#cite_note-GT-29">[29]</a> Hardy–Littlewood's big Omega <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> (with or without the +, − or ± subscripts) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"> notations.<a href="#cite_note-Ivic-22">[22]</a> The small omega <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> notation is not used as often in analysis.<a href="#cite_note-30">[30]</a> <div class="mw-heading mw-heading3"><h3 id="Use_in_computer_science">Use in computer science</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=25" title="Edit section: Use in computer science">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></div> Informally, especially in computer science, the big O notation often can be used somewhat differently to describe an asymptotic <a href="/wiki/Upper_and_lower_bounds#Tight_bounds" title="Upper and lower bounds">tight</a> bound where using big Theta Θ notation might be more factually appropriate in a given context.<a href="#cite_note-31">[31]</a> For example, when considering a function T(n) = 73n3 + 22n2 + 58, all of the following are generally acceptable, but tighter bounds (such as numbers 2 and 3 below) are usually strongly preferred over looser bounds (such as number 1 below). <ol><li>T(n) = O(n100)</li> <li>T(n) = O(n3)</li> <li>T(n) = Θ(n3)</li></ol> The equivalent English statements are respectively: <ol><li>T(n) grows asymptotically no faster than n100</li> <li>T(n) grows asymptotically no faster than n3</li> <li>T(n) grows asymptotically as fast as n3.</li></ol> So while all three statements are true, progressively more information is contained in each. In some fields, however, the big O notation (number 2 in the lists above) would be used more commonly than the big Theta notation (items numbered 3 in the lists above). For example, if T(n) represents the running time of a newly developed algorithm for input size n, the inventors and users of the algorithm might be more inclined to put an upper asymptotic bound on how long it will take to run without making an explicit statement about the lower asymptotic bound. <div class="mw-heading mw-heading3"><h3 id="Other_notation">Other notation</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=26" title="Edit section: Other notation">edit</a>]</div> In their book <a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a>, <a href="/wiki/Thomas_H._Cormen" title="Thomas H. Cormen">Cormen</a>, <a href="/wiki/Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson</a>, <a href="/wiki/Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Rivest</a> and <a href="/wiki/Clifford_Stein" title="Clifford Stein">Stein</a> consider the set of functions f which satisfy <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=O(g(n))\quad (n\to \infty )~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=O(g(n))\quad (n\to \infty )~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0646d976911ba7bb59922696d6978c8a0dc5f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.176ex; height:2.843ex;" alt="{\displaystyle f(n)=O(g(n))\quad (n\to \infty )~.}"></dd></dl> In a correct notation this set can, for instance, be called O(g), where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(g)=\{f:{\text{there exist positive constants}}~c~{\text{and}}~n_{0}~{\text{such that}}~0\leq f(n)\leq cg(n){\text{ for all }}n\geq n_{0}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>there exist positive constants</mtext> </mrow> <mtext> </mtext> <mi>c</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mtext> </mtext> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>such that</mtext> </mrow> <mtext> </mtext> <mn>0</mn> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>c</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(g)=\{f:{\text{there exist positive constants}}~c~{\text{and}}~n_{0}~{\text{such that}}~0\leq f(n)\leq cg(n){\text{ for all }}n\geq n_{0}\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b5656aaec472c2053d8261bd1b68c71eae8f1c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:93.621ex; height:2.843ex;" alt="{\displaystyle O(g)=\{f:{\text{there exist positive constants}}~c~{\text{and}}~n_{0}~{\text{such that}}~0\leq f(n)\leq cg(n){\text{ for all }}n\geq n_{0}\}.}"><a href="#cite_note-32">[32]</a> The authors state that the use of equality operator (=) to denote set membership rather than the set membership operator (∈) is an abuse of notation, but that doing so has advantages.<a href="#cite_note-clrs3-6">[6]</a> Inside an equation or inequality, the use of asymptotic notation stands for an anonymous function in the set O(g), which eliminates lower-order terms, and helps to reduce inessential clutter in equations, for example:<a href="#cite_note-33">[33]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n^{2}+3n+1=2n^{2}+O(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n^{2}+3n+1=2n^{2}+O(n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eedc305f6b5d915801f7046b89be85833efa5f73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.186ex; height:3.176ex;" alt="{\displaystyle 2n^{2}+3n+1=2n^{2}+O(n).}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Extensions_to_the_Bachmann–Landau_notations">Extensions to the Bachmann–Landau notations</h3>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=27" title="Edit section: Extensions to the Bachmann–Landau notations">edit</a>]</div> Another notation sometimes used in computer science is <a href="/wiki/%C3%95" title="Õ">Õ</a> (read soft-O), which hides polylogarithmic factors. There are two definitions in use: some authors use f(n) = Õ(g(n)) as <a href="/wiki/Shorthand" title="Shorthand">shorthand</a> for f(n) = O(g(n) <a href="/wiki/Polylogarithmic_function" title="Polylogarithmic function">logk n</a>) for some k, while others use it as shorthand for f(n) = O(g(n) logk g(n)).<a href="#cite_note-34">[34]</a> When g(n) is polynomial in n, there is no difference; however, the latter definition allows one to say, e.g. that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n2^{n}={\tilde {O}}(2^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>O</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n2^{n}={\tilde {O}}(2^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ba0a1a6539cb13a71f780c881e36968c6a9d60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.838ex; height:3.176ex;" alt="{\displaystyle n2^{n}={\tilde {O}}(2^{n})}"> while the former definition allows for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log ^{k}n={\tilde {O}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⁡</mo> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>O</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log ^{k}n={\tilde {O}}(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e914ff8543f85d1e7f2f0aff80fab95847cd4a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.686ex; height:3.176ex;" alt="{\displaystyle \log ^{k}n={\tilde {O}}(1)}"> for any constant k. Some authors write O* for the same purpose as the latter definition.