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<span class="vector-toc-numb">1</span> <span>Definitie</span> </div> </a> <button aria-controls="toc-Definitie-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Definitie-subkopje inklappen</span> </button> <ul id="toc-Definitie-sublist" class="vector-toc-list"> <li id="toc-Notatie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notatie"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Notatie</span> </div> </a> <ul id="toc-Notatie-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geschiedenis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geschiedenis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Geschiedenis</span> </div> </a> <ul id="toc-Geschiedenis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voorbeelden" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voorbeelden"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Voorbeelden</span> </div> </a> <ul id="toc-Voorbeelden-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Informatica" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Informatica"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Informatica</span> </div> </a> <ul id="toc-Informatica-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="Inhoud" 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="Inhoudsopgave omschakelen" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Inhoudsopgave omschakelen</span> </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"><span class="mw-page-title-main">Grote-O-notatie</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input 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Beschikbaar in 36 talen" > <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" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">36 talen</span> </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 – Arabisch" lang="ar" hreflang="ar" data-title="تمثيل O الكبرى" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></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 – Azerbeidzjaans" lang="az" hreflang="az" data-title="Böyük O işarələr sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbeidzjaans" class="interlanguage-link-target"><span>Azərbaycanca</span></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 – Belarussisch" lang="be" hreflang="be" data-title="О-натацыя" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></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 – Bengaals" lang="bn" hreflang="bn" data-title="বড় O লিখনপদ্ধতি" data-language-autonym="বাংলা" data-language-local-name="Bengaals" class="interlanguage-link-target"><span>বাংলা</span></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 – Catalaans" lang="ca" hreflang="ca" data-title="Notació de Landau" data-language-autonym="Català" data-language-local-name="Catalaans" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Landauova_notace" title="Landauova notace – Tsjechisch" lang="cs" hreflang="cs" data-title="Landauova notace" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Landau-Symbole" title="Landau-Symbole – Duits" lang="de" hreflang="de" data-title="Landau-Symbole" data-language-autonym="Deutsch" data-language-local-name="Duits" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Big_O_notation" title="Big O notation – Engels" lang="en" hreflang="en" data-title="Big O notation" data-language-autonym="English" data-language-local-name="Engels" class="interlanguage-link-target"><span>English</span></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"><span>Esperanto</span></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 – Spaans" lang="es" hreflang="es" data-title="Cota superior asintótica" data-language-autonym="Español" data-language-local-name="Spaans" class="interlanguage-link-target"><span>Español</span></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 – Baskisch" lang="eu" hreflang="eu" data-title="Goi borne asintotiko" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></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 – Perzisch" lang="fa" hreflang="fa" data-title="نماد O بزرگ" data-language-autonym="فارسی" data-language-local-name="Perzisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Comparaison_asymptotique" title="Comparaison asymptotique – Frans" lang="fr" hreflang="fr" data-title="Comparaison asymptotique" data-language-autonym="Français" data-language-local-name="Frans" class="interlanguage-link-target"><span>Français</span></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 – Hebreeuws" lang="he" hreflang="he" data-title="סימון אסימפטוטי" data-language-autonym="עברית" data-language-local-name="Hebreeuws" class="interlanguage-link-target"><span>עברית</span></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"><span>हिन्दी</span></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 – Hongaars" lang="hu" hreflang="hu" data-title="O jelölés" data-language-autonym="Magyar" data-language-local-name="Hongaars" class="interlanguage-link-target"><span>Magyar</span></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 – Indonesisch" lang="id" hreflang="id" data-title="Notasi O besar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/O-grande" title="O-grande – Italiaans" lang="it" hreflang="it" data-title="O-grande" data-language-autonym="Italiano" data-language-local-name="Italiaans" class="interlanguage-link-target"><span>Italiano</span></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 – Japans" lang="ja" hreflang="ja" data-title="ランダウの記号" data-language-autonym="日本語" data-language-local-name="Japans" class="interlanguage-link-target"><span>日本語</span></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 – Georgisch" lang="ka" hreflang="ka" data-title="ასიმპტოტური აღნიშვნა O-დიდი" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target"><span>ქართული</span></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 – Koreaans" lang="ko" hreflang="ko" data-title="점근 표기법" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></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 – Noors - Bokmål" lang="nb" hreflang="nb" data-title="Stor