Formule de Newton-Cotes

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ordre_de_la_méthode"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ordre de la méthode</span> </div> </a> <ul id="toc-Ordre_de_la_méthode-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convergence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Convergence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Convergence</span> </div> </a> <ul id="toc-Convergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liens_externes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Liens_externes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Liens externes</span> </div> </a> <ul id="toc-Liens_externes-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sommaire" 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="Basculer la table des matières" > <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">Basculer la table des matières</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">Formule de Newton-Cotes</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="Aller à un article dans une autre langue. Disponible en 18 langues." > <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-18" 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">18 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B5%D9%8A%D8%BA_%D9%86%D9%8A%D9%88%D8%AA%D9%86-%D9%83%D9%88%D8%AA%D8%B3" title="صيغ نيوتن-كوتس – arabe" lang="ar" hreflang="ar" data-title="صيغ نيوتن-كوتس" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/F%C3%B3rmules_de_Newton-Cotes" title="Fórmules de Newton-Cotes – catalan" lang="ca" hreflang="ca" data-title="Fórmules de Newton-Cotes" 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/Newtonovy%E2%80%93Cotesovy_vzorce" title="Newtonovy–Cotesovy vzorce – tchèque" lang="cs" hreflang="cs" data-title="Newtonovy–Cotesovy vzorce" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Newton-Cotes-Formeln" title="Newton-Cotes-Formeln – allemand" lang="de" hreflang="de" data-title="Newton-Cotes-Formeln" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas – anglais" lang="en" hreflang="en" data-title="Newton–Cotes formulas" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/F%C3%B3rmulas_de_Newton-Cotes" title="Fórmulas de Newton-Cotes – espagnol" lang="es" hreflang="es" data-title="Fórmulas de Newton-Cotes" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%95%D7%A1%D7%97%D7%90%D7%95%D7%AA_%D7%A0%D7%99%D7%95%D7%98%D7%95%D7%9F-%D7%A7%D7%95%D7%98%D7%A1" title="נוסחאות ניוטון-קוטס – hébreu" lang="he" hreflang="he" data-title="נוסחאות ניוטון-קוטס" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Newton%E2%80%93Cotes-formula" title="Newton–Cotes-formula – hongrois" lang="hu" hreflang="hu" data-title="Newton–Cotes-formula" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Formule_di_Newton-Cotes" title="Formule di Newton-Cotes – italien" lang="it" hreflang="it" data-title="Formule di Newton-Cotes" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E3%83%BB%E3%82%B3%E3%83%BC%E3%83%84%E3%81%AE%E5%85%AC%E5%BC%8F" title="ニュートン・コーツの公式 – japonais" lang="ja" hreflang="ja" data-title="ニュートン・コーツの公式" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9A%E1%9E%BC%E1%9E%94%E1%9E%98%E1%9E%93%E1%9F%92%E1%9E%8F%E1%9E%89%E1%9E%BC%E1%9E%8F%E1%9E%BB%E1%9E%93-%E1%9E%80%E1%9E%BC%E1%9E%8F%E1%9F%92%E1%9E%9F" title="រូបមន្តញូតុន-កូត្ស – khmer" lang="km" hreflang="km" data-title="រូបមន្តញូតុន-កូត្ស" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%89%B4%ED%84%B4-%EC%BD%94%EC%B8%A0_%EA%B3%B5%EC%8B%9D" title="뉴턴-코츠 공식 – coréen" lang="ko" hreflang="ko" data-title="뉴턴-코츠 공식" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Formule_van_Newton-Cotes" title="Formule van Newton-Cotes – néerlandais" lang="nl" hreflang="nl" data-title="Formule van Newton-Cotes" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Metody_Newtona-Cotesa" title="Metody Newtona-Cotesa – polonais" lang="pl" hreflang="pl" data-title="Metody Newtona-Cotesa" data-language-autonym="Polski" data-language-local-name="polonais" 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/F%C3%B3rmulas_de_Newton-Cotes" title="Fórmulas de Newton-Cotes – portugais" lang="pt" hreflang="pt" data-title="Fórmulas de Newton-Cotes" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D1%8B_%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B0_%E2%80%94_%D0%9A%D0%BE%D1%82%D1%81%D0%B0" title="Формулы Ньютона — Котса – russe" lang="ru" hreflang="ru" data-title="Формулы Ньютона — Котса" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%8A%D1%83%D1%82%D0%BD-%D0%9A%D0%BE%D1%83%D1%82%D1%81_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B5" title="Њутн-Коутс формуле – serbe" 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Simpson_rule.png/330px-Simpson_rule.