<a href="#cite_note-35">[35]</a> Essentially, it is big O notation, ignoring <a href="/wiki/Polylogarithmic_function" title="Polylogarithmic function">logarithmic factors</a> because the <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">growth-rate</a> effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since logk n is always o(nε) for any constant k and any ε > 0). Also, the <a href="/wiki/L-notation" title="L-notation">L notation</a>, defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>α</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59b5d9f7b7c3e8763737cb136642c142bd27da00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.401ex; height:3.676ex;" alt="{\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }},}"></dd></dl> is convenient for functions that are between <a href="/wiki/Time_complexity#Polynomial_time" title="Time complexity">polynomial</a> and <a href="/wiki/Time_complexity#Exponential_time" title="Time complexity">exponential</a> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7e60afd0a48ba01cb2fee90f0199c28dce0298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.721ex; height:2.176ex;" alt="{\displaystyle \ln n}">. <div class="mw-heading mw-heading2"><h2 id="Generalizations_and_related_usages">Generalizations and related usages</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=28" title="Edit section: Generalizations and related usages">edit</a>]</div> The generalization to functions taking values in any <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. A generalization to functions g taking values in any <a href="/wiki/Topological_group" title="Topological group">topological group</a> is also possible[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]. The "limiting process" x → xo can also be generalized by introducing an arbitrary <a href="/wiki/Filter_base" class="mw-redirect" title="Filter base">filter base</a>, i.e. to directed <a href="/wiki/Net_(mathematics)" title="Net (mathematics)">nets</a> f and g. The o notation can be used to define <a href="/wiki/Derivative" title="Derivative">derivatives</a> and <a href="/wiki/Differentiability" class="mw-redirect" title="Differentiability">differentiability</a> in quite general spaces, and also (asymptotical) equivalence of functions, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\sim g\iff (f-g)\in o(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∼</mo> <mi>g</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\sim g\iff (f-g)\in o(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/288861b9af5eb92d645701a42b4887578d6aeed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.328ex; height:2.843ex;" alt="{\displaystyle f\sim g\iff (f-g)\in o(g)}"></dd></dl> which is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> and a more restrictive notion than the relationship "f is Θ(g)" from above. (It reduces to lim f / g = 1 if f and g are positive real valued functions.) For example, 2x is Θ(x), but 2x − x is not o(x). <div class="mw-heading mw-heading2"><h2 id="History_(Bachmann–Landau,_Hardy,_and_Vinogradov_notations)">History (Bachmann–Landau, Hardy, and Vinogradov notations)</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=29" title="Edit section: History (Bachmann–Landau, Hardy, and Vinogradov notations)">edit</a>]</div> The symbol O was first introduced by number theorist <a href="/wiki/Paul_Bachmann" class="mw-redirect" title="Paul Bachmann">Paul Bachmann</a> in 1894, in the second volume of his book Analytische Zahlentheorie ("<a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>").<a href="#cite_note-Bachmann-1">[1]</a> The number theorist <a href="/wiki/Edmund_Landau" title="Edmund Landau">Edmund Landau</a> adopted it, and was thus inspired to introduce in 1909 the notation o;<a href="#cite_note-Landau-2">[2]</a> hence both are now called Landau symbols. These notations were used in applied mathematics during the 1950s for asymptotic analysis.<a href="#cite_note-36">[36]</a> The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> (in the sense "is not an o of") was introduced in 1914 by Hardy and Littlewood.<a href="#cite_note-HL-19">[19]</a> Hardy and Littlewood also introduced in 1916 the symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{R}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb36a87fe7819b8c5f859475abb8e5e92152a886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.158ex; height:2.509ex;" alt="{\displaystyle \Omega _{R}}"> ("right") and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ca7869e6dabfee30cbede644d9597b1e644422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \Omega _{L}}"> ("left"),<a href="#cite_note-HL2-20">[20]</a> precursors of the modern symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aece5d68daf75b9115c8bda0d66c4a0c1d462888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \Omega _{+}}"> ("is not smaller than a small o of") and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aca6086114a9cedeb6893d24e0a5c894e853de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \Omega _{-}}"> ("is not larger than a small o of"). Thus the Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols". This notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> became commonly used in number theory at least since the 1950s.<a href="#cite_note-titchmarsh-37">[37]</a> The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }">, although it had been used before with different meanings,<a href="#cite_note-Wild-27">[27]</a> was given its modern definition by Landau in 1909<a href="#cite_note-38">[38]</a> and by Hardy in 1910.<a href="#cite_note-Hardy-39">[39]</a> Just above on the same page of his tract Hardy defined the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \asymp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≍</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \asymp }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57855824799fd82a250ecfe10596a751bb1b091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \asymp }">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\asymp g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≍</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\asymp g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee00f45070774eacea191324a222395bc050895" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle f(x)\asymp g(x)}"> means that both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O(g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O(g(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e69c69cb5dc8e33a50a094deff53ec988fa6aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.354ex; height:2.843ex;" alt="{\displaystyle f(x)=O(g(x))}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=O(f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=O(f(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/003c7cb944326248afb5cc4233db0ef354229feb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.354ex; height:2.843ex;" alt="{\displaystyle g(x)=O(f(x))}"> are satisfied. The notation is still currently used in analytic number theory.<a href="#cite_note-40">[40]</a><a href="#cite_note-GT-29">[29]</a> In his tract Hardy also proposed the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\asymp \!\!\!\!\!\!-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>≍</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\asymp \!\!\!\!\!\!-}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/362539d1e3e203e2f4d61b2bcf6bafb24620aa4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-left: -0.127ex; width:2.453ex; height:2.176ex;" alt="{\displaystyle \,\asymp \!\!\!\!\!\!-}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\asymp \!\!\!\!\!\!-g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≍</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>−</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\asymp \!\!\!\!\!\!-g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/069538a19179ea99ff9cff29619d6db6842a3ae2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.979ex; height:2.509ex;" alt="{\displaystyle f\asymp \!\!\!\!\!\!