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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conditioning"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Conditioning</span> </div> </a> <ul id="toc-Conditioning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discretization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discretization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Discretization</span> </div> </a> <ul id="toc-Discretization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generation_and_propagation_of_errors" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generation_and_propagation_of_errors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Generation and propagation of errors</span> </div> </a> <button aria-controls="toc-Generation_and_propagation_of_errors-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>Toggle Generation and propagation of errors subsection</span> </button> <ul id="toc-Generation_and_propagation_of_errors-sublist" class="vector-toc-list"> <li id="toc-Round-off" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Round-off"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Round-off</span> </div> </a> <ul id="toc-Round-off-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Truncation_and_discretization_error" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Truncation_and_discretization_error"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Truncation and discretization error</span> </div> </a> <ul id="toc-Truncation_and_discretization_error-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_stability_and_well-posed_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerical_stability_and_well-posed_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Numerical stability and well-posed problems</span> </div> </a> <ul id="toc-Numerical_stability_and_well-posed_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Areas_of_study" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Areas_of_study"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Areas of study</span> </div> </a> <button aria-controls="toc-Areas_of_study-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>Toggle Areas of study subsection</span> </button> <ul id="toc-Areas_of_study-sublist" class="vector-toc-list"> <li id="toc-Computing_values_of_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computing_values_of_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Computing values of functions</span> </div> </a> <ul id="toc-Computing_values_of_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpolation,_extrapolation,_and_regression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpolation,_extrapolation,_and_regression"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Interpolation, extrapolation, and regression</span> </div> </a> <ul id="toc-Interpolation,_extrapolation,_and_regression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solving_equations_and_systems_of_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solving_equations_and_systems_of_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Solving equations and systems of equations</span> </div> </a> <ul id="toc-Solving_equations_and_systems_of_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solving_eigenvalue_or_singular_value_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solving_eigenvalue_or_singular_value_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Solving eigenvalue or singular value problems</span> </div> </a> <ul id="toc-Solving_eigenvalue_or_singular_value_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Optimization</span> </div> </a> <ul id="toc-Optimization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Evaluating_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Evaluating_integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Evaluating integrals</span> </div> </a> <ul id="toc-Evaluating_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Differential equations</span> </div> </a> <ul id="toc-Differential_equations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Software" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Software"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Software</span> </div> </a> <ul id="toc-Software-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"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </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"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-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>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Journals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Journals"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Journals</span> </div> </a> <ul id="toc-Journals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Online_texts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Online_texts"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Online texts</span> </div> </a> <ul id="toc-Online_texts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Online_course_material" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Online_course_material"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Online course material</span> </div> </a> <ul id="toc-Online_course_material-sublist" class="vector-toc-list"> </ul> </li> </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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</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">Numerical analysis</span></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 88 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-88" 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">88 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Numeriese_analise" title="Numeriese analise – Afrikaans" lang="af" hreflang="af" data-title="Numeriese analise" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Numerische_Mathematik" title="Numerische Mathematik – Alemannic" lang="gsw" hreflang="gsw" data-title="Numerische Mathematik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%B9%D8%AF%D8%AF%D9%8A" title="تحليل عددي – Arabic" lang="ar" hreflang="ar" data-title="تحليل عددي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Anal%C3%ADs_numerica" title="Analís numerica – Aragonese" lang="an" hreflang="an" data-title="Analís numerica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A6%BE%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="সাংখ্যিক বিশ্লেষণ – Assamese" lang="as" hreflang="as" data-title="সাংখ্যিক বিশ্লেষণ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Anal%C3%ADs_numb%C3%A9ricu" title="Analís numbéricu – Asturian" lang="ast" hreflang="ast" data-title="Analís numbéricu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fd%C9%99di_analiz" title="Ədədi analiz – Azerbaijani" lang="az" hreflang="az" data-title="Ədədi analiz" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%BE%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="সাংখ্যিক বিশ্লেষণ – Bangla" lang="bn" hreflang="bn" data-title="সাংখ্যিক বিশ্লেষণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2%CD%98-ta%CC%8Dt_k%C3%A1i-sek" title="Sò͘-ta̍t kái-sek – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-ta̍t kái-sek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%BA%D0%B0%D0%BD%D0%BB%D1%8B_%D1%8B%D1%81%D1%83%D0%BB%D0%B4%D0%B0%D1%80" title="Һанлы ысулдар – Bashkir" lang="ba" hreflang="ba" data-title="Һанлы ысулдар" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D1%96%D0%BA%D0%B0%D0%B2%D1%8B%D1%8F_%D0%BC%D0%B5%D1%82%D0%B0%D0%B4%D1%8B" title="Лікавыя метады – Belarusian" lang="be" hreflang="be" data-title="Лікавыя метады" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%96%D0%BA%D0%B0%D0%B2%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Лікавы аналіз – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лікавы аналіз" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Числен анализ – Bulgarian" lang="bg" hreflang="bg" data-title="Числен анализ" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Numeri%C4%8Dka_analiza" title="Numerička analiza – Bosnian" lang="bs" hreflang="bs" data-title="Numerička analiza" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_num%C3%A8rica" title="Anàlisi numèrica – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi numèrica" data-language-autonym="Català" data-language-local-name="Catalan" 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/Numerick%C3%A1_matematika" title="Numerická matematika – Czech" lang="cs" hreflang="cs" data-title="Numerická matematika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-tum mw-list-item"><a href="https://tum.wikipedia.org/wiki/Kusanthula_Vipendero" title="Kusanthula Vipendero – Tumbuka" lang="tum" hreflang="tum" data-title="Kusanthula Vipendero" data-language-autonym="ChiTumbuka" data-language-local-name="Tumbuka" class="interlanguage-link-target"><span>ChiTumbuka</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dadansoddiad_rhifiadol" title="Dadansoddiad rhifiadol – Welsh" lang="cy" hreflang="cy" data-title="Dadansoddiad rhifiadol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Numerisk_analyse" title="Numerisk analyse – Danish" lang="da" hreflang="da" data-title="Numerisk analyse" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Numerische_Mathematik" title="Numerische Mathematik – German" lang="de" hreflang="de" data-title="Numerische Mathematik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Arvutusmatemaatika" title="Arvutusmatemaatika – Estonian" lang="et" hreflang="et" data-title="Arvutusmatemaatika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CE%B7%CF%84%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7" title="Αριθμητική ανάλυση – Greek" lang="el" hreflang="el" data-title="Αριθμητική ανάλυση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_num%C3%A9rico" title="Análisis numérico – Spanish" lang="es" hreflang="es" data-title="Análisis numérico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Cifereca_analitiko" title="Cifereca analitiko – Esperanto" lang="eo" hreflang="eo" data-title="Cifereca analitiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbakizko_analisi" title="Zenbakizko analisi – Basque" lang="eu" hreflang="eu" data-title="Zenbakizko analisi" data-language-autonym="Euskara" data-language-local-name="Basque" 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/%D8%A2%D9%86%D8%A7%D9%84%DB%8C%D8%B2_%D8%B9%D8%AF%D8%AF%DB%8C" title="آنالیز عددی – Persian" lang="fa" hreflang="fa" data-title="آنالیز عددی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Numerical_analysis" title="Numerical analysis – Fiji Hindi" lang="hif" hreflang="hif" data-title="Numerical analysis" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_num%C3%A9rique" title="Analyse numérique – French" lang="fr" hreflang="fr" data-title="Analyse numérique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Anail%C3%ADs_uimhri%C3%BAil" title="Anailís