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Simpson_rule.png/440px-Simpson_rule.png 2x" data-file-width="3052" data-file-height="1989" /></a><figcaption>La courbe noire est la courbe représentative de la fonction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. La surface orange représente une approximation <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 \int _{a}^{b}f(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cac1fab0e1353f0e514fe66f83ff8c0fb3419fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.752ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)dx}"></span> à l'aide d'une interpolation polynomiale aux points répartis uniformément <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a+b}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+b}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1325e0aa44cdaf4b2e765a44c7109e6b9ed74e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.904ex; height:5.343ex;" alt="{\displaystyle {\frac {a+b}{2}}}"></span>et <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> (<a href="/wiki/M%C3%A9thode_de_Simpson" title="Méthode de Simpson">Méthode de Simpson</a>). Il s'agit d'un cas particulier de la formule de Newton-Cotes.</figcaption></figure> <p>En <a href="/wiki/Analyse_num%C3%A9rique" title="Analyse numérique">analyse numérique</a>, les <b>formules de Newton-Cotes</b>, du nom d'<a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> et de <a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a>, servent au <a href="/wiki/Calcul_num%C3%A9rique_d%27une_int%C3%A9grale" title="Calcul numérique d'une intégrale">calcul numérique d'une intégrale</a> sur un intervalle réel <span class="texhtml">[<i>a</i>, <i>b</i>]</span>, ceci à l’aide d’une <a href="/wiki/Interpolation_polynomiale" title="Interpolation polynomiale">interpolation polynomiale</a> de la fonction en des points répartis uniformément. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Méthodologie"><span id="M.C3.A9thodologie"></span>Méthodologie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_de_Newton-Cotes&veaction=edit&section=1" title="Modifier la section : Méthodologie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_de_Newton-Cotes&action=edit&section=1" title="Modifier le code source de la section : Méthodologie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La fonction <span class="texhtml"><i>f</i></span> est évaluée en des points équidistants <span class="texhtml"><i>x<sub>i</sub> = a + i</i>Δ</span>, pour <span class="texhtml"><i>i </i>= 0, … , <i>n</i></span> et <span class="texhtml">Δ = (<i>b – a</i>)/<i>n</i></span>. La formule de degré <span class="texhtml"><i>n</i></span> est définie ainsi : </p> <center><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 \int _{a}^{b}f(x)~{\rm {d}}x\approx \sum _{i=0}^{n}w_{i}\,f(x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>≈</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)~{\rm {d}}x\approx \sum _{i=0}^{n}w_{i}\,f(x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb9cafdb3e63be6fccfebc78649ad587bb560df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.318ex; height:7.009ex;" alt="{\displaystyle \int _{a}^{b}f(x)~{\rm {d}}x\approx \sum _{i=0}^{n}w_{i}\,f(x_{i})}"></span></center> <p>où les <span class="texhtml"><i>w<sub>i</sub></i></span> sont appelés les <i>coefficients de <a href="/wiki/Quadrature_(math%C3%A9matiques)" title="Quadrature (mathématiques)">quadrature</a></i>. Ils se déduisent d'une <a href="/wiki/Interpolation_lagrangienne" title="Interpolation lagrangienne">base de polynômes de Lagrange</a> et sont indépendants de la fonction <span class="texhtml"><i>f</i></span>. </p><p>Plus précisément, si <span class="texhtml"><i>L</i>(<i>x</i>)</span> est l'interpolation lagrangienne aux points <span class="texhtml">(<i>x<sub>i</sub></i>, <i>f</i>(<i>x<sub>i</sub></i>))</span> et <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 l_{i}(X)=\prod _{j=0,j\neq i}^{n}{\frac {X-x_{j}}{x_{i}-x_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>X</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{i}(X)=\prod _{j=0,j\neq i}^{n}{\frac {X-x_{j}}{x_{i}-x_{j}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db12a219daef0d892c20a3548902ea75d334302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.572ex; height:7.343ex;" alt="{\displaystyle l_{i}(X)=\prod _{j=0,j\neq i}^{n}{\frac {X-x_{j}}{x_{i}-x_{j}}}}"></span>, alors : </p> <center><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 {\begin{aligned}\int _{a}^{b}f(x)~{\rm {d}}x\approx \int _{a}^{b}L(x)~{\rm {d}}x&=\int _{a}^{b}\sum _{i=0}^{n}f(x_{i})\,l_{i}(x)~{\rm {d}}x\\&=\sum _{i=0}^{n}\int _{a}^{b}f(x_{i})l_{i}(x)~{\rm {d}}x\\&=\sum _{i=0}^{n}f(x_{i})\underbrace {\int _{a}^{b}l_{i}(x)~{\rm {d}}x} _{w_{i}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>≈</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </munder> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{a}^{b}f(x)~{\rm {d}}x\approx \int _{a}^{b}L(x)~{\rm {d}}x&=\int _{a}^{b}\sum _{i=0}^{n}f(x_{i})\,l_{i}(x)~{\rm {d}}x\\&=\sum _{i=0}^{n}\int _{a}^{b}f(x_{i})l_{i}(x)~{\rm {d}}x\\&=\sum _{i=0}^{n}f(x_{i})\underbrace {\int _{a}^{b}l_{i}(x)~{\rm {d}}x} _{w_{i}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a1f1c79a9885bd74847575cba81d1e0f86e553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.3ex; margin-bottom: -0.204ex; width:51.301ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}\int _{a}^{b}f(x)~{\rm {d}}x\approx \int _{a}^{b}L(x)~{\rm {d}}x&=\int _{a}^{b}\sum _{i=0}^{n}f(x_{i})\,l_{i}(x)~{\rm {d}}x\\&=\sum _{i=0}^{n}\int _{a}^{b}f(x_{i})l_{i}(x)~{\rm {d}}x\\&=\sum _{i=0}^{n}f(x_{i})\underbrace {\int _{a}^{b}l_{i}(x)~{\rm {d}}x} _{w_{i}}.\end{aligned}}}"></span></center> <p>Ainsi ; <span style="display: block; margin-left:1.6em;"><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 w_{i}=\int _{a}^{b}\prod _{j=0,j\neq i}^{n}{\frac {x-x_{j}}{x_{i}-x_{j}}}~{\rm {d}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}=\int _{a}^{b}\prod _{j=0,j\neq i}^{n}{\frac {x-x_{j}}{x_{i}-x_{j}}}~{\rm {d}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5532a4af78e838d40cc6ef7463f7b930a5e64e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.392ex; height:7.509ex;" alt="{\displaystyle w_{i}=\int _{a}^{b}\prod _{j=0,j\neq i}^{n}{\frac {x-x_{j}}{x_{i}-x_{j}}}~{\rm {d}}x.}"></span></span> Le <a href="/wiki/Int%C3%A9gration_par_changement_de_variable" title="Intégration par changement de variable">changement</a> de <a href="/wiki/Variable_(math%C3%A9matiques)" title="Variable (mathématiques)">variable</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\frac {x-a}{\Delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi mathvariant="normal">Δ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {x-a}{\Delta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9410a037dfee24e076512ba849a1ad91d62e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.49ex; height:5.176ex;" alt="{\displaystyle y={\frac {x-a}{\Delta }}}"></span> conduit à l'expression<sup id="cite_ref-Wolfram_1-0" class="reference"><a href="#cite_note-Wolfram-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>: <span style="display: block; margin-left:1.6em;"><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 w_{i}={\frac {(b-a)}{n}}{\frac {(-1)^{n-i}}{i!(n-i)!}}\int _{0}^{n}\prod _{k=0,k\neq i}^{n}(y-k)~{\rm {d}}y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mi>i</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>k</mi> <mo>≠</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}={\frac {(b-a)}{n}}{\frac {(-1)^{n-i}}{i!(n-i)!}}\int _{0}^{n}\prod _{k=0,k\neq i}^{n}(y-k)~{\rm {d}}y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74cd096d8e2b76e385628ea9068a57a8084ac6b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.32ex; height:7.509ex;" alt="{\displaystyle w_{i}={\frac {(b-a)}{n}}{\frac {(-1)^{n-i}}{i!(n-i)!}}\int _{0}^{n}\prod _{k=0,k\neq i}^{n}(y-k)~{\rm {d}}y.}"></span></span> </p> <div class="mw-heading mw-heading3"><h3 id="Application_pour_n_=_1"><span id="Application_pour_n_.3D_1"></span>Application pour <i>n </i>= 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_de_Newton-Cotes&veaction=edit&section=2" title="Modifier la section : Application pour n = 1" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_de_Newton-Cotes&action=edit&section=2" title="Modifier le code source de la section : Application pour n = 1"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En calculant l'expression précédente lorsque <span class="texhtml"><i>n</i> = 1</span> et <span class="texhtml"><i>i</i> = 0</span>, on obtient </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}w_{0}&=(b-a){\frac {(-1)^{1-0}}{0!