-g}"> means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\sim Kg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∼</mo> <mi>K</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\sim Kg}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3923ac7fa2211ce5236777910a23ae71061cc10e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.559ex; height:2.509ex;" alt="{\displaystyle f\sim Kg}"> for some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\not =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\not =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1424e3c491415e77856462bf80d4015d66afb3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.327ex; height:2.676ex;" alt="{\displaystyle K\not =0}">. In the 1970s the big O was popularized in computer science by <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>, who proposed the different notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\Theta (g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\Theta (g(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecd70e6ee84d7584d91b2d5d7cda885fcbe2f11a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.388ex; height:2.843ex;" alt="{\displaystyle f(x)=\Theta (g(x))}"> for Hardy's <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\asymp g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≍</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\asymp g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee00f45070774eacea191324a222395bc050895" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle f(x)\asymp g(x)}">, and proposed a different definition for the Hardy and Littlewood Omega notation.<a href="#cite_note-knuth-24">[24]</a> Two other symbols coined by Hardy were (in terms of the modern O notation) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\preccurlyeq g\iff f=O(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≼</mo> <mi>g</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>f</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\preccurlyeq g\iff f=O(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/284c00309f89e2f9f1022c935b110d020ef66b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.466ex; height:2.843ex;" alt="{\displaystyle f\preccurlyeq g\iff f=O(g)}">   and   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\prec g\iff f=o(g);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≺</mo> <mi>g</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>f</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\prec g\iff f=o(g);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4cee5e7aba783c4a64db2022ff63767baecb9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.467ex; height:2.843ex;" alt="{\displaystyle f\prec g\iff f=o(g);}"></dd></dl> (Hardy however never defined or used the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prec \!\!\prec }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≺</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>≺</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prec \!\!\prec }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a283e08c1b4c6e1505c7c13cd4382b19a2844b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.842ex; height:1.843ex;" alt="{\displaystyle \prec \!\!\prec }">, nor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ll }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ll }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10563f31a4178c71bef006ea5c601e908f984136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \ll }">, as it has been sometimes reported). Hardy introduced the symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \preccurlyeq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \preccurlyeq }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bc8c68be5ec5d812bb700383d1911b5f94a1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.343ex;" alt="{\displaystyle \preccurlyeq }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prec }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≺</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prec }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59707ac9078525b52a9e21a1baf9ab787af7a9aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \prec }"> (as well as the already mentioned other symbols) in his 1910 tract "Orders of Infinity", and made use of them only in three papers (1910–1913). In his nearly 400 remaining papers and books he consistently used the Landau symbols O and o. Hardy's symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \preccurlyeq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \preccurlyeq }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bc8c68be5ec5d812bb700383d1911b5f94a1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.343ex;" alt="{\displaystyle \preccurlyeq }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prec }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≺</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prec }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59707ac9078525b52a9e21a1baf9ab787af7a9aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \prec }"> (as well as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\asymp \!\!\!\!\!\!-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>≍</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\asymp \!\!\!\!\!\!-}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/362539d1e3e203e2f4d61b2bcf6bafb24620aa4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-left: -0.127ex; width:2.453ex; height:2.176ex;" alt="{\displaystyle \,\asymp \!\!\!\!\!\!-}">) are not used anymore. On the other hand, in the 1930s,<a href="#cite_note-41">[41]</a> the Russian number theorist <a href="/wiki/Ivan_Matveyevich_Vinogradov" class="mw-redirect" title="Ivan Matveyevich Vinogradov">Ivan Matveyevich Vinogradov</a> introduced his notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ll }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ll }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10563f31a4178c71bef006ea5c601e908f984136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \ll }">, which has been increasingly used in number theory instead of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"> notation. We have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\ll g\iff f=O(g),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≪</mo> <mi>g</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>f</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\ll g\iff f=O(g),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bed035cc5f8f5dfaf9e38b92cfdb16b87e481929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.628ex; height:2.843ex;" alt="{\displaystyle f\ll g\iff f=O(g),}"></dd></dl> and frequently both notations are used in the same paper. The big-O originally stands for "order of" ("Ordnung", Bachmann 1894), and is thus a Latin letter. Neither Bachmann nor Landau ever call it "Omicron". The symbol was much later on (1976) viewed by Knuth as a capital <a href="/wiki/Omicron" title="Omicron">omicron</a>,<a href="#cite_note-knuth-24">[24]</a> probably in reference to his definition of the symbol <a href="/wiki/Omega" title="Omega">Omega</a>. The digit <a href="/wiki/0" title="0">zero</a> should not be used. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=30" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Asymptotic_computational_complexity" title="Asymptotic computational complexity">Asymptotic computational complexity</a></li> <li><a href="/wiki/Asymptotic_expansion" title="Asymptotic expansion">Asymptotic expansion</a>: Approximation of functions generalizing Taylor's formula</li> <li><a href="/wiki/Asymptotically_optimal_algorithm" title="Asymptotically optimal algorithm">Asymptotically optimal algorithm</a>: A phrase frequently used to describe an algorithm that has an upper bound asymptotically within a constant of a lower bound for the problem</li> <li><a href="/wiki/Big_O_in_probability_notation" title="Big O in probability notation">Big O in probability notation</a>: Op, op</li> <li><a href="/wiki/Limit_inferior_and_limit_superior" title="Limit inferior and limit superior">Limit inferior and limit superior</a>: An explanation of some of the limit notation used in this article</li> <li><a href="/wiki/Master_theorem_(analysis_of_algorithms)" title="Master theorem (analysis of algorithms)">Master theorem (analysis of algorithms)</a>: For analyzing divide-and-conquer recursive algorithms using Big O notation</li> <li><a href="/wiki/Nachbin%27s_theorem" title="Nachbin's theorem">Nachbin's theorem</a>: A precise method of bounding <a href="/wiki/Complex_analytic" class="mw-redirect" title="Complex analytic">complex analytic</a> functions so that the domain of convergence of <a href="/wiki/Integral_transform" title="Integral transform">integral transforms</a> can be stated</li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/Computational_complexity_of_mathematical_operations" title="Computational complexity of mathematical operations">Computational complexity of mathematical operations</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References_and_notes">References and notes</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=31" title="Edit section: References and notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Bachmann-1">^ <a href="#cite_ref-Bachmann_1-0">a</a> <a href="#cite_ref-Bachmann_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBachmann1894" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Paul_Bachmann" class="mw-redirect" title="Paul Bachmann">Bachmann, Paul</a> (1894). <a rel="nofollow" class="external text" href="https://archive.org/stream/dieanalytischeza00bachuoft#page/402/mode/2up">Analytische Zahlentheorie</a> [Analytic Number Theory] (in German). Vol. 2. Leipzig: Teubner.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytische+Zahlentheorie&rft.place=Leipzig&rft.pub=Teubner&rft.date=1894&rft.aulast=Bachmann&rft.aufirst=Paul&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fdieanalytischeza00bachuoft%23page%2F402%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-Landau-2">^ <a href="#cite_ref-Landau_2-0">a</a> <a href="#cite_ref-Landau_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau1909" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Edmund_Landau" title="Edmund Landau">Landau, Edmund</a> (1909). <a rel="nofollow" class="external text" href="https://archive.org/details/handbuchderlehre01landuoft">Handbuch der Lehre von der Verteilung der Primzahlen</a> [Handbook on the theory of the distribution of the primes] (in German). Vol. 1. Leipzig: B. G. Teubner. p. 61.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbuch+der+Lehre+von+der+Verteilung+der+Primzahlen&rft.place=Leipzig&rft.pages=61&rft.pub=B.+G.+Teubner&rft.date=1909&rft.aulast=Landau&rft.aufirst=Edmund&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhandbuchderlehre01landuoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> Also see page 883 in vol. 2 of the book (not available from the link given). </li> <li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <a href="#cite_ref-:0_3-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormenLeisersonRivest1990" class="citation book cs1"><a href="/wiki/Thomas_H._Cormen" title="Thomas H. Cormen">Cormen, Thomas H.</a>; <a href="/wiki/Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson, Charles E.</a>; <a href="/wiki/Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Rivest, Ronald L.</a> (1990). "Growth of Functions". <a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a> (1st ed.). MIT Press and McGraw-Hill. pp. 23–41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-53091-0" title="Special:BookSources/978-0-262-53091-0"><bdi>978-0-262-53091-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Growth+of+Functions&rft.btitle=Introduction+to+Algorithms&rft.pages=23-41&rft.edition=1st&rft.pub=MIT+Press+and+McGraw-Hill&rft.date=1990&rft.isbn=978-0-262-53091-0&rft.aulast=Cormen&rft.aufirst=Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-LandauO-4"><a href="#cite_ref-LandauO_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau1909" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Edmund_Landau" title="Edmund Landau">Landau, Edmund</a> (1909). <a rel="nofollow" class="external text" href="https://archive.org/stream/handbuchderlehre01landuoft#page/31/mode/2up">Handbuch der Lehre von der Verteilung der Primzahlen</a> [Handbook on the theory of the distribution of the primes] (in German). Leipzig: B.G. Teubner. p. 31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbuch+der+Lehre+von+der+Verteilung+der+Primzahlen&rft.place=Leipzig&rft.pages=31&rft.pub=B.G.+Teubner&rft.date=1909&rft.aulast=Landau&rft.aufirst=Edmund&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fhandbuchderlehre01landuoft%23page%2F31%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSipser1997" class="citation book cs1">Sipser, Michael (1997). Introduction to the Theory of Computation. Boston, MA: PWS Publishing. p. 227, def. 7.2.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Computation&rft.place=Boston%2C+MA&rft.pages=227%2C+def.-7.2&rft.pub=PWS+Publishing&rft.date=1997&rft.aulast=Sipser&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-clrs3-6">^ <a href="#cite_ref-clrs3_6-0">a</a> <a href="#cite_ref-clrs3_6-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormen,_Thomas_H.Leiserson,_Charles_E.Rivest,_Ronald_L.2009" class="citation book cs1">Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoal00corm_805">Introduction to Algorithms</a> (3rd ed.). Cambridge/MA: MIT Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoal00corm_805/page/n65">45</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-53305-8" title="Special:BookSources/978-0-262-53305-8"><bdi>978-0-262-53305-8</bdi></a>. <q>Because θ(g(n)) is a set, we could write "f(n) ∈ θ(g(n))" to indicate that f(n) is a member of θ(g(n)). Instead, we will usually write f(n) = θ(g(n)) to express the same notion. You might be confused because we abuse equality in this way, but we shall see later in this section that doing so has its advantages.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Algorithms&rft.place=Cambridge%2FMA&rft.pages=45&rft.edition=3rd&rft.pub=MIT+Press&rft.date=2009&rft.isbn=978-0-262-53305-8&rft.au=Cormen%2C+Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoal00corm_805&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFCormenLeisersonRivestStein2009">Cormen et al. (2009)</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoal00corm_805/page/n73">p. 53</a> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHowell" class="citation web cs1">Howell, Rodney. <a rel="nofollow" class="external text" href="http://people.cis.ksu.edu/~rhowell/asymptotic.pdf">"On Asymptotic Notation with Multiple Variables"</a> (PDF). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150424012920/http://people.cis.ksu.edu/~rhowell/asymptotic.pdf">Archived</a> (PDF) from the original on 2015-04-24. Retrieved 2015-04-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=On+Asymptotic+Notation+with+Multiple+Variables&rft.aulast=Howell&rft.aufirst=Rodney&rft_id=http%3A%2F%2Fpeople.cis.ksu.edu%2F~rhowell%2Fasymptotic.