uimhriúil – Irish" lang="ga" hreflang="ga" data-title="Anailís uimhriúil" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/An%C3%A1lise_num%C3%A9rica" title="Análise numérica – Galician" lang="gl" hreflang="gl" data-title="Análise numérica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EC%B9%98%ED%95%B4%EC%84%9D%ED%95%99" title="수치해석학 – Korean" lang="ko" hreflang="ko" data-title="수치해석학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B9%D5%BE%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%A9%D5%B8%D5%A4%D5%B6%D5%A5%D6%80" title="Թվային մեթոդներ – Armenian" lang="hy" hreflang="hy" data-title="Թվային մեթոդներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%AE%E0%A4%95_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="संख्यात्मक विश्लेषण – Hindi" lang="hi" hreflang="hi" data-title="संख्यात्मक विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Numeri%C4%8Dka_analiza" title="Numerička analiza – Croatian" lang="hr" hreflang="hr" data-title="Numerička analiza" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_numerik" title="Analisis numerik – Indonesian" lang="id" hreflang="id" data-title="Analisis numerik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/T%C3%B6luleg_greining" title="Töluleg greining – Icelandic" lang="is" hreflang="is" data-title="Töluleg greining" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Analisi_numerica" title="Analisi numerica – Italian" lang="it" hreflang="it" data-title="Analisi numerica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%A0%D7%95%D7%9E%D7%A8%D7%99%D7%AA" title="אנליזה נומרית – Hebrew" lang="he" hreflang="he" data-title="אנליזה נומרית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98%E1%83%97%E1%83%98_%E1%83%90%E1%83%9C%E1%83%90%E1%83%9A%E1%83%98%E1%83%96%E1%83%98" title="რიცხვითი ანალიზი – Georgian" lang="ka" hreflang="ka" data-title="რიცხვითი ანალიზი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%95%D1%81%D0%B5%D0%BF%D1%82%D0%B5%D1%83_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D1%81%D1%8B" title="Есептеу математикасы – Kazakh" lang="kk" hreflang="kk" data-title="Есептеу математикасы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Uchambuzi_namba" title="Uchambuzi namba – Swahili" lang="sw" hreflang="sw" data-title="Uchambuzi namba" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Analysis_numerica" title="Analysis numerica – Latin" lang="la" hreflang="la" data-title="Analysis numerica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Skait%C4%BCo%C5%A1anas_matem%C4%81tika" title="Skaitļošanas matemātika – Latvian" lang="lv" hreflang="lv" data-title="Skaitļošanas matemātika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Numeresch_Mathematik" title="Numeresch Mathematik – Luxembourgish" lang="lb" hreflang="lb" data-title="Numeresch Mathematik" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skai%C4%8Diavimo_metodai" title="Skaičiavimo metodai – Lithuanian" lang="lt" hreflang="lt" data-title="Skaičiavimo metodai" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Numerikus_anal%C3%ADzis" title="Numerikus analízis – Hungarian" lang="hu" hreflang="hu" data-title="Numerikus analízis" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%98%D1%87%D0%B5%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Бројчена анализа – Macedonian" lang="mk" hreflang="mk" data-title="Бројчена анализа" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%B5%E0%B4%BF%E0%B4%B6%E0%B4%95%E0%B4%B2%E0%B4%A8%E0%B4%82" title="സംഖ്യാവിശകലനം – Malayalam" lang="ml" hreflang="ml" data-title="സംഖ്യാവിശകലനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Analisis_berangka" title="Analisis berangka – Malay" lang="ms" hreflang="ms" data-title="Analisis berangka" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BE%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Тоон анализ – Mongolian" lang="mn" hreflang="mn" data-title="Тоон анализ" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Numerieke_wiskunde" title="Numerieke wiskunde – Dutch" lang="nl" hreflang="nl" data-title="Numerieke wiskunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E5%80%A4%E8%A7%A3%E6%9E%90" title="数値解析 – Japanese" lang="ja" hreflang="ja" data-title="数値解析" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Numeerisk_matematiik" title="Numeerisk matematiik – Northern Frisian" lang="frr" hreflang="frr" data-title="Numeerisk matematiik" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Numerisk_analyse" title="Numerisk analyse – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Numerisk analyse" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Numerisk_analyse" title="Numerisk analyse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Numerisk analyse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Analisi_numerica" title="Analisi numerica – Occitan" lang="oc" hreflang="oc" data-title="Analisi numerica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Hisoblash_matematikasi" title="Hisoblash matematikasi – Uzbek" lang="uz" hreflang="uz" data-title="Hisoblash matematikasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A9%B0%E0%A8%95%E0%A9%80_%E0%A8%B5%E0%A8%BF%E0%A8%B8%E0%A8%BC%E0%A8%B2%E0%A9%87%E0%A8%B8%E0%A8%BC%E0%A8%A3" title="ਅੰਕੀ ਵਿਸ਼ਲੇਸ਼ਣ – Punjabi" lang="pa" hreflang="pa" data-title="ਅੰਕੀ ਵਿਸ਼ਲੇਸ਼ਣ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%D9%85%D8%A8%D8%B1%DB%8C_%D8%A7%D9%86%DB%8C%D9%84%DB%8C%D8%B3%D8%B2" title="نمبری انیلیسز – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نمبری انیلیسز" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_numeryczna" title="Analiza numeryczna – Polish" lang="pl" hreflang="pl" data-title="Analiza numeryczna" data-language-autonym="Polski" data-language-local-name="Polish" 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/An%C3%A1lise_num%C3%A9rica" title="Análise numérica – Portuguese" lang="pt" hreflang="pt" data-title="Análise numérica" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Analiz%C4%83_numeric%C4%83" title="Analiză numerică – Romanian" lang="ro" hreflang="ro" data-title="Analiză numerică" data-language-autonym="Română" data-language-local-name="Romanian" 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/%D0%A7%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8B%D0%B5_%D0%BC%D0%B5%D1%82%D0%BE%D0%B4%D1%8B" title="Численные методы – Russian" lang="ru" hreflang="ru" data-title="Численные методы" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/An%C3%A0lisi_num%C3%A8rica" title="Anàlisi numèrica – Sardinian" lang="sc" hreflang="sc" data-title="Anàlisi numèrica" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Numerical_analysis" title="Numerical analysis – Scots" lang="sco" hreflang="sco" data-title="Numerical analysis" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Analiza_numerike" title="Analiza numerike – Albanian" lang="sq" hreflang="sq" data-title="Analiza numerike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B6%B8%E0%B6%BA_%E0%B7%80%E0%B7%92%E0%B7%81%E0%B7%8A%E0%B6%BD%E0%B7%9A%E0%B7%82%E0%B6%AB%E0%B6%BA" title="සංඛ්‍යාමය විශ්ලේෂණය – Sinhala" lang="si" hreflang="si" data-title="සංඛ්‍යාමය විශ්ලේෂණය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Numerical_analysis" title="Numerical analysis – Simple English" lang="en-simple" hreflang="en-simple" data-title="Numerical analysis" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Numerick%C3%A1_matematika" title="Numerická matematika – Slovak" lang="sk" hreflang="sk" data-title="Numerická matematika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Numeri%C4%8Dna_matematika" title="Numerična matematika – Slovenian" lang="sl" hreflang="sl" data-title="Numerična matematika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li 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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">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Methods for numerical approximations</div> <p class="mw-empty-elt"> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ybc7289-bw.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Ybc7289-bw.jpg/250px-Ybc7289-bw.jpg" decoding="async" width="250" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/0b/Ybc7289-bw.jpg 1.5x" data-file-width="338" data-file-height="315" /></a><figcaption>Babylonian clay tablet <a href="/wiki/YBC_7289" title="YBC 7289">YBC 7289</a> (c. 1800–1600 BCE) with annotations. The approximation of the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a> is four <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> figures, which is about six <a href="/wiki/Decimal" title="Decimal">decimal</a> figures. 1 + 24/60 + 51/60<sup>2</sup> + 10/60<sup>3</sup> = 1.41421296...<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p><b>Numerical analysis</b> is the study of <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> that use numerical <a href="/wiki/Approximation" title="Approximation">approximation</a> (as opposed to <a href="/wiki/Symbolic_computation" class="mw-redirect" title="Symbolic computation">symbolic manipulations</a>) for the problems of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> (as distinguished from <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a>). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> as found in <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a> (predicting the motions of planets, stars and galaxies), <a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a> in data analysis,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">stochastic differential equations</a> and <a href="/wiki/Markov_chain" title="Markov chain">Markov chains</a> for simulating living cells in medicine and biology. </p><p>Before modern computers, <a href="/wiki/Numerical_method" title="Numerical method">numerical methods</a> often relied on hand <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> formulas, using data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.<sup id="cite_ref-20c_5-0" class="reference"><a href="#cite_note-20c-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The numerical point of view goes back to the earliest mathematical writings. A tablet from the <a href="/wiki/Yale_Babylonian_Collection" title="Yale Babylonian Collection">Yale Babylonian Collection</a> (<a href="/wiki/YBC_7289" title="YBC 7289">YBC 7289</a>), gives a <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> numerical approximation of the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>, the length of the <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> in a <a href="/wiki/Unit_square" title="Unit square">unit square</a>. </p><p>Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=1" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to a wide variety of hard problems, many of which are infeasible to solve symbolically: </p> <ul><li>Advanced numerical methods are essential in making <a href="/wiki/Numerical_weather_prediction" title="Numerical weather prediction">numerical weather prediction</a> feasible.