O-notasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Noors - Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></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 – Pools" lang="pl" hreflang="pl" data-title="Asymptotyczne tempo wzrostu" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grande-O" title="Grande-O – Portugees" lang="pt" hreflang="pt" data-title="Grande-O" data-language-autonym="Português" data-language-local-name="Portugees" class="interlanguage-link-target"><span>Português</span></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 – Roemeens" lang="ro" hreflang="ro" data-title="Notația Big O" data-language-autonym="Română" data-language-local-name="Roemeens" class="interlanguage-link-target"><span>Română</span></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 – Russisch" lang="ru" hreflang="ru" data-title="«O» большое и «o» малое" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></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"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/O_notacija" title="O notacija – Sloveens" lang="sl" hreflang="sl" data-title="O notacija" data-language-autonym="Slovenščina" data-language-local-name="Sloveens" class="interlanguage-link-target"><span>Slovenščina</span></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 – Servisch" lang="sr" hreflang="sr" data-title="Велико О" data-language-autonym="Српски / srpski" data-language-local-name="Servisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ordo" title="Ordo – Zweeds" lang="sv" hreflang="sv" data-title="Ordo" data-language-autonym="Svenska" data-language-local-name="Zweeds" class="interlanguage-link-target"><span>Svenska</span></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"><span>ไทย</span></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 – Turks" lang="tr" hreflang="tr" data-title="Büyük O gösterimi" data-language-autonym="Türkçe" data-language-local-name="Turks" class="interlanguage-link-target"><span>Türkçe</span></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 – Oekraïens" lang="uk" hreflang="uk" data-title="Нотація Ландау" data-language-autonym="Українська" data-language-local-name="Oekraïens" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/K%C3%BD_hi%E1%BB%87u_O_l%E1%BB%9Bn" title="Ký hiệu O lớn – Vietnamees" lang="vi" hreflang="vi" data-title="Ký hiệu O lớn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamees" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%A7O%E7%AC%A6%E5%8F%B7" title="大O符号 – Chinees" lang="zh" hreflang="zh" data-title="大O符号" data-language-autonym="中文" data-language-local-name="Chinees" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%A7_O_%E7%AC%A6%E8%99%9F" title="大 O 符號 – Kantonees" lang="yue" hreflang="yue" data-title="大 O 符號" data-language-autonym="粵語" data-language-local-name="Kantonees" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q269878#sitelinks-wikipedia" 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data-event-name="pinnable-header.vector-appearance.pin">naar zijbalk verplaatsen</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">verbergen</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Uit Wikipedia, de vrije encyclopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="nl" dir="ltr"><p>In de <a href="/wiki/Wiskunde" title="Wiskunde">wiskunde</a> is de <b>grote-O-notatie</b>, ook het <b>grote-O-symbool</b>, een van de <b>Landau-symbolen</b> waarmee op compacte wijze aangegeven kan worden dat een <a href="/wiki/Functie_(wiskunde)" title="Functie (wiskunde)">functie</a> asymptotisch gedomineerd wordt door een andere functie. Meestal is de dominerende functie van een eenvoudige vorm, zodat een overzichtelijke indruk verkregen wordt van het asymptotische gedrag van de doelfunctie. </p><p>Een andere manier om het asymptotische gedrag van de ene functie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> te vergelijken met een andere functie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is het bepalen van de relevante <a href="/wiki/Limiet" title="Limiet">limiet</a> van het quotiënt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)/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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)/g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b315e81de2a653c2f6c2a03f980c0e5c3ae267b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.835ex; height:2.843ex;" alt="{\displaystyle f(x)/g(x)}"></span>. Bij de grote-O-notatie gaat het slechts om een <a href="/wiki/Ongelijkheid_(wiskunde)" title="Ongelijkheid (wiskunde)">ongelijkheid</a>, dus dat is een zwakkere eigenschap. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitie">Definitie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grote-O-notatie&veaction=edit&section=1" title="Bewerk dit kopje: Definitie" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grote-O-notatie&action=edit&section=1" title="De broncode bewerken van de sectie: Definitie"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Men zegt dat de functie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a10a3ad05781f5cf9c2d875a02227e21a8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"></span> zich voor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> gedraagt als grote O van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdfd1e16b7f932cdc2716a1b6bbe345089b250cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.023ex; height:2.509ex;" alt="{\displaystyle g:\mathbb {R} \to \mathbb {R} }"></span>, en noteert </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O(g(x))\ (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> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O(g(x))\ (x\to \infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d1909788dc8c5fd50c827400fae28007d26b66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.011ex; height:2.843ex;" alt="{\displaystyle f(x)=O(g(x))\ (x\to \infty )}"></span></dd></dl> <p>of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\in O(g(x))\ (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> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\in