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-deBruijn-9">^ <a href="#cite_ref-deBruijn_9-0">a</a> <a href="#cite_ref-deBruijn_9-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Bruijn1958" class="citation book cs1"><a href="/wiki/N._G._de_Bruijn" class="mw-redirect" title="N. G. de Bruijn">de Bruijn, N.G.</a> (1958). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_tnwmvHmVwMC&q=%22The+trouble+is%22&pg=PA5">Asymptotic Methods in Analysis</a>. Amsterdam: North-Holland. pp. 5–7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-64221-5" title="Special:BookSources/978-0-486-64221-5"><bdi>978-0-486-64221-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230117051949/https://books.google.com/books?id=_tnwmvHmVwMC&q=%22The+trouble+is%22&pg=PA5">Archived</a> from the original on 2023-01-17. Retrieved 2021-09-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Asymptotic+Methods+in+Analysis&rft.place=Amsterdam&rft.pages=5-7&rft.pub=North-Holland&rft.date=1958&rft.isbn=978-0-486-64221-5&rft.aulast=de+Bruijn&rft.aufirst=N.G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_tnwmvHmVwMC%26q%3D%2522The%2Btrouble%2Bis%2522%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-Concrete_Mathematics-10">^ <a href="#cite_ref-Concrete_Mathematics_10-0">a</a> <a href="#cite_ref-Concrete_Mathematics_10-1">b</a> <a href="#cite_ref-Concrete_Mathematics_10-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrahamKnuthPatashnik1994" class="citation book cs1"><a href="/wiki/Ronald_Graham" title="Ronald Graham">Graham, Ronald</a>; <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a>; <a href="/wiki/Oren_Patashnik" title="Oren Patashnik">Patashnik, Oren</a> (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pntQAAAAMAAJ">Concrete Mathematics</a> (2 ed.). Reading, Massachusetts: Addison–Wesley. p. 446. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-55802-9" title="Special:BookSources/978-0-201-55802-9"><bdi>978-0-201-55802-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230117051955/https://books.google.com/books?id=pntQAAAAMAAJ">Archived</a> from the original on 2023-01-17. Retrieved 2016-09-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concrete+Mathematics&rft.place=Reading%2C+Massachusetts&rft.pages=446&rft.edition=2&rft.pub=Addison%E2%80%93Wesley&rft.date=1994&rft.isbn=978-0-201-55802-9&rft.aulast=Graham&rft.aufirst=Ronald&rft.au=Knuth%2C+Donald&rft.au=Patashnik%2C+Oren&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpntQAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonald_Knuth1998" class="citation journal cs1">Donald Knuth (June–July 1998). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/199806/commentary.pdf">"Teach Calculus with Big O"</a> (PDF). <a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a>. 45 (6): 687. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211014070416/https://www.ams.org/notices/199806/commentary.pdf">Archived</a> (PDF) from the original on 2021-10-14. Retrieved 2021-09-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=Teach+Calculus+with+Big+O&rft.volume=45&rft.issue=6&rft.pages=687&rft.date=1998-06%2F1998-07&rft.au=Donald+Knuth&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F199806%2Fcommentary.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> (<a rel="nofollow" class="external text" href="http://www-cs-staff.stanford.edu/~knuth/ocalc.tex">Unabridged version</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080513234708/http://www-cs-staff.stanford.edu/~knuth/ocalc.tex">Archived</a> 2008-05-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>) </li> <li id="cite_note-KnuthArt-12"><a href="#cite_ref-KnuthArt_12-0">^</a> Donald E. Knuth, The art of computer programming. Vol. 1. Fundamental algorithms, third edition, Addison Wesley Longman, 1997. Section 1.2.11.1. </li> <li id="cite_note-ConcreteMath-13"><a href="#cite_ref-ConcreteMath_13-0">^</a> Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science (2nd ed.), Addison-Wesley, 1994. Section 9.2, p. 443. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Sivaram Ambikasaran and Eric Darve, An <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(N\log N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(N\log N)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8658b7724f2492980eab8528378dc975a9a072ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.532ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(N\log N)}"> Fast Direct Solver for Partial Hierarchically Semi-Separable Matrices, J. Scientific Computing 57 (2013), no. 3, 477–501. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> Saket Saurabh and Meirav Zehavi, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k,n-k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k,n-k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ffc83943cba6e01a096c690f9d676fad4043f09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.501ex; height:2.843ex;" alt="{\displaystyle (k,n-k)}">-Max-Cut: An <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}^{*}(2^{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}^{*}(2^{p})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d068915ba2c03baee0e3671bc56e3057bc9441e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.935ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}^{*}(2^{p})}">-Time Algorithm and a Polynomial Kernel, Algorithmica 80 (2018), no. 12, 3844–3860. </li> <li id="cite_note-Landausmallo-16">^ <a href="#cite_ref-Landausmallo_16-0">a</a> <a href="#cite_ref-Landausmallo_16-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau1909" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Edmund_Landau" title="Edmund Landau">Landau, Edmund</a> (1909). <a rel="nofollow" class="external text" href="https://archive.org/stream/handbuchderlehre01landuoft#page/61/mode/2up">Handbuch der Lehre von der Verteilung der Primzahlen</a> [Handbook on the theory of the distribution of the primes] (in German). Leipzig: B. G. Teubner. p. 61.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbuch+der+Lehre+von+der+Verteilung+der+Primzahlen&rft.place=Leipzig&rft.pages=61&rft.pub=B.+G.+Teubner&rft.date=1909&rft.aulast=Landau&rft.aufirst=Edmund&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fhandbuchderlehre01landuoft%23page%2F61%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-Introduction_to_Algorithms-17"><a href="#cite_ref-Introduction_to_Algorithms_17-0">^</a> Thomas H. Cormen et al., 2001, <a rel="nofollow" class="external text" href="http://highered.mcgraw-hill.com/sites/0070131511/">Introduction to Algorithms, Second Edition, Ch. 3.1</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090116115944/http://highered.mcgraw-hill.com/sites/0070131511/">Archived</a> 2009-01-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormenLeisersonRivestStein2009" class="citation book cs1">Cormen TH, Leiserson CE, Rivest RL, Stein C (2009). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/676697295">Introduction to algorithms</a> (3rd ed.). Cambridge, Mass.: MIT Press. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-27083-0" title="Special:BookSources/978-0-262-27083-0"><bdi>978-0-262-27083-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/676697295">676697295</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+algorithms&rft.place=Cambridge%2C+Mass.&rft.pages=48&rft.edition=3rd&rft.pub=MIT+Press&rft.date=2009&rft_id=info%3Aoclcnum%2F676697295&rft.isbn=978-0-262-27083-0&rft.aulast=Cormen&rft.aufirst=TH&rft.au=Leiserson%2C+CE&rft.au=Rivest%2C+RL&rft.au=Stein%2C+C&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F676697295&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-HL-19">^ <a href="#cite_ref-HL_19-0">a</a> <a href="#cite_ref-HL_19-1">b</a> <a href="#cite_ref-HL_19-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyLittlewood1914" class="citation journal cs1"><a href="/wiki/Godfrey_Harold_Hardy" class="mw-redirect" title="Godfrey Harold Hardy">Hardy, G.H.