</li> <li>Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.</li> <li>Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> numerically.</li> <li>In the financial field, (private investment funds) and other financial institutions use <a href="/wiki/Quantitative_finance" class="mw-redirect" title="Quantitative finance">quantitative finance</a> tools from numerical analysis to attempt to calculate the value of <a href="/wiki/Share_capital" title="Share capital">stocks</a> and <a href="/wiki/Derivative_(finance)" title="Derivative (finance)">derivatives</a> more precisely than other market participants.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of <a href="/wiki/Operations_research" title="Operations research">operations research</a>.</li> <li>Insurance companies use numerical programs for <a href="/wiki/Actuary" title="Actuary">actuarial</a> analysis.</li></ul> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The field of numerical analysis predates the invention of modern computers by many centuries. <a href="/wiki/Linear_interpolation" title="Linear interpolation">Linear interpolation</a> was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis,<sup id="cite_ref-20c_5-1" class="reference"><a href="#cite_note-20c-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> as is obvious from the names of important algorithms like <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>, <a href="/wiki/Lagrange_polynomial" title="Lagrange polynomial">Lagrange interpolation polynomial</a>, <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>, or <a href="/wiki/Euler%27s_method" class="mw-redirect" title="Euler's method">Euler's method</a>. The origins of modern numerical analysis are often linked to a 1947 paper by <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> and <a href="/wiki/Herman_Goldstine" title="Herman Goldstine">Herman Goldstine</a>,<sup id="cite_ref-watson_7-0" class="reference"><a href="#cite_note-watson-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> but others consider modern numerical analysis to go back to work by <a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">E. T. Whittaker</a> in 1912.<sup id="cite_ref-watson_7-1" class="reference"><a href="#cite_note-watson-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Handbook_of_Mathematical_Functions,_by_Abramowitz_and_Stegun,_cover.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Handbook_of_Mathematical_Functions%2C_by_Abramowitz_and_Stegun%2C_cover.jpg/128px-Handbook_of_Mathematical_Functions%2C_by_Abramowitz_and_Stegun%2C_cover.jpg" decoding="async" width="128" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Handbook_of_Mathematical_Functions%2C_by_Abramowitz_and_Stegun%2C_cover.jpg/192px-Handbook_of_Mathematical_Functions%2C_by_Abramowitz_and_Stegun%2C_cover.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Handbook_of_Mathematical_Functions%2C_by_Abramowitz_and_Stegun%2C_cover.jpg/256px-Handbook_of_Mathematical_Functions%2C_by_Abramowitz_and_Stegun%2C_cover.jpg 2x" data-file-width="3184" data-file-height="4124" /></a><figcaption>NIST publication</figcaption></figure> <p>To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a> publication edited by <a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Abramowitz and Stegun</a>, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy. </p><p>The <a href="/wiki/Mechanical_calculator" title="Mechanical calculator">mechanical calculator</a> was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis,<sup id="cite_ref-20c_5-2" class="reference"><a href="#cite_note-20c-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> since now longer and more complicated calculations could be done. </p><p>The <a href="/wiki/Leslie_Fox_Prize_for_Numerical_Analysis" title="Leslie Fox Prize for Numerical Analysis">Leslie Fox Prize for Numerical Analysis</a> was initiated in 1985 by the <a href="/wiki/Institute_of_Mathematics_and_its_Applications" title="Institute of Mathematics and its Applications">Institute of Mathematics and its Applications</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Key_concepts">Key concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=3" title="Edit section: Key concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Direct_and_iterative_methods">Direct and iterative methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=4" title="Edit section: Direct and iterative methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in <a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">infinite precision arithmetic</a>. Examples include <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>, the <a href="/wiki/QR_decomposition" title="QR decomposition">QR factorization</a> method for solving <a href="/wiki/System_of_linear_equations" title="System of linear equations">systems of linear equations</a>, and the <a href="/wiki/Simplex_method" class="mw-redirect" title="Simplex method">simplex method</a> of <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a>. In practice, <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">finite precision</a> is used and the result is an approximation of the true solution (assuming <a href="/wiki/Numerically_stable" class="mw-redirect" title="Numerically stable">stability</a>). </p><p>In contrast to direct methods, <a href="/wiki/Iterative_method" title="Iterative method">iterative methods</a> are not expected to terminate in a finite number of steps, even if infinite precision were possible. Starting from an initial guess, iterative methods form successive approximations that <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">converge</a> to the exact solution only in the limit. A convergence test, often involving <a href="/wiki/Residual_(numerical_analysis)" title="Residual (numerical analysis)">the residual</a>, is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the <a href="/wiki/Bisection_method" title="Bisection method">bisection method</a>, and <a href="/wiki/Jacobi_iteration" class="mw-redirect" title="Jacobi iteration">Jacobi iteration</a>. In computational matrix algebra, iterative methods are generally needed for large problems.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. <a href="/wiki/GMRES" class="mw-redirect" title="GMRES">GMRES</a> and the <a href="/wiki/Conjugate_gradient_method" title="Conjugate gradient method">conjugate gradient method</a>. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method. </p><p>As an example, consider the problem of solving </p> <dl><dd>3<i>x</i><sup>3</sup> + 4 = 28</dd></dl> <p>for the unknown quantity <i>x</i>. </p> <table style="margin:auto; text-align:right"> <caption>Direct method </caption> <tbody><tr> <td></td> <td>3<i>x</i><sup>3</sup> + 4 = 28. </td></tr> <tr> <td><i>Subtract 4</i></td> <td>3<i>x</i><sup>3</sup> = 24. </td></tr> <tr> <td><i>Divide by 3</i></td> <td><i>x</i><sup>3</sup> =  8. </td></tr> <tr> <td><i>Take cube roots</i></td> <td><i>x</i> =  2. </td></tr></tbody></table> <p>For the iterative method, apply the <a href="/wiki/Bisection_method" title="Bisection method">bisection method</a> to <i>f</i>(<i>x</i>) = 3<i>x</i><sup>3</sup> − 24. The initial values are <i>a</i> = 0, <i>b</i> = 3, <i>f</i>(<i>a</i>) = −24, <i>f</i>(<i>b</i>) = 57. </p> <table style="margin:auto;" class="wikitable"> <caption>Iterative method </caption> <tbody><tr> <th><i>a</i></th> <th><i>b</i></th> <th>mid</th> <th><i>f</i>(mid) </th></tr> <tr> <td>0</td> <td>3</td> <td>1.5</td> <td>−13.875 </td></tr> <tr> <td>1.5</td> <td>3</td> <td>2.25</td> <td>10.17... </td></tr> <tr> <td>1.5</td> <td>2.25</td> <td>1.875</td> <td>−4.22... </td></tr> <tr> <td>1.875</td> <td>2.25</td> <td>2.0625</td> <td>2.32... </td></tr></tbody></table> <p>From this table it can be concluded that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2. </p> <div class="mw-heading mw-heading3"><h3 id="Conditioning">Conditioning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=5" title="Edit section: Conditioning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ill-conditioned problem: Take the function <span class="texhtml texhtml-big" style="font-size:100%;"><i>f</i>(<i>x</i>) = 1/(<i>x</i> − 1)</span>. Note that <i>f</i>(1.1) = 10 and <i>f</i>(1.001) = 1000: a change in <i>x</i> of less than 0.1 turns into a change in <i>f</i>(<i>x</i>) of nearly 1000. Evaluating <i>f</i>(<i>x</i>) near <i>x</i> = 1 is an ill-conditioned problem. </p><p>Well-conditioned problem: By contrast, evaluating the same function <span class="texhtml texhtml-big" style="font-size:100%;"><i>f</i>(<i>x</i>) = 1/(<i>x</i> − 1)</span> near <i>x</i> = 10 is a well-conditioned problem. For instance, <i>f</i>(10) = 1/9 ≈ 0.111 and <i>f</i>(11) = 0.1: a modest change in <i>x</i> leads to a modest change in <i>f</i>(<i>x</i>). </p> <div class="mw-heading mw-heading3"><h3 id="Discretization">Discretization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=6" title="Edit section: Discretization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called '<a href="/wiki/Discretization" title="Discretization">discretization</a>'. For example, the solution of a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a <a href="/wiki/Continuum_(set_theory)" title="Continuum (set theory)">continuum</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generation_and_propagation_of_errors">Generation and propagation of errors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=7" title="Edit section: Generation and propagation of errors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Error_propagation" class="mw-redirect" title="Error propagation">Error propagation</a></div> <p>The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem. </p> <div class="mw-heading mw-heading3"><h3 id="Round-off">Round-off</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=8" title="Edit section: Round-off"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Round-off_error" title="Round-off error">Round-off errors</a> arise because it is impossible to represent all <a href="/wiki/Real_number" title="Real number">real numbers</a> exactly on a machine with finite memory (which is what all practical <a href="/wiki/Digital_computer" class="mw-redirect" title="Digital computer">digital computers</a> are). </p> <div class="mw-heading mw-heading3"><h3 id="Truncation_and_discretization_error">Truncation and discretization error</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=9" title="Edit section: Truncation and discretization error"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Truncation_error" title="Truncation error">Truncation errors</a> are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces a <a href="/wiki/Discretization_error" title="Discretization error">discretization error</a> because the solution of the discrete problem does not coincide with the solution of the continuous problem. In the