O(g(x))\ (x\to \infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2505489b35fcb1c9daf97f294b1949afc5de638f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.753ex; height:2.843ex;" alt="{\displaystyle f(x)\in O(g(x))\ (x\to \infty )}"></span>,</dd></dl> <p>als er een positief getal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\in \mathbb {R} ,\ M>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>M</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\in \mathbb {R} ,\ M>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d64e5f6980bbbe8984df6e20f3f4d29923aa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.279ex; height:2.509ex;" alt="{\displaystyle M\in \mathbb {R} ,\ M>0}"></span> en een getal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in \mathbb {R} }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4efa9cde595361b1ea89743b7080654c3c8614f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.903ex; height:2.509ex;" alt="{\displaystyle x_{0}\in \mathbb {R} }"></span> is, zodanig dat voor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>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>x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a572da461e16d8d92e8886a4a49fe71cecf943" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.176ex;" alt="{\displaystyle x>x_{0}}"></span> geldt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math 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> <mspace width="thinmathspace" /> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq M\,|g(x)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df70381f13623bb1d7b581dd37a9fd572ea99a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.188ex; height:2.843ex;" alt="{\displaystyle |f(x)|\leq M\,|g(x)|}"></span></dd></dl> <p>Dit houdt in dat voor <a href="/wiki/Voldoend_grote" class="mw-redirect" title="Voldoend grote">voldoend grote</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span> in <a href="/wiki/Absolute_waarde" title="Absolute waarde">absolute waarde</a> begrensd wordt door <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\,|g(x)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mspace width="thinmathspace" /> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\,|g(x)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85df71b661449b232f535725fc10729894fbf7fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.378ex; height:2.843ex;" alt="{\displaystyle M\,|g(x)|}"></span>. </p><p>Analoog wordt de notatie gebruikt om het gedrag van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in de buurt van een punt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044c60e01b54c7116ee355431f37ed846badc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} }"></span> te karakteriseren. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=O(g(x))\ (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> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=O(g(x))\ (x\to a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea39cd2deace3c6d503a6d5d596d1bc1cea2feb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.917ex; height:2.843ex;" alt="{\displaystyle f(x)=O(g(x))\ (x\to a)}"></span></dd></dl> <p>of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\in O(g(x))\ (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> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\in O(g(x))\ (x\to a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e122403d07ab94674ade3b9239fdb6ffe6e5a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.659ex; height:2.843ex;" alt="{\displaystyle f(x)\in O(g(x))\ (x\to a)}"></span>,</dd></dl> <p>als er positieve getallen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,\delta \in \mathbb {R} ,\ M,\delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>,</mo> <mi>δ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>M</mi> <mo>,</mo> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M,\delta \in \mathbb {R} ,\ M,\delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efcd94990dceb86e6630cfe8c7f178409cfeff26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.444ex; height:2.676ex;" alt="{\displaystyle M,\delta \in \mathbb {R} ,\ M,\delta >0}"></span> zijn, zodanig dat voor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-a|<\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-a|<\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d0084efcea36d9735ea90c17db93e6e5a8cb8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.841ex; height:2.843ex;" alt="{\displaystyle |x-a|<\delta }"></span> geldt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math 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> <mspace width="thinmathspace" /> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq M\,|g(x)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df70381f13623bb1d7b581dd37a9fd572ea99a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.188ex; height:2.843ex;" alt="{\displaystyle |f(x)|\leq M\,|g(x)|}"></span></dd></dl> <p>Dit houdt in dat voor waarden van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in de buurt van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span> in absolute waarde begrensd wordt door <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\,|g(x)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mspace width="thinmathspace" /> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\,|g(x)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85df71b661449b232f535725fc10729894fbf7fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.378ex; height:2.843ex;" alt="{\displaystyle M\,|g(x)|}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Notatie">Notatie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grote-O-notatie&veaction=edit&section=2" title="Bewerk dit kopje: Notatie" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grote-O-notatie&action=edit&section=2" title="De broncode bewerken van de sectie: Notatie"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De notatie met het '='-teken is historisch bepaald, maar maakt eigenlijk misbruik van dat teken. In wezen wordt in de