</a>; <a href="/wiki/John_Edensor_Littlewood" title="John Edensor Littlewood">Littlewood, J.E.</a> (1914). <a rel="nofollow" class="external text" href="http://projecteuclid.org/download/pdf_1/euclid.acta/1485887376">"Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic θ functions"</a>. <a href="/wiki/Acta_Mathematica" title="Acta Mathematica">Acta Mathematica</a>. 37: 225. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02401834">10.1007/BF02401834</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181212063403/https://projecteuclid.org/download/pdf_1/euclid.acta/1485887376">Archived</a> from the original on 2018-12-12. 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SIGACT News. 8 (2): 18–24. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1008328.1008329">10.1145/1008328.1008329</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5230246">5230246</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIGACT+News&rft.atitle=Big+Omicron+and+big+Omega+and+big+Theta&rft.volume=8&rft.issue=2&rft.pages=18-24&rft.date=1976-04%2F1976-06&rft_id=info%3Adoi%2F10.1145%2F1008328.1008329&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5230246%23id-name%3DS2CID&rft.aulast=Knuth&rft.aufirst=Donald&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F1008328.1008329&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-Balcázar-25"><a href="#cite_ref-Balcázar_25-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBalcázarGabarró" class="citation journal cs1">Balcázar, José L.; Gabarró, Joaquim. <a rel="nofollow" class="external text" href="http://archive.numdam.org/article/ITA_1989__23_2_177_0.pdf">"Nonuniform complexity classes specified by lower and upper bounds"</a> (PDF). RAIRO – Theoretical Informatics and Applications – Informatique Théorique et Applications. 23 (2): 180. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0988-3754">0988-3754</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170314153158/http://archive.numdam.org/article/ITA_1989__23_2_177_0.pdf">Archived</a> (PDF) from the original on 14 March 2017. Retrieved 14 March 2017 – via Numdam.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=RAIRO+%E2%80%93+Theoretical+Informatics+and+Applications+%E2%80%93+Informatique+Th%C3%A9orique+et+Applications&rft.atitle=Nonuniform+complexity+classes+specified+by+lower+and+upper+bounds&rft.volume=23&rft.issue=2&rft.pages=180&rft.issn=0988-3754&rft.aulast=Balc%C3%A1zar&rft.aufirst=Jos%C3%A9+L.&rft.au=Gabarr%C3%B3%2C+Joaquim&rft_id=http%3A%2F%2Farchive.numdam.org%2Farticle%2FITA_1989__23_2_177_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-Cucker-26"><a href="#cite_ref-Cucker_26-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCuckerBürgisser2013" class="citation book cs1">Cucker, Felipe; Bürgisser, Peter (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SNu4BAAAQBAJ&pg=PA467">"A.1 Big Oh, Little Oh, and Other Comparisons"</a>. Condition: The Geometry of Numerical Algorithms. Berlin, Heidelberg: Springer. pp. 467–468. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-38896-5">10.1007/978-3-642-38896-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-38896-5" title="Special:BookSources/978-3-642-38896-5"><bdi>978-3-642-38896-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A.1+Big+Oh%2C+Little+Oh%2C+and+Other+Comparisons&rft.btitle=Condition%3A+The+Geometry+of+Numerical+Algorithms&rft.place=Berlin%2C+Heidelberg&rft.pages=467-468&rft.pub=Springer&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-3-642-38896-5&rft.isbn=978-3-642-38896-5&rft.aulast=Cucker&rft.aufirst=Felipe&rft.au=B%C3%BCrgisser%2C+Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSNu4BAAAQBAJ%26pg%3DPA467&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-Wild-27">^ <a href="#cite_ref-Wild_27-0">a</a> <a href="#cite_ref-Wild_27-1">b</a> <a href="#cite_ref-Wild_27-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVitányiMeertens1985" class="citation journal cs1"><a href="/wiki/Paul_Vitanyi" class="mw-redirect" title="Paul Vitanyi">Vitányi, Paul</a>; <a href="/wiki/Lambert_Meertens" title="Lambert Meertens">Meertens, Lambert</a> (April 1985). <a rel="nofollow" class="external text" href="http://www.kestrel.edu/home/people/meertens/publications/papers/Big_Omega_contra_the_wild_functions.pdf">"Big Omega versus the wild functions"</a> (PDF). ACM SIGACT News. 16 (4): 56–59. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.694.3072">10.1.1.694.3072</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F382242.382835">10.1145/382242.382835</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11700420">11700420</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160310012405/http://www.kestrel.edu/home/people/meertens/publications/papers/Big_Omega_contra_the_wild_functions.pdf">Archived</a> (PDF) from the original on 2016-03-10. Retrieved 2017-03-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGACT+News&rft.atitle=Big+Omega+versus+the+wild+functions&rft.volume=16&rft.issue=4&rft.pages=56-59&rft.date=1985-04&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.694.3072%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11700420%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F382242.382835&rft.aulast=Vit%C3%A1nyi&rft.aufirst=Paul&rft.au=Meertens%2C+Lambert&rft_id=http%3A%2F%2Fwww.kestrel.edu%2Fhome%2Fpeople%2Fmeertens%2Fpublications%2Fpapers%2FBig_Omega_contra_the_wild_functions.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormenLeisersonRivestStein2001" class="citation book cs1"><a href="/wiki/Thomas_H._Cormen" title="Thomas H. Cormen">Cormen, Thomas H.</a>; <a href="/wiki/Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson, Charles E.</a>; <a href="/wiki/Ron_Rivest" title="Ron Rivest">Rivest, Ronald L.</a>; <a href="/wiki/Clifford_Stein" title="Clifford Stein">Stein, Clifford</a> (2001) [1990]. <a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a> (2nd ed.). MIT Press and McGraw-Hill. pp. 41–50. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-03293-7" title="Special:BookSources/0-262-03293-7"><bdi>0-262-03293-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Algorithms&rft.pages=41-50&rft.edition=2nd&rft.pub=MIT+Press+and+McGraw-Hill&rft.date=2001&rft.isbn=0-262-03293-7&rft.aulast=Cormen&rft.aufirst=Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rft.au=Stein%2C+Clifford&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-GT-29">^ <a href="#cite_ref-GT_29-0">a</a> <a href="#cite_ref-GT_29-1">b</a> Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, « Notation », page xxiii. American Mathematical Society, Providence RI, 2015. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> for example it is omitted in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHildebrand" class="citation web cs1">Hildebrand, A.J. <a rel="nofollow" class="external text" href="http://www.math.uiuc.edu/~ajh/595ama/ama-ch2.pdf">"Asymptotic Notations"</a> (PDF). Department of Mathematics. Asymptotic Methods in Analysis. Math 595, Fall 2009. Urbana, IL: University of Illinois. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170314153801/http://www.math.uiuc.edu/~ajh/595ama/ama-ch2.pdf">Archived</a> (PDF) from the original on 14 March 2017. Retrieved 14 March 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Asymptotic+Methods+in+Analysis&rft.atitle=Asymptotic+Notations&rft.aulast=Hildebrand&rft.aufirst=A.J.