example above to compute the solution of <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 3x^{3}+4=28}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mn>28</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{3}+4=28}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb27471bb2fc45e4efb1852e3563fe0871d8c882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.973ex; height:2.843ex;" alt="{\displaystyle 3x^{3}+4=28}"></span>, after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01. </p><p>Once an error is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type <span class="nowrap">⁠<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+b+c+d+e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo>+</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b+c+d+e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/697d815a1f48e45ef5b0b569d2ef502ebb8876fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.895ex; height:2.343ex;" alt="{\displaystyle a+b+c+d+e}"></span>⁠</span> is even more inexact. </p><p>A truncation error is created when a mathematical procedure is approximated. To integrate a function exactly, an infinite sum of regions must be found, but numerically only a finite sum of regions can be found, and hence the approximation of the exact solution. Similarly, to differentiate a function, the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen. </p> <div class="mw-heading mw-heading3"><h3 id="Numerical_stability_and_well-posed_problems">Numerical stability and well-posed problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=10" title="Edit section: Numerical stability and well-posed problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An algorithm is called <i><a href="/wiki/Numerically_stable" class="mw-redirect" title="Numerically stable">numerically stable</a></i> if an error, whatever its cause, does not grow to be much larger during the calculation.<sup id="cite_ref-stab_13-0" class="reference"><a href="#cite_note-stab-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This happens if the problem is <i><a href="/wiki/Well-conditioned" class="mw-redirect" title="Well-conditioned">well-conditioned</a></i>, meaning that the solution changes by only a small amount if the problem data are changed by a small amount.<sup id="cite_ref-stab_13-1" class="reference"><a href="#cite_note-stab-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> To the contrary, if a problem is 'ill-conditioned', then any small error in the data will grow to be a large error.<sup id="cite_ref-stab_13-2" class="reference"><a href="#cite_note-stab-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Both the original problem and the algorithm used to solve that problem can be well-conditioned or ill-conditioned, and any combination is possible. So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem. </p> <div class="mw-heading mw-heading2"><h2 id="Areas_of_study">Areas of study</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=11" title="Edit section: Areas of study"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The field of numerical analysis includes many sub-disciplines. Some of the major ones are: </p> <div class="mw-heading mw-heading3"><h3 id="Computing_values_of_functions">Computing values of functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=12" title="Edit section: Computing values of functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="float: right; width: 250px; clear: right; margin-left: 1em;"> <tbody><tr> <td> <p>Interpolation: Observing that the temperature varies from 20 degrees Celsius at 1:00 to 14 degrees at 3:00, a linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm. </p><p>Extrapolation: If the <a href="/wiki/Gross_domestic_product" title="Gross domestic product">gross domestic product</a> of a country has been growing an average of 5% per year and was 100 billion last year, it might be extrapolated that it will be 105 billion this year. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Linear-regression.svg" class="mw-file-description" title="A line through 20 points"><img alt="A line through 20 points" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Linear-regression.svg/100px-Linear-regression.svg.png" decoding="async" width="100" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Linear-regression.svg/150px-Linear-regression.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Linear-regression.svg/200px-Linear-regression.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>A line through 20 points</figcaption></figure> <p>Regression: In linear regression, given <i>n</i> points, a line is computed that passes as close as possible to those <i>n</i> points. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:LemonadeJuly2006.JPG" class="mw-file-description" title="How much for a glass of lemonade?"><img alt="How much for a glass of lemonade?" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/LemonadeJuly2006.JPG/100px-LemonadeJuly2006.JPG" decoding="async" width="100" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/LemonadeJuly2006.JPG/150px-LemonadeJuly2006.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/LemonadeJuly2006.JPG/200px-LemonadeJuly2006.JPG 2x" data-file-width="332" data-file-height="356" /></a><figcaption>How much for a glass of lemonade?</figcaption></figure> <p>Optimization: Suppose lemonade is sold at a <a href="/wiki/Lemonade_stand" title="Lemonade stand">lemonade stand</a>, at $1.00 per glass, that 197 glasses of lemonade can be sold per day, and that for each increase of $0.01, one less glass of lemonade will be sold per day. If $1.485 could be charged, profit would be maximized, but due to the constraint of having to charge a whole-cent amount, charging $1.48 or $1.49 per glass will both yield the maximum income of $220.52 per day. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/File:Wind-particle.png" class="mw-file-description" title="Wind direction in blue, true trajectory in black, Euler method in red"><img alt="Wind direction in blue, true trajectory in black, Euler method in red" src="//upload.wikimedia.org/wikipedia/commons/e/ef/Wind-particle.png" decoding="async" width="94" height="101" class="mw-file-element" data-file-width="94" data-file-height="101" /></a><figcaption>Wind direction in blue, true trajectory in black, Euler method in red</figcaption></figure> <p>Differential equation: If 100 fans are set up to blow air from one end of the room to the other and then a feather is dropped into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the <a href="/wiki/Euler_method" title="Euler method">Euler method</a> for solving an ordinary differential equation. </p> </td></tr></tbody></table> <p>One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the <a href="/wiki/Horner_scheme" class="mw-redirect" title="Horner scheme">Horner scheme</a>, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control <a href="/wiki/Round-off_error" title="Round-off error">round-off errors</a> arising from the use of <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Interpolation,_extrapolation,_and_regression"><span id="Interpolation.2C_extrapolation.2C_and_regression"></span>Interpolation, extrapolation, and regression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=13" title="Edit section: Interpolation, extrapolation, and regression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Interpolation" title="Interpolation">Interpolation</a> solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points? </p><p><a href="/wiki/Extrapolation" title="Extrapolation">Extrapolation</a> is very similar to interpolation, except that now the value of the unknown function at a point which is outside the given points must be found.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a> is also similar, but it takes into account that the data are imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The <a href="/wiki/Numerical_methods_for_linear_least_squares" title="Numerical methods for linear least squares">least squares</a>-method is one way to achieve this. </p> <div class="mw-heading mw-heading3"><h3 id="Solving_equations_and_systems_of_equations">Solving equations and systems of equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=14" title="Edit section: Solving equations and systems of equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation <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 2x+5=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x+5=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c0adb58ed2acc4d9b540e5f2df6dd913d77fce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.756ex; height:2.343ex;" alt="{\displaystyle 2x+5=3}"></span> is linear while <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 2x^{2}+5=3}"> <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> <mn>5</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}+5=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f96893bb2543d1c02eb664f26e9c6cfa7b5bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.81ex; height:2.843ex;" alt="{\displaystyle 2x^{2}+5=3}"></span> is not. </p><p>Much effort has been put in the development of methods for solving <a href="/wiki/Systems_of_linear_equations" class="mw-redirect" title="Systems of linear equations">systems of linear equations</a>. Standard direct methods, i.e., methods that use some <a href="/wiki/Matrix_decomposition" title="Matrix decomposition">matrix decomposition</a> are <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>, <a href="/wiki/LU_decomposition" title="LU decomposition">LU decomposition</a>, <a href="/wiki/Cholesky_decomposition" title="Cholesky decomposition">Cholesky decomposition</a> for <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> (or <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">hermitian</a>) and <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite matrix</a>, and <a href="/wiki/QR_decomposition" title="QR decomposition">QR decomposition</a> for non-square matrices. Iterative methods such as the <a href="/wiki/Jacobi_method" title="Jacobi method">Jacobi method</a>, <a href="/wiki/Gauss%E2%80%93Seidel_method" title="Gauss–Seidel method">Gauss–Seidel method</a>, <a href="/wiki/Successive_over-relaxation" title="Successive over-relaxation">successive over-relaxation</a> and <a href="/wiki/Conjugate_gradient_method" title="Conjugate gradient method">conjugate gradient method</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> are usually preferred for large systems. General iterative methods can be developed using a <a href="/wiki/Matrix_splitting" title="Matrix splitting">matrix splitting</a>. </p><p><a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">Root-finding algorithms</a> are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is <a href="/wiki/Derivative" title="Derivative">differentiable</a> and the derivative is known, then Newton's method is a popular choice.