uitspraak: "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(g(x))}"> <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> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c47693b874bd7439510aa7e9002c95a2b5ed3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.838ex; height:2.843ex;" alt="{\displaystyle O(g(x))}"></span>" het woordje 'is' als '=' geschreven. De modernere notatie met '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \in }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \in }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe4d5b0a594c1da89b5e78e7dfbeed90bdcc32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:1.843ex;" alt="{\displaystyle \in }"></span>'-teken is daarentegen wel correct, als met <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(g(x))}"> <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> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c47693b874bd7439510aa7e9002c95a2b5ed3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.838ex; height:2.843ex;" alt="{\displaystyle O(g(x))}"></span> de verzameling functies aangeduid wordt die de bedoelde eigenschap bezitten. Deze verzameling functies is een lineaire ruimte, zoals eenvoudig aangetoond kan worden. </p><p>Met <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(x)=f_{2}(x)+O(g(x))}"> <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 stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <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_{1}(x)=f_{2}(x)+O(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2984a110e030e3159e2593cfe405349efafd024a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.441ex; height:2.843ex;" alt="{\displaystyle f_{1}(x)=f_{2}(x)+O(g(x))}"></span> wordt bedoeld <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(x)-f_{2}(x)=O(g(x))}"> <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 stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <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_{1}(x)-f_{2}(x)=O(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77421d572b3dc343c830e91e372f9f4226b23f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.441ex; height:2.843ex;" alt="{\displaystyle f_{1}(x)-f_{2}(x)=O(g(x))}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Geschiedenis">Geschiedenis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grote-O-notatie&veaction=edit&section=3" title="Bewerk dit kopje: Geschiedenis" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grote-O-notatie&action=edit&section=3" title="De broncode bewerken van de sectie: Geschiedenis"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Het symbool (grote) O, dat een afkorting is van 'orde', werd voor het eerst gebruikt door de Duitse <a href="/wiki/Getaltheorie" title="Getaltheorie">getaltheoreticus</a> <a href="/wiki/Paul_Bachmann" title="Paul Bachmann">Paul Bachmann</a> in de in 1894 verschenen 2e druk van zijn boek <i>Analytische Zahlentheorie</i>. Bekendheid kreeg de notatie door de eveneens Duitse getaltheoreticus <a href="/wiki/Edmund_Landau" title="Edmund Landau">Edmund Landau</a>, wiens naam met de symboliek verbonden is. </p> <div class="mw-heading mw-heading2"><h2 id="Voorbeelden">Voorbeelden</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grote-O-notatie&veaction=edit&section=4" title="Bewerk dit kopje: Voorbeelden" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grote-O-notatie&action=edit&section=4" title="De broncode bewerken van de sectie: Voorbeelden"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Asymptotisch geldt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}+2x^{2}-4x+3\in O(x^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}+2x^{2}-4x+3\in O(x^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7dc5a458068d22dffb05a1a6f09e4dc7d078b7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.913ex; height:3.176ex;" alt="{\displaystyle x^{3}+2x^{2}-4x+3\in O(x^{3})}"></span>,</dd></dl> <p>wat ook te begrijpen is, omdat voor grote <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> de term met de hoogste macht, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2d4389a3b6f20cb8a118506601a68c2263143a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{3}}"></span>, overheerst. </p><p>Met de grote-O notatie kan de restterm beschreven worden in de benadering van een functie. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}=1+x+{\tfrac {1}{2}}x^{2}+O(x^{3})\quad (x\to 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <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>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}=1+x+{\tfrac {1}{2}}x^{2}+O(x^{3})\quad (x\to 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0544be4c1f9ae093a52bf484f2d309598a4c60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:36.614ex; height:3.509ex;" alt="{\displaystyle e^{x}=1+x+{\tfrac {1}{2}}x^{2}+O(x^{3})\quad (x\to 0)}"></span></dd></dl> <p>waarmee uitgedrukt wordt dat het verschil <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}-(1+x+{\tfrac {1}{2}}x^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <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 e^{x}-(1+x+{\tfrac {1}{2}}x^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3474a730b0942463da1833e8e40acc51140eed2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.121ex; height:3.509ex;" alt="{\displaystyle e^{x}-(1+x+{\tfrac {1}{2}}x^{2})}"></span> voor waarden van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> dicht bij 0 en zekere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> kleiner is dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |Mx^{3}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>M</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |Mx^{3}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b2996d5e4d1a6f2a3d593f9aab13dc2795d54e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.12ex; height:3.176ex;" alt="{\displaystyle |Mx^{3}|}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Informatica">Informatica</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grote-O-notatie&veaction=edit&section=5" title="Bewerk