&rft_id=http%3A%2F%2Fwww.math.uiuc.edu%2F~ajh%2F595ama%2Fama-ch2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <a href="#CITEREFCormenLeisersonRivestStein2009">Cormen et al. (2009)</a>, p. 64: "Many people continue to use the O-notation where the Θ-notation is more technically precise." </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormen,_Thomas_H.Leiserson,_Charles_E.Rivest,_Ronald_L.2009" class="citation book cs1">Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. (2009). Introduction to Algorithms (3rd ed.). Cambridge/MA: MIT Press. p. 47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-53305-8" title="Special:BookSources/978-0-262-53305-8"><bdi>978-0-262-53305-8</bdi></a>. <q>When we have only an asymptotic upper bound, we use O-notation. For a given function g(n), we denote by O(g(n)) (pronounced "big-oh of g of n" or sometimes just "oh of g of n") the set of functions O(g(n)) = { f(n) : there exist positive constants c and n0 such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0}</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Algorithms&rft.place=Cambridge%2FMA&rft.pages=47&rft.edition=3rd&rft.pub=MIT+Press&rft.date=2009&rft.isbn=978-0-262-53305-8&rft.au=Cormen%2C+Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormen,_Thomas_H.Leiserson,_Charles_E.Rivest,_Ronald_L.2009" class="citation book cs1">Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoal00corm_805">Introduction to Algorithms</a> (3rd ed.). Cambridge/MA: MIT Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoal00corm_805/page/n69">49</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-53305-8" title="Special:BookSources/978-0-262-53305-8"><bdi>978-0-262-53305-8</bdi></a>. <q>When the asymptotic notation stands alone (that is, not within a larger formula) on the right-hand side of an equation (or inequality), as in n = O(n2), we have already defined the equal sign to mean set membership: n ∈ O(n2). In general, however, when asymptotic notation appears in a formula, we interpret it as standing for some anonymous function that we do not care to name. For example, the formula 2n2 + 3n + 1 = 2n2 + θ(n) means that 2n2 + 3n + 1 = 2n2 + f(n), where f(n) is some function in the set θ(n). In this case, we let f(n) = 3n + 1, which is indeed in θ(n). Using asymptotic notation in this manner can help eliminate inessential detail and clutter in an equation.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Algorithms&rft.place=Cambridge%2FMA&rft.pages=49&rft.edition=3rd&rft.pub=MIT+Press&rft.date=2009&rft.isbn=978-0-262-53305-8&rft.au=Cormen%2C+Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoal00corm_805&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormenLeisersonRivestStein2022" class="citation book cs1">Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2022). <a rel="nofollow" class="external text" href="https://mitpress.mit.edu/9780262046305/introduction-to-algorithms/">Introduction to Algorithms</a> (4th ed.). Cambridge, Mass.: The MIT Press. pp. 74–75. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262046305" title="Special:BookSources/9780262046305"><bdi>9780262046305</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Algorithms&rft.place=Cambridge%2C+Mass.&rft.pages=74-75&rft.edition=4th&rft.pub=The+MIT+Press&rft.date=2022&rft.isbn=9780262046305&rft.aulast=Cormen&rft.aufirst=Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rft.au=Stein%2C+Clifford&rft_id=https%3A%2F%2Fmitpress.mit.edu%2F9780262046305%2Fintroduction-to-algorithms%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndreas_Björklund_and_Thore_Husfeldt_and_Mikko_Koivisto2009" class="citation journal cs1">Andreas Björklund and Thore Husfeldt and Mikko Koivisto (2009). <a rel="nofollow" class="external text" href="https://www.cs.helsinki.fi/u/mkhkoivi/publications/sicomp-2009.pdf">"Set partitioning via inclusion-exclusion"</a> (PDF). <a href="/wiki/SIAM_Journal_on_Computing" title="SIAM Journal on Computing">SIAM Journal on Computing</a>. 39 (2): 546–563. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F070683933">10.1137/070683933</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220203095918/https://www.cs.helsinki.fi/u/mkhkoivi/publications/sicomp-2009.pdf">Archived</a> (PDF) from the original on 2022-02-03. Retrieved 2022-02-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Computing&rft.atitle=Set+partitioning+via+inclusion-exclusion&rft.volume=39&rft.issue=2&rft.pages=546-563&rft.date=2009&rft_id=info%3Adoi%2F10.1137%2F070683933&rft.au=Andreas+Bj%C3%B6rklund+and+Thore+Husfeldt+and+Mikko+Koivisto&rft_id=https%3A%2F%2Fwww.cs.helsinki.fi%2Fu%2Fmkhkoivi%2Fpublications%2Fsicomp-2009.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> See sect.2.3, p.551. </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErdelyi1956" class="citation book cs1">Erdelyi, A. (1956). Asymptotic Expansions. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60318-6" title="Special:BookSources/978-0-486-60318-6"><bdi>978-0-486-60318-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Asymptotic+Expansions&rft.pub=Courier+Corporation&rft.date=1956&rft.isbn=978-0-486-60318-6&rft.aulast=Erdelyi&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988">. </li> <li id="cite_note-titchmarsh-37"><a href="#cite_ref-titchmarsh_37-0">^</a> E. C. Titchmarsh, The Theory of the Riemann Zeta-Function (Oxford; Clarendon Press, 1951) </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau1909" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Edmund_Landau" title="Edmund Landau">Landau, Edmund</a> (1909). <a rel="nofollow" class="external text" href="https://archive.org/details/handbuchderlehre01landuoft">Handbuch der Lehre von der Verteilung der Primzahlen</a> [Handbook on the theory of the distribution of the primes] (in German). Leipzig: B. G. Teubner. p. 62.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbuch+der+Lehre+von+der+Verteilung+der+Primzahlen&rft.place=Leipzig&rft.pages=62&rft.pub=B.+G.+Teubner&rft.date=1909&rft.aulast=Landau&rft.aufirst=Edmund&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhandbuchderlehre01landuoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-Hardy-39"><a href="#cite_ref-Hardy_39-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1910" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a> (1910). <a rel="nofollow" class="external text" href="https://archive.org/details/ordersofinfinity00harduoft">Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond</a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 2.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Orders+of+Infinity%3A+The+%27Infinit%C3%A4rcalc%C3%BCl%27+of+Paul+du+Bois-Reymond&rft.pages=2&rft.pub=Cambridge+University+Press&rft.date=1910&rft.aulast=Hardy&rft.aufirst=G.+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fordersofinfinity00harduoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyWright2008" class="citation book cs1">Hardy, G. H.; <a href="/wiki/E._M._Wright" title="E. M. Wright">Wright, E. M.</a> (2008) [1st ed. 1938]. "1.6. Some notations". An Introduction to the Theory of Numbers. Revised by <a href="/wiki/Roger_Heath-Brown" title="Roger Heath-Brown">D. R. Heath-Brown</a> and <a href="/wiki/Joseph_H._Silverman" title="Joseph H. Silverman">J. H. Silverman</a>, with a foreword by <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a> (6th ed.). Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-921985-8" title="Special:BookSources/978-0-19-921985-8"><bdi>978-0-19-921985-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.6.