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Linearization" title="Linearization">Linearization</a> is another technique for solving nonlinear equations. </p> <div class="mw-heading mw-heading3"><h3 id="Solving_eigenvalue_or_singular_value_problems">Solving eigenvalue or singular value problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=15" title="Edit section: Solving eigenvalue or singular value problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several important problems can be phrased in terms of <a href="/wiki/Eigenvalue_decomposition" class="mw-redirect" title="Eigenvalue decomposition">eigenvalue decompositions</a> or <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decompositions</a>. For instance, the <a href="/wiki/Image_compression" title="Image compression">spectral image compression</a> algorithm<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> is based on the singular value decomposition. The corresponding tool in statistics is called <a href="/wiki/Principal_component_analysis" title="Principal component analysis">principal component analysis</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Optimization">Optimization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=16" title="Edit section: Optimization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Mathematical optimization</a></div> <p>Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some <a href="/wiki/Constraint_(mathematics)" title="Constraint (mathematics)">constraints</a>. </p><p>The field of optimization is further split in several subfields, depending on the form of the <a href="/wiki/Objective_function" class="mw-redirect" title="Objective function">objective function</a> and the constraint. For instance, <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a> deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the <a href="/wiki/Simplex_algorithm" title="Simplex algorithm">simplex method</a>. </p><p>The method of <a href="/wiki/Lagrange_multipliers" class="mw-redirect" title="Lagrange multipliers">Lagrange multipliers</a> can be used to reduce optimization problems with constraints to unconstrained optimization problems. </p> <div class="mw-heading mw-heading3"><h3 id="Evaluating_integrals">Evaluating integrals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=17" title="Edit section: Evaluating integrals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></div> <p>Numerical integration, in some instances also known as numerical <a href="/wiki/Quadrature_(mathematics)" class="mw-redirect" title="Quadrature (mathematics)">quadrature</a>, asks for the value of a definite <a href="/wiki/Integral" title="Integral">integral</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Popular methods use one of the <a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a> (like the midpoint rule or <a href="/wiki/Simpson%27s_rule" title="Simpson's rule">Simpson's rule</a>) or <a href="/wiki/Gaussian_quadrature" title="Gaussian quadrature">Gaussian quadrature</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo</a> or <a href="/wiki/Quasi-Monte_Carlo_method" title="Quasi-Monte Carlo method">quasi-Monte Carlo methods</a> (see <a href="/wiki/Monte_Carlo_integration" title="Monte Carlo integration">Monte Carlo integration</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>), or, in modestly large dimensions, the method of <a href="/wiki/Sparse_grid" title="Sparse grid">sparse grids</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Differential_equations">Differential equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=18" title="Edit section: Differential equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Numerical_ordinary_differential_equations" class="mw-redirect" title="Numerical ordinary differential equations">Numerical ordinary differential equations</a> and <a href="/wiki/Numerical_partial_differential_equations" class="mw-redirect" title="Numerical partial differential equations">Numerical partial differential equations</a></div> <p>Numerical analysis is also concerned with computing (in an approximate way) the solution of <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>, both <a href="/wiki/Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a> and <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> This can be done by a <a href="/wiki/Finite_element_method" title="Finite element method">finite element method</a>,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> a <a href="/wiki/Finite_difference" title="Finite difference">finite difference</a> method,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> or (particularly in engineering) a <a href="/wiki/Finite_volume_method" title="Finite volume method">finite volume method</a>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The theoretical justification of these methods often involves theorems from <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. This reduces the problem to the solution of an algebraic equation. </p> <div class="mw-heading mw-heading2"><h2 id="Software">Software</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=19" title="Edit section: Software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">List of numerical-analysis software</a> and <a href="/wiki/Comparison_of_numerical-analysis_software" title="Comparison of numerical-analysis software">Comparison of numerical-analysis software</a></div> <p>Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The <a href="/wiki/Netlib" title="Netlib">Netlib</a> repository contains various collections of software routines for numerical problems, mostly in <a href="/wiki/Fortran" title="Fortran">Fortran</a> and <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a>. Commercial products implementing many different numerical algorithms include the <a href="/wiki/IMSL_Numerical_Libraries" title="IMSL Numerical Libraries">IMSL</a> and <a href="/wiki/Numerical_Algorithms_Group" class="mw-redirect" title="Numerical Algorithms Group">NAG</a> libraries; a <a href="/wiki/Free_software" title="Free software">free-software</a> alternative is the <a href="/wiki/GNU_Scientific_Library" title="GNU Scientific Library">GNU Scientific Library</a>. </p><p>Over the years the <a href="/wiki/Royal_Statistical_Society" title="Royal Statistical Society">Royal Statistical Society</a> published numerous algorithms in its <a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_C_(Applied_Statistics)" class="mw-redirect" title="Journal of the Royal Statistical Society, Series C (Applied Statistics)"><i>Applied Statistics</i></a> (code for these "AS" functions is <a rel="nofollow" class="external text" href="https://jblevins.org/mirror/amiller/#apstat">here</a>); <a href="/wiki/Association_for_Computing_Machinery" title="Association for Computing Machinery">ACM</a> similarly, in its <i><a href="/wiki/Transactions_on_Mathematical_Software" class="mw-redirect" title="Transactions on Mathematical Software">Transactions on Mathematical Software</a></i> ("TOMS" code is <a rel="nofollow" class="external text" href="https://jblevins.org/mirror/amiller/#toms">here</a>). The <a href="/wiki/Naval_Surface_Warfare_Center" title="Naval Surface Warfare Center">Naval Surface Warfare Center</a> several times published its <a rel="nofollow" class="external text" href="https://apps.dtic.mil/sti/pdfs/ADA476840.pdf"><i>Library of Mathematics Subroutines</i></a> (code <a rel="nofollow" class="external text" href="https://jblevins.org/mirror/amiller/#nswc">here</a>). </p><p>There are several popular numerical computing applications such as <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>,<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gh_30-0" class="reference"><a href="#cite_note-gh-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bf_31-0" class="reference"><a href="#cite_note-bf-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> <a href="/wiki/TK_Solver" title="TK Solver">TK Solver</a>, <a href="/wiki/S-PLUS" title="S-PLUS">S-PLUS</a>, and <a href="/wiki/IDL_(programming_language)" title="IDL (programming language)">IDL</a><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> as well as free and open-source alternatives such as <a href="/wiki/FreeMat" title="FreeMat">FreeMat</a>, <a href="/wiki/Scilab" title="Scilab">Scilab</a>,<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> <a href="/wiki/GNU_Octave" title="GNU Octave">GNU Octave</a> (similar to Matlab), and <a href="/wiki/IT%2B%2B" title="IT++">IT++</a> (a C++ library). There are also programming languages such as <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> (similar to S-PLUS), <a href="/wiki/Julia_(programming_language)" title="Julia (programming language)">Julia</a>,<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> with libraries such as <a href="/wiki/NumPy" title="NumPy">NumPy</a>, <a href="/wiki/SciPy" title="SciPy">SciPy</a><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/SymPy" title="SymPy">SymPy</a>. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>Many <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a> such as <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> also benefit from the availability of <a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">arbitrary-precision arithmetic</a> which can provide more accurate results.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p><p>Also, any <a href="/wiki/Spreadsheet" title="Spreadsheet">spreadsheet</a> <a href="/wiki/Software" title="Software">software</a> can be used to solve simple problems relating to numerical analysis. <a href="/wiki/Microsoft_Excel" title="Microsoft Excel">Excel</a>, for example, has hundreds of <a href="/wiki/Microsoft_Excel#Functions" title="Microsoft Excel">available functions</a>, including for matrices, which may be used in conjunction with its <a href="/wiki/Microsoft_Excel#Add-ins" title="Microsoft Excel">built in "solver"</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Category:Numerical_analysts" title="Category:Numerical analysts">Category:Numerical analysts</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></li> <li><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Computational_science" title="Computational science">Computational science</a></li> <li><a href="/wiki/Computational_physics" title="Computational physics">Computational physics</a></li> <li><a href="/wiki/Gordon_Bell_Prize" title="Gordon Bell Prize">Gordon Bell Prize</a></li> <li><a href="/wiki/Interval_arithmetic" title="Interval arithmetic">Interval arithmetic</a></li> <li><a href="/wiki/List_of_numerical_analysis_topics" title="List of numerical analysis topics">List of numerical analysis topics</a></li> <li><a href="/wiki/Local_linearization_method" title="Local linearization method">Local linearization method</a></li> <li><a href="/wiki/Numerical_differentiation" title="Numerical differentiation">Numerical differentiation</a></li> <li><a href="/wiki/Numerical_Recipes" title="Numerical Recipes">Numerical Recipes</a></li> <li><a href="/wiki/Probabilistic_numerics" title="Probabilistic numerics">Probabilistic numerics</a></li> <li><a href="/wiki/Symbolic-numeric_computation" title="Symbolic-numeric computation">Symbolic-numeric computation</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=21" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=23" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120813054036/http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html">"Photograph, illustration, and description of the <i>root(2)</i> tablet from the Yale Babylonian Collection"</a>. Archived from <a rel="nofollow" class="external text" href="http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html">the original</a> on 13 August 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">2 October</span> 2006</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Photograph%2C+illustration%2C+and+description+of+the+root%282%29+tablet+from+the+Yale+Babylonian+Collection&rft_id=http%3A%2F%2Fit.stlawu.edu%2F%257Edmelvill%2Fmesomath%2Ftablets%2FYBC7289.