dit kopje: Informatica" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grote-O-notatie&action=edit&section=5" title="De broncode bewerken van de sectie: Informatica"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De notatie wordt in de <a href="/wiki/Informatica" title="Informatica">informatica</a> gebruikt om de (tijds)<a href="/wiki/Complexiteitsgraad" title="Complexiteitsgraad">complexiteitsgraad</a> van <a href="/wiki/Algoritme" title="Algoritme">algoritmen</a> uit te drukken in termen van een <a href="/wiki/Geheel_getal" title="Geheel getal">geheel getal</a> als <a href="/wiki/Parameter" title="Parameter">parameter</a>, waarbij het gaat om het aantal elementaire bewerkingen als functie hiervan, als de parameter naar oneindig gaat. Deze notatie is een 'slechtste-geval' maat. Ze geeft enkel informatie over het maximaal aantal elementaire bewerkingen dat een algoritme bij gegeven invoergrootte zal uitvoeren. De notatie is machine-onafhankelijk. </p><p>Algoritmen met een complexiteit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{k}),\ \ k\in \mathbb {N} }"> <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>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{k}),\ \ k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf1a71394a43f48ccbb3e2dd3acb3da1c88d078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.991ex; height:3.176ex;" alt="{\displaystyle O(n^{k}),\ \ k\in \mathbb {N} }"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> de invoergrootte, heten polynomiaal. <a href="/wiki/Algoritme" title="Algoritme">Algoritmen</a> met een complexiteit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k^{n}),\ \ k\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(k^{n}),\ \ k\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac4eb4c06b95d31c6efe5060f1844e823399811d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.937ex; height:2.843ex;" alt="{\displaystyle O(k^{n}),\ \ k\in \mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda43bd4034dc2d04cd562005d0af81d3d2dbc6" 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>1}"></span> heten exponentieel. Merk opnieuw op dat polynomiale algoritmen exponentieel zijn, maar omgekeerd niet. </p><p>Merk op dat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span> tevens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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})}"></span> impliceert. Immers als het algoritme trager of gelijk aan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> groeit, dan zeker trager dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}"></span>. Dit is algemeen geldig in de grote-O-notatie: een lagere complexiteit impliceert een hogere complexiteit. </p><p>Veel voorkomende ordes voor functies zijn, in volgorde van stijgende complexiteit: </p> <table class="wikitable"> <tbody><tr> <th>O-notatie</th> <th>Naam</th> <th>Omschrijving</th> <th>Voorbeelden </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span></td> <td>constant</td> <td>tijd onafhankelijk van de grootte van het probleem, dus is de tijd om het op te lossen nooit meer dan een bepaalde bovengrens, ook al is het probleem nog zo groot</td> <td> <ul><li>Bepalen of een binair getal even of oneven is</li> <li>(-1)<sup>n</sup> berekenen</li> <li>Een opzoektabel met constante grootte gebruiken</li></ul> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span></td> <td>logaritmisch</td> <td>evenredig aan logaritme van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> 10 maal zo groot wordt, neemt de tijd met een constante toe</td> <td> <ul><li><a href="/wiki/Bisectie" title="Bisectie">Binair zoeken</a></li></ul> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span></td> <td>lineair</td> <td>tijd evenredig aan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span></td> <td> <ul><li>Een item opzoeken in een ongeordende lijst</li></ul> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2320768fb54880ca4356e61f60eb02a3f9d9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="{\displaystyle O(n\log n)}"></span></td> <td>loglineair</td> <td>tijd evenredig aan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot log(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot log(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bb0a2f60be2493154338183ba50b7e3d9a7a2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.215ex; height:2.843ex;" alt="{\displaystyle n\cdot log(n)}"></span></td> <td> <ul><li>Beste <a href="/wiki/Sorteeralgoritme" title="Sorteeralgoritme">sorteeralgoritmen</a> die bekend zijn</li> <li><a href="/wiki/Fouriertransformatie" title="Fouriertransformatie">snelle fouriertransformatie</a></li></ul> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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})}"></span></td> <td>kwadratisch</td> <td>tijd neemt kwadratisch toe met de grootte van het probleem</td> <td> <ul><li>Twee getallen met n cijfers vermenigvuldigen zoals op school</li> <li>Simpele sorteeralgoritmen zoals <a href="/wiki/Bubblesort" title="Bubblesort">bubblesort</a></li></ul> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(2^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <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 O(2^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b1a4ff0bc4f81ebf79f28260c6fb54ee08ff8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.964ex; height:2.843ex;" alt="{\displaystyle O(2^{n})}"></span></td> <td>exponentieel</td> <td>tijd neemt exponentieel toe met de grootte van het probleem</td> <td> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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!)}"></span></td> <td>factorieel</td> <td>tijd is evenredig met n faculteit</td> <td> <ul><li>Berekenen van permutaties</li> <li><a href="/wiki/Brute_force_(methode)" title="Brute force (methode)">Brute force</a> oplossen van het <a href="/wiki/Handelsreizigersprobleem" title="Handelsreizigersprobleem">handelsreizigersprobleem</a></li></ul> </td></tr></tbody></table>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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