+Some+notations&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.place=Oxford&rft.edition=6th&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=978-0-19-921985-8&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.au=Wright%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> See for instance "A new estimate for G(n) in Waring's problem" (Russian). Doklady Akademii Nauk SSSR 5, No 5-6 (1934), 249–253. Translated in English in: Selected works / Ivan Matveevič Vinogradov; prepared by the Steklov Mathematical Institute of the Academy of Sciences of the USSR on the occasion of his 90th birthday. Springer-Verlag, 1985. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=32" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1910" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a> (1910). <a rel="nofollow" class="external text" href="https://archive.org/details/ordersofinfinity00harduoft">Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond</a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Orders+of+Infinity%3A+The+%27Infinit%C3%A4rcalc%C3%BCl%27+of+Paul+du+Bois-Reymond&rft.pub=Cambridge+University+Press&rft.date=1910&rft.aulast=Hardy&rft.aufirst=G.+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fordersofinfinity00harduoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1997" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1997). "1.2.11: Asymptotic Representations". Fundamental Algorithms. The Art of Computer Programming. Vol. 1 (3rd ed.). Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-89683-1" title="Special:BookSources/978-0-201-89683-1"><bdi>978-0-201-89683-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.2.11%3A+Asymptotic+Representations&rft.btitle=Fundamental+Algorithms&rft.series=The+Art+of+Computer+Programming&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1997&rft.isbn=978-0-201-89683-1&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormenLeisersonRivestStein2001" class="citation book cs1"><a href="/wiki/Thomas_H._Cormen" title="Thomas H. Cormen">Cormen, Thomas H.</a>; <a href="/wiki/Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson, Charles E.</a>; <a href="/wiki/Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Rivest, Ronald L.</a>; <a href="/wiki/Clifford_Stein" title="Clifford Stein">Stein, Clifford</a> (2001). "3.1: Asymptotic notation". <a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a> (2nd ed.). MIT Press and McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-03293-3" title="Special:BookSources/978-0-262-03293-3"><bdi>978-0-262-03293-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.1%3A+Asymptotic+notation&rft.btitle=Introduction+to+Algorithms&rft.edition=2nd&rft.pub=MIT+Press+and+McGraw-Hill&rft.date=2001&rft.isbn=978-0-262-03293-3&rft.aulast=Cormen&rft.aufirst=Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rft.au=Stein%2C+Clifford&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSipser1997" class="citation book cs1"><a href="/wiki/Michael_Sipser" title="Michael Sipser">Sipser, Michael</a> (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00sips_928">Introduction to the Theory of Computation</a>. PWS Publishing. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00sips_928/page/n239">226</a>–228. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-94728-6" title="Special:BookSources/978-0-534-94728-6"><bdi>978-0-534-94728-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Computation&rft.pages=226-228&rft.pub=PWS+Publishing&rft.date=1997&rft.isbn=978-0-534-94728-6&rft.aulast=Sipser&rft.aufirst=Michael&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth00sips_928&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAvigadDonnelly2004" class="citation conference cs1">Avigad, Jeremy; Donnelly, Kevin (2004). <a rel="nofollow" class="external text" href="http://www.andrew.cmu.edu/~avigad/Papers/bigo.pdf">Formalizing O notation in Isabelle/HOL</a> (PDF). International Joint Conference on Automated Reasoning. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-25984-8_27">10.1007/978-3-540-25984-8_27</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Formalizing+O+notation+in+Isabelle%2FHOL&rft.date=2004&rft_id=info%3Adoi%2F10.1007%2F978-3-540-25984-8_27&rft.aulast=Avigad&rft.aufirst=Jeremy&rft.au=Donnelly%2C+Kevin&rft_id=http%3A%2F%2Fwww.andrew.cmu.edu%2F~avigad%2FPapers%2Fbigo.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlack2005" class="citation web cs1">Black, Paul E. (11 March 2005). Black, Paul E. (ed.). <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/bigOnotation.html">"big-O notation"</a>. Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dictionary+of+Algorithms+and+Data+Structures&rft.atitle=big-O+notation&rft.date=2005-03-11&rft.aulast=Black&rft.aufirst=Paul+E.&rft_id=https%3A%2F%2Fxlinux.nist.gov%2Fdads%2FHTML%2FbigOnotation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlack2004" class="citation web cs1">Black, Paul E. (17 December 2004). Black, Paul E. (ed.). <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/littleOnotation.html">"little-o notation"</a>. Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dictionary+of+Algorithms+and+Data+Structures&rft.atitle=little-o+notation&rft.date=2004-12-17&rft.aulast=Black&rft.aufirst=Paul+E.&rft_id=https%3A%2F%2Fxlinux.nist.gov%2Fdads%2FHTML%2FlittleOnotation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlack2004" class="citation web cs1">Black, Paul E. (17 December 2004). Black, Paul E. (ed.). <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/omegaCapital.html">"Ω"</a>. Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dictionary+of+Algorithms+and+Data+Structures&rft.atitle=%CE%A9&rft.date=2004-12-17&rft.aulast=Black&rft.aufirst=Paul+E.&rft_id=https%3A%2F%2Fxlinux.nist.gov%2Fdads%2FHTML%2FomegaCapital.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlack2004" class="citation web cs1">Black, Paul E. (17 December 2004). Black, Paul E. (ed.). <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/omega.html">"ω"</a>. Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dictionary+of+Algorithms+and+Data+Structures&rft.atitle=%CF%89&rft.date=2004-12-17&rft.aulast=Black&rft.aufirst=Paul+E.&rft_id=https%3A%2F%2Fxlinux.nist.gov%2Fdads%2FHTML%2Fomega.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlack2004" class="citation web cs1">Black, Paul E. (17 December 2004). Black, Paul E. (ed.). <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/theta.html">"Θ"</a>. Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dictionary+of+Algorithms+and+Data+Structures&rft.atitle=%CE%98&rft.date=2004-12-17&rft.aulast=Black&rft.aufirst=Paul+E.&rft_id=https%3A%2F%2Fxlinux.nist.gov%2Fdads%2FHTML%2Ftheta.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABig+O+notation" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Big_O_notation&action=edit&section=33" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output 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for</a></li> <li><a rel="nofollow" class="external text" href="https://autarkaw.org/2013/01/30/making-sense-of-the-big-oh/">An example of Big O in accuracy of central divided difference scheme for first derivative</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181007223123/https://autarkaw.org/2013/01/30/making-sense-of-the-big-oh/">Archived</a> 2018-10-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://discrete.gr/complexity/">A Gentle Introduction to Algorithm Complexity Analysis</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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Big O notation - Wikipedia