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDemmel1997" class="citation book cs1">Demmel, J.W. (1997). <a rel="nofollow" class="external text" href="https://epubs.siam.org/doi/epdf/10.1137/1.9781611971446.fm"><i>Applied numerical linear algebra</i></a>. <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611971446">10.1137/1.9781611971446</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61197-144-6" title="Special:BookSources/978-1-61197-144-6"><bdi>978-1-61197-144-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+numerical+linear+algebra&rft.pub=SIAM&rft.date=1997&rft_id=info%3Adoi%2F10.1137%2F1.9781611971446&rft.isbn=978-1-61197-144-6&rft.aulast=Demmel&rft.aufirst=J.W.&rft_id=https%3A%2F%2Fepubs.siam.org%2Fdoi%2Fepdf%2F10.1137%2F1.9781611971446.fm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCiarletMiaraThomas1989" class="citation book cs1">Ciarlet, P.G.; Miara, B.; Thomas, J.M. (1989). <i>Introduction to numerical linear algebra and optimization</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521327886" title="Special:BookSources/9780521327886"><bdi>9780521327886</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/877155729">877155729</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+numerical+linear+algebra+and+optimization&rft.pub=Cambridge+University+Press&rft.date=1989&rft_id=info%3Aoclcnum%2F877155729&rft.isbn=9780521327886&rft.aulast=Ciarlet&rft.aufirst=P.G.&rft.au=Miara%2C+B.&rft.au=Thomas%2C+J.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrefethenBau_III1997" class="citation book cs1">Trefethen, Lloyd; Bau III, David (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4Mou5YpRD_kC&pg=PR7"><i>Numerical Linear Algebra</i></a>. SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-361-9" title="Special:BookSources/978-0-89871-361-9"><bdi>978-0-89871-361-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Linear+Algebra&rft.pub=SIAM&rft.date=1997&rft.isbn=978-0-89871-361-9&rft.aulast=Trefethen&rft.aufirst=Lloyd&rft.au=Bau+III%2C+David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4Mou5YpRD_kC%26pg%3DPR7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-20c-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-20c_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-20c_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-20c_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrezinskiWuytack2012" class="citation book cs1">Brezinski, C.; Wuytack, L. 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Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-59858-5" title="Special:BookSources/978-0-444-59858-5"><bdi>978-0-444-59858-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+analysis%3A+Historical+developments+in+the+20th+century&rft.pub=Elsevier&rft.date=2012&rft.isbn=978-0-444-59858-5&rft.aulast=Brezinski&rft.aufirst=C.&rft.au=Wuytack%2C+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Ddt3Z1yu2VxwC%26pg%3DPP6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"> Stephen Blyth. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SXbcAAAAQBAJ">"An Introduction to Quantitative Finance"</a>. 2013. page VII.</span> </li> <li id="cite_note-watson-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-watson_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-watson_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWatson2010" class="citation book cs1">Watson, G.A. (2010). <a rel="nofollow" class="external text" href="https://core.ac.uk/download/pdf/206717434.pdf">"The history and development of numerical analysis in Scotland: a personal perspective"</a> <span class="cs1-format">(PDF)</span>. <i>The Birth of Numerical Analysis</i>. World Scientific. pp. 161–177. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789814469456" title="Special:BookSources/9789814469456"><bdi>9789814469456</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+history+and+development+of+numerical+analysis+in+Scotland%3A+a+personal+perspective&rft.btitle=The+Birth+of+Numerical+Analysis&rft.pages=161-177&rft.pub=World+Scientific&rft.date=2010&rft.isbn=9789814469456&rft.aulast=Watson&rft.aufirst=G.A.&rft_id=https%3A%2F%2Fcore.ac.uk%2Fdownload%2Fpdf%2F206717434.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBultheelCools2010" class="citation book cs1"><a href="/wiki/Adhemar_Bultheel" title="Adhemar Bultheel">Bultheel, Adhemar</a>; Cools, Ronald, eds. 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Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-46900-3" title="Special:BookSources/978-0-486-46900-3"><bdi>978-0-486-46900-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+solution+of+partial+differential+equations+by+the+finite+element+method&rft.pub=Courier+Corporation&rft.date=2012&rft.isbn=978-0-486-46900-3&rft.aulast=Johnson&rft.aufirst=C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0IFCAwAAQBAJ%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrennerScott2013" class="citation book cs1">Brenner, S.; Scott, R. 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Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-3658-8" title="Special:BookSources/978-1-4757-3658-8"><bdi>978-1-4757-3658-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+mathematical+theory+of+finite+element+methods&rft.edition=2nd&rft.pub=Springer&rft.date=2013&rft.isbn=978-1-4757-3658-8&rft.aulast=Brenner&rft.aufirst=S.&rft.au=Scott%2C+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DServBwAAQBAJ%26pg%3DPR11&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrangFix2018" class="citation book cs1">Strang, G.; Fix, G.J. 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Wellesley-Cambridge Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780980232783" title="Special:BookSources/9780980232783"><bdi>9780980232783</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/1145780513">1145780513</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+analysis+of+the+finite+element+method&rft.edition=2nd&rft.pub=Wellesley-Cambridge+Press&rft.date=2018&rft_id=info%3Aoclcnum%2F1145780513&rft.isbn=9780980232783&rft.aulast=Strang&rft.aufirst=G.&rft.au=Fix%2C+G.J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fanalysisoffinite0000stra&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrikwerda2004" class="citation book cs1">Strikwerda, J.C. 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Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-43418-8" title="Special:BookSources/978-1-139-43418-8"><bdi>978-1-139-43418-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Volume+Methods+for+Hyperbolic+Problems&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-1-139-43418-8&rft.aulast=LeVeque&rft.aufirst=Randall&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmfAfAwAAQBAJ%26pg%3DPT6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuarteroniSaleriGervasio2014" class="citation book cs1">Quarteroni, A.; Saleri, F.; Gervasio, P. 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(2001). SciPy: Open source scientific tools for Python.</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBressert2012" class="citation book cs1">Bressert, E. (2012). <i>SciPy and NumPy: an overview for developers</i>. O'Reilly. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781306810395" title="Special:BookSources/9781306810395"><bdi>9781306810395</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=SciPy+and+NumPy%3A+an+overview+for+developers&rft.pub=O%27Reilly&rft.date=2012&rft.isbn=9781306810395&rft.aulast=Bressert&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlanco-Silva2013" class="citation book cs1">Blanco-Silva, F.J. (2013). <i>Learning SciPy for numerical and scientific computing</i>. Packt. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781782161639" title="Special:BookSources/9781782161639"><bdi>9781782161639</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Learning+SciPy+for+numerical+and+scientific+computing&rft.pub=Packt&rft.date=2013&rft.isbn=9781782161639&rft.aulast=Blanco-Silva&rft.aufirst=F.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.sciviews.org/benchmark/">Speed comparison of various number crunching packages</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061005024002/http://www.sciviews.org/benchmark/">Archived</a> 5 October 2006 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.scientificweb.com/ncrunch/ncrunch5.pdf">Comparison of mathematical programs for data analysis</a> <a rel="nofollow" class="external text" href="http://arquivo.pt/wayback/20160518062220/http://www.scientificweb.com/ncrunch/ncrunch5.pdf">Archived</a> 18 May 2016 at the Portuguese Web Archive Stefan Steinhaus, ScientificWeb.com</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaeder1997" class="citation book cs1">Maeder, R.E. (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/programminginmat0000maed_l2m6"><i>Programming in mathematica</i></a> (3rd ed.). Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780201854497" title="Special:BookSources/9780201854497"><bdi>9780201854497</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/1311056676">1311056676</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Programming+in+mathematica&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1997&rft_id=info%3Aoclcnum%2F1311056676&rft.isbn=9780201854497&rft.aulast=Maeder&rft.aufirst=R.E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprogramminginmat0000maed_l2m6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfram1999" class="citation book cs1">Wolfram, Stephen (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Xny77v_QPkEC&pg=PR19"><i>The MATHEMATICA® book, version 4</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781579550042" title="Special:BookSources/9781579550042"><bdi>9781579550042</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+MATHEMATICA%C2%AE+book%2C+version+4&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=9781579550042&rft.aulast=Wolfram&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXny77v_QPkEC%26pg%3DPR19&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShawTigg1993" class="citation book cs1">Shaw, W.T.; Tigg, J. (1993). <a rel="nofollow" class="external text" href="http://www.gbv.de/dms/bowker/toc/9780201542172.pdf"><i>Applied Mathematica: getting started, getting it done</i></a> <span class="cs1-format">(PDF)</span>. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-54217-2" title="Special:BookSources/978-0-201-54217-2"><bdi>978-0-201-54217-2</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/28149048">28149048</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Mathematica%3A+getting+started%2C+getting+it+done&rft.pub=Addison-Wesley&rft.date=1993&rft_id=info%3Aoclcnum%2F28149048&rft.isbn=978-0-201-54217-2&rft.aulast=Shaw&rft.aufirst=W.T.&rft.au=Tigg%2C+J.&rft_id=http%3A%2F%2Fwww.gbv.de%2Fdms%2Fbowker%2Ftoc%2F9780201542172.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarascoRomano2001" class="citation book cs1">Marasco, A.; Romano, A. (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iFRqemnmMqUC&pg=PR7"><i>Scientific Computing with Mathematica: Mathematical Problems for Ordinary Differential Equations</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4205-1" title="Special:BookSources/978-0-8176-4205-1"><bdi>978-0-8176-4205-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scientific+Computing+with+Mathematica%3A+Mathematical+Problems+for+Ordinary+Differential+Equations&rft.pub=Springer&rft.date=2001&rft.isbn=978-0-8176-4205-1&rft.aulast=Marasco&rft.aufirst=A.&rft.au=Romano%2C+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiFRqemnmMqUC%26pg%3DPR7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=24" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolub,_Gene_H.Charles_F._Van_Loan1986" class="citation book cs1"><a href="/wiki/Gene_H._Golub" title="Gene H. Golub">Golub, Gene H.</a>; <a href="/wiki/Charles_F._Van_Loan" title="Charles F. Van Loan">Charles F. Van Loan</a> (1986). <i>Matrix Computations</i> (3rd ed.). Johns Hopkins University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8018-5413-X" title="Special:BookSources/0-8018-5413-X"><bdi>0-8018-5413-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Computations&rft.edition=3rd&rft.pub=Johns+Hopkins+University+Press&rft.date=1986&rft.isbn=0-8018-5413-X&rft.au=Golub%2C+Gene+H.&rft.au=Charles+F.+Van+Loan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRalston_AnthonyRabinowitz_Philips2001" class="citation book cs1">Ralston Anthony; Rabinowitz Philips (2001). <i>A First Course in Numerical Analysis</i> (2nd ed.). Dover publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0486414546" title="Special:BookSources/978-0486414546"><bdi>978-0486414546</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Numerical+Analysis&rft.edition=2nd&rft.pub=Dover+publications&rft.date=2001&rft.isbn=978-0486414546&rft.au=Ralston+Anthony&rft.au=Rabinowitz+Philips&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHigham2002" class="citation book cs1"><a href="/wiki/Nicholas_Higham" title="Nicholas Higham">Higham, Nicholas J.</a> (2002) [1996]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/accuracystabilit0000high"><i>Accuracy and Stability of Numerical Algorithms</i></a></span>. Society for Industrial and Applied Mathematics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-355-2" title="Special:BookSources/0-89871-355-2"><bdi>0-89871-355-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Accuracy+and+Stability+of+Numerical+Algorithms&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=2002&rft.isbn=0-89871-355-2&rft.aulast=Higham&rft.aufirst=Nicholas+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Faccuracystabilit0000high&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHildebrand1974" class="citation book cs1"><a href="/wiki/Francis_B._Hildebrand" title="Francis B. Hildebrand">Hildebrand, F. B.</a> (1974). <i>Introduction to Numerical Analysis</i> (2nd ed.). McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-028761-9" title="Special:BookSources/0-07-028761-9"><bdi>0-07-028761-9</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+Numerical+Analysis&rft.edition=2nd&rft.pub=McGraw-Hill&rft.date=1974&rft.isbn=0-07-028761-9&rft.aulast=Hildebrand&rft.aufirst=F.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></li> <li>David Kincaid and Ward Cheney: <i>Numerical Analysis : Mathematics of Scientific Computing</i>, 3rd Ed., AMS, ISBN 978-0-8218-4788-6 (2002).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeader2004" class="citation book cs1"><a href="/wiki/Jeffery_J._Leader" title="Jeffery J. Leader">Leader, Jeffery J.</a> (2004). <i>Numerical Analysis and Scientific Computation</i>. Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-73499-0" title="Special:BookSources/0-201-73499-0"><bdi>0-201-73499-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Analysis+and+Scientific+Computation&rft.pub=Addison+Wesley&rft.date=2004&rft.isbn=0-201-73499-0&rft.aulast=Leader&rft.aufirst=Jeffery+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilkinson1988" class="citation book cs1"><a href="/wiki/James_H._Wilkinson" title="James H. Wilkinson">Wilkinson, J.H.</a> (1988) [1965]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/algebraiceigenva0000wilk"><i>The Algebraic Eigenvalue Problem</i></a></span>. Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853418-1" title="Special:BookSources/978-0-19-853418-1"><bdi>978-0-19-853418-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Algebraic+Eigenvalue+Problem&rft.pub=Clarendon+Press&rft.date=1988&rft.isbn=978-0-19-853418-1&rft.aulast=Wilkinson&rft.aufirst=J.H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebraiceigenva0000wilk&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKahan1972" class="citation conference cs1"><a href="/wiki/William_Kahan" title="William Kahan">Kahan, W.</a> (1972). <i>A survey of error-analysis</i>. Proc. IFIP Congress 71 in Ljubljana. <i>Info. Processing 71</i>. Vol. 2. North-Holland. pp. 1214–39. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7204-2063-0" title="Special:BookSources/978-0-7204-2063-0"><bdi>978-0-7204-2063-0</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/25116949">25116949</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=conference&rft.jtitle=Info.+Processing+71&rft.atitle=A+survey+of+error-analysis&rft.volume=2&rft.pages=1214-39&rft.date=1972&rft_id=info%3Aoclcnum%2F25116949&rft.isbn=978-0-7204-2063-0&rft.aulast=Kahan&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span> (examples of the importance of accurate arithmetic).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrefethen2008" class="citation book cs1"><a href="/wiki/Lloyd_N._Trefethen" class="mw-redirect" title="Lloyd N. Trefethen">Trefethen, Lloyd N.</a> (2008). <a rel="nofollow" class="external text" href="http://people.maths.ox.ac.uk/trefethen/NAessay.pdf">"IV.21 Numerical analysis"</a> <span class="cs1-format">(PDF)</span>. In Leader, I.; Gowers, T.; Barrow-Green, J. (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GLumDwAAQBAJ&pg=PR5"><i>Princeton Companion of Mathematics</i></a>. Princeton University Press. pp. 604–614. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11880-2" title="Special:BookSources/978-0-691-11880-2"><bdi>978-0-691-11880-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=IV.21+Numerical+analysis&rft.btitle=Princeton+Companion+of+Mathematics&rft.pages=604-614&rft.pub=Princeton+University+Press&rft.date=2008&rft.isbn=978-0-691-11880-2&rft.aulast=Trefethen&rft.aufirst=Lloyd+N.&rft_id=http%3A%2F%2Fpeople.maths.ox.ac.uk%2Ftrefethen%2FNAessay.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=25" title="Edit section: 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data-file-height="355" /></span></span></span><span class="sister-link"><a href="https://en.wikiquote.org/wiki/Numerical_analysis" class="extiw" title="q:Numerical analysis">Quotations</a> from Wikiquote</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/41px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/54px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span></span><span class="sister-link"><a href="https://en.wikibooks.org/wiki/Numerical_Methods" class="extiw" title="b:Numerical Methods">Textbooks</a> from Wikibooks</span></li></ul></div></div> </div> <div class="mw-heading mw-heading3"><h3 id="Journals">Journals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=26" title="Edit section: Journals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i><a href="/wiki/Numerische_Mathematik" title="Numerische Mathematik">Numerische Mathematik</a></i>, volumes 1–..., <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/numerical+and+computational+mathematics/journal/211">Springer</a>, 1959– <ul><li><a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN362160546">volumes 1–66, 1959–1994</a> (searchable; pages are images). <span class="languageicon">(in English and German)</span></li></ul></li> <li><i><a href="/w/index.php?title=Journal_on_Numerical_Analysis&action=edit&redlink=1" class="new" title="Journal on Numerical Analysis (page does not exist)">Journal on Numerical Analysis</a></i> <a rel="nofollow" class="external text" href="https://epubs.siam.org/journal/sjnaam">(SINUM)</a>, volumes 1–..., SIAM, 1964–</li></ul> <div class="mw-heading mw-heading3"><h3 id="Online_texts">Online texts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=27" title="Edit section: Online texts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Numerical_analysis">"Numerical analysis"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Numerical+analysis&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DNumerical_analysis&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumerical+analysis" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150905141405/http://www.nr.com/oldverswitcher.html"><i>Numerical Recipes</i></a>, William H. Press (free, downloadable previous editions)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120225082123/http://kr.cs.ait.ac.th/~radok/math/mat7/stepsa.htm"><i>First Steps in Numerical Analysis</i></a> (<a href="/wiki/Internet_Archive" title="Internet Archive">archived</a>), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170801213333/http://www.phy.ornl.gov/csep/CSEP/TEXTOC.html"><i>CSEP</i> (Computational Science Education Project)</a>, <a href="/wiki/U.S._Department_of_Energy" class="mw-redirect" title="U.S. Department of Energy">U.S. Department of Energy</a> (<a href="/wiki/Internet_Archive" title="Internet Archive">archived 2017-08-01</a>)</li> <li><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/3">Numerical Methods</a>, ch 3. in the <i><a href="/wiki/Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">Digital Library of Mathematical Functions</a></i></li> <li><a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_875.htm">Numerical Interpolation, Differentiation and Integration</a>, ch 25. in the <i>Handbook of Mathematical Functions</i> (<a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Abramowitz and Stegun</a>)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Online_course_material">Online course material</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_analysis&action=edit&section=28" title="Edit section: Online course material"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/user/fdl/people/sd103/lectures/nummeth98/index.htm#L_1_Title_Page">Numerical Methods</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20090728181209/http://www.damtp.cam.ac.uk/user/fdl/people/sd103/lectures/nummeth98/index.htm#L_1_Title_Page">Archived</a> 28 July 2009 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>), Stuart Dalziel <a href="/wiki/University_of_Cambridge" title="University of Cambridge">University of Cambridge</a></li> <li><a rel="nofollow" class="external text" href="http://www.math.upenn.edu/~wilf/DeturckWilf.pdf">Lectures on Numerical Analysis</a>, Dennis Deturck and Herbert S. Wilf <a href="/wiki/University_of_Pennsylvania" title="University of Pennsylvania">University of Pennsylvania</a></li> <li><a rel="nofollow" class="external text" href="http://johndfenton.com/Lectures/Numerical-Methods/Numerical-Methods.pdf">Numerical methods</a>, John D. Fenton <a href="/wiki/University_of_Karlsruhe" class="mw-redirect" title="University of Karlsruhe">University of Karlsruhe</a></li> <li><a rel="nofollow" class="external text" href="http://www-teaching.physics.ox.ac.uk/computing/NumericalMethods/NMfP.pdf">Numerical Methods for Physicists</a>, Anthony O’Hare <a href="/wiki/Oxford_University" class="mw-redirect" title="Oxford University">Oxford University</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120225082123/http://kr.cs.ait.ac.th/~radok/math/mat7/stepsa.htm">Lectures in Numerical Analysis</a> (<a href="/wiki/Internet_Archive" title="Internet Archive">archived</a>), R. Radok <a href="/wiki/Mahidol_University" title="Mahidol University">Mahidol University</a></li> <li><a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/mechanical-engineering/2-993j-introduction-to-numerical-analysis-for-engineering-13-002j-spring-2005/">Introduction to Numerical Analysis for Engineering</a>, Henrik Schmidt <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a></li> <li><a rel="nofollow" class="external text" href="http://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/"><i>Numerical Analysis for Engineering</i></a>, D. W. Harder <a href="/wiki/University_of_Waterloo" title="University of Waterloo">University of Waterloo</a></li> <li><a rel="nofollow" class="external text" href="https://www.math.umd.edu/~diom/courses/AMSC466/Levy-notes.pdf">Introduction to Numerical Analysis</a>, Doron Levy <a href="/wiki/University_of_Maryland" class="mw-redirect" title="University of Maryland">University of Maryland</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070310212643/http://math.fullerton.edu/mathews/n2003/NumericalUndergradMod.html">Numerical Analysis - Numerical Methods</a> (archived), John H. 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href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a 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class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_branches_of_physics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Branches_of_physics" title="Template:Branches of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Branches_of_physics" title="Template talk:Branches of physics"><abbr title="Discuss this 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mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li></ul></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Classical electromagnetism</a></li> <li><a href="/wiki/Classical_optics" class="mw-redirect" title="Classical optics">Classical optics</a> <ul><li><a href="/wiki/Geometrical_optics" title="Geometrical optics">Ray</a></li> <li><a href="/wiki/Physical_optics" title="Physical optics">Wave</a></li></ul></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Modern_physics" 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physics">Molecular</a></li> <li><a href="/wiki/Optics#Modern_optics" title="Optics">Modern optics</a></li></ul></li> <li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a> <ul><li><a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li> <li><a href="/wiki/Crystallography" title="Crystallography">Crystallography</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Applied_and_interdisciplinary_physics" title="Category:Applied and interdisciplinary physics">Interdisciplinary</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Atmospheric_physics" title="Atmospheric physics">Atmospheric physics</a></li> <li><a href="/wiki/Biophysics" title="Biophysics">Biophysics</a></li> <li><a href="/wiki/Chemical_physics" title="Chemical physics">Chemical physics</a></li> <li><a href="/wiki/Geophysics" title="Geophysics">Geophysics</a></li> <li><a href="/wiki/Materials_science" title="Materials science">Materials science</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Medical_physics" title="Medical physics">Medical physics</a></li> <li><a href="/wiki/Physical_oceanography" title="Physical oceanography">Ocean physics</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_physics" title="History of physics">History of physics</a></li> <li><a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a></li> <li><a href="/wiki/Philosophy_of_physics" title="Philosophy of physics">Philosophy of physics</a></li> <li><a href="/wiki/Physics_education" title="Physics education">Physics education</a></li> <li><a href="/wiki/Timeline_of_fundamental_physics_discoveries" title="Timeline of fundamental physics discoveries">Timeline of physics discoveries</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Computer_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Computer_science" title="Template:Computer science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Computer_science" title="Template talk:Computer science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Computer_science" title="Special:EditPage/Template:Computer science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Computer_science" style="font-size:114%;margin:0 4em"><a href="/wiki/Computer_science" title="Computer science">Computer science</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Note: This template roughly follows the 2012 <a href="/wiki/ACM_Computing_Classification_System" title="ACM Computing Classification System">ACM Computing Classification System</a>.</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_hardware" title="Computer hardware">Hardware</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Printed_circuit_board" title="Printed circuit board">Printed circuit board</a></li> <li><a href="/wiki/Peripheral" title="Peripheral">Peripheral</a></li> <li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a></li> <li><a href="/wiki/Very_Large_Scale_Integration" class="mw-redirect" title="Very Large Scale Integration">Very Large Scale Integration</a></li> <li><a href="/wiki/System_on_a_chip" title="System on a chip">Systems on Chip (SoCs)</a></li> <li><a href="/wiki/Green_computing" title="Green computing">Energy consumption (Green computing)</a></li> <li><a href="/wiki/Electronic_design_automation" title="Electronic design automation">Electronic design automation</a></li> <li><a href="/wiki/Hardware_acceleration" title="Hardware acceleration">Hardware acceleration</a></li> <li><a href="/wiki/Processor_(computing)" title="Processor (computing)">Processor</a></li> <li><a href="/wiki/List_of_computer_size_categories" title="List of computer size categories">Size</a> / <a href="/wiki/Form_factor_(design)" title="Form factor (design)">Form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Computer systems organization</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_architecture" title="Computer architecture">Computer architecture</a></li> <li><a href="/wiki/Computational_complexity" title="Computational complexity">Computational complexity</a></li> <li><a href="/wiki/Dependability" title="Dependability">Dependability</a></li> <li><a href="/wiki/Embedded_system" title="Embedded system">Embedded system</a></li> <li><a href="/wiki/Real-time_computing" title="Real-time computing">Real-time computing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_network" title="Computer network">Networks</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Network_architecture" title="Network architecture">Network architecture</a></li> <li><a href="/wiki/Network_protocol" class="mw-redirect" title="Network protocol">Network protocol</a></li> <li><a href="/wiki/Networking_hardware" title="Networking hardware">Network components</a></li> <li><a href="/wiki/Network_scheduler" title="Network scheduler">Network scheduler</a></li> <li><a href="/wiki/Network_performance" title="Network performance">Network performance evaluation</a></li> <li><a href="/wiki/Network_service" title="Network service">Network service</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software organization</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpreter_(computing)" title="Interpreter (computing)">Interpreter</a></li> <li><a href="/wiki/Middleware" title="Middleware">Middleware</a></li> <li><a href="/wiki/Virtual_machine" title="Virtual machine">Virtual machine</a></li> <li><a href="/wiki/Operating_system" title="Operating system">Operating system</a></li> <li><a href="/wiki/Software_quality" title="Software quality">Software quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Programming_language_theory" title="Programming language theory">Software notations</a> and <a href="/wiki/Programming_tool" title="Programming tool">tools</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Programming_paradigm" title="Programming paradigm">Programming paradigm</a></li> <li><a href="/wiki/Programming_language" title="Programming language">Programming language</a></li> <li><a href="/wiki/Compiler_construction" class="mw-redirect" title="Compiler construction">Compiler</a></li> <li><a href="/wiki/Domain-specific_language" title="Domain-specific language">Domain-specific language</a></li> <li><a href="/wiki/Modeling_language" title="Modeling language">Modeling language</a></li> <li><a href="/wiki/Software_framework" title="Software framework">Software framework</a></li> <li><a href="/wiki/Integrated_development_environment" title="Integrated development environment">Integrated development environment</a></li> <li><a href="/wiki/Software_configuration_management" title="Software configuration 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