\,(1-0)!}}\int _{0}^{1}\prod _{k=0,k\neq 0}^{1}(y-k)~{\rm {d}}y\\&=-(b-a)\int _{0}^{1}(y-1)~{\rm {d}}y\\&=-(b-a)\left[{\frac {(y-1)^{2}}{2}}\right]_{0}^{1}\\&={\frac {b-a}{2}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>0</mn> </mrow> </msup> </mrow> <mrow> <mn>0</mn> <mo>!</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>k</mi> <mo>≠</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msubsup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}w_{0}&=(b-a){\frac {(-1)^{1-0}}{0!\,(1-0)!}}\int _{0}^{1}\prod _{k=0,k\neq 0}^{1}(y-k)~{\rm {d}}y\\&=-(b-a)\int _{0}^{1}(y-1)~{\rm {d}}y\\&=-(b-a)\left[{\frac {(y-1)^{2}}{2}}\right]_{0}^{1}\\&={\frac {b-a}{2}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de6b93f5103b0a24b6f920e733e776aa6e16bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.338ex; width:44.808ex; height:27.843ex;" alt="{\displaystyle {\begin{aligned}w_{0}&=(b-a){\frac {(-1)^{1-0}}{0!\,(1-0)!}}\int _{0}^{1}\prod _{k=0,k\neq 0}^{1}(y-k)~{\rm {d}}y\\&=-(b-a)\int _{0}^{1}(y-1)~{\rm {d}}y\\&=-(b-a)\left[{\frac {(y-1)^{2}}{2}}\right]_{0}^{1}\\&={\frac {b-a}{2}}.\end{aligned}}}"></span></dd></dl> <p>On obtient de la même manière <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 w_{1}={\frac {b-a}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1}={\frac {b-a}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b99f680f4c8ce97f1290f296eb96a43922f8a4ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.721ex; height:5.343ex;" alt="{\displaystyle w_{1}={\frac {b-a}{2}}}"></span>. On a ainsi retrouvé les coefficients de quadrature de la <a href="/wiki/M%C3%A9thode_des_trap%C3%A8zes" title="Méthode des trapèzes">méthode des trapèzes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Premières_formules_de_Newton-Cotes"><span id="Premi.C3.A8res_formules_de_Newton-Cotes"></span>Premières formules de Newton-Cotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_de_Newton-Cotes&veaction=edit&section=3" title="Modifier la section : Premières formules de Newton-Cotes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_de_Newton-Cotes&action=edit&section=3" title="Modifier le code source de la section : Premières formules de Newton-Cotes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soit un intervalle <span class="texhtml">[<i>a</i>, <i>b</i>]</span> séparé en <i>n</i> intervalles de longueur <span class="texhtml">Δ = (<i>b – a</i>)/<i>n</i></span>. On note <span class="texhtml"><i>f<sub>i</sub></i> = <i>f</i>(<i>a + i</i> Δ)</span> et ξ un élément indéterminé de <span class="texhtml">]<i>a</i>, <i>b</i>[</span>. Les formules relatives aux premiers degrés sont résumées dans le tableau suivant : </p> <center> <table border="" cellpadding="5" cellspacing="0" align="center"> <tbody><tr align="center"> <td><b>Degré</b></td> <td><b>Nom commun</b></td> <td><b>Formule</b> </td> <td><b>Terme d'erreur</b> </td></tr> <tr align="center"> <td>1</td> <td><a href="/wiki/M%C3%A9thode_des_trap%C3%A8zes" title="Méthode des trapèzes">Méthode des trapèzes</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{2}}(f_{0}+f_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{2}}(f_{0}+f_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/898874be93dd6abb959fcbef070b3ddcb08c96f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.941ex; height:5.343ex;" alt="{\displaystyle {\frac {b-a}{2}}(f_{0}+f_{1})}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {(b-a)^{3}}{12}}\,f^{(2)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>12</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {(b-a)^{3}}{12}}\,f^{(2)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/832a38281b02d44403f66937c749653209eefe07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.456ex; height:5.843ex;" alt="{\displaystyle -{\frac {(b-a)^{3}}{12}}\,f^{(2)}(\xi )}"></span> </td></tr> <tr align="center"> <td>2</td> <td><a href="/wiki/M%C3%A9thode_de_Simpson" title="Méthode de Simpson">Méthode de Simpson 1/3</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{6}}(f_{0}+4f_{1}+f_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{6}}(f_{0}+4f_{1}+f_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80da787700ee8d6df4377f5b2108594c8063f0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.137ex; height:5.343ex;" alt="{\displaystyle {\frac {b-a}{6}}(f_{0}+4f_{1}+f_{2})}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {(b-a)^{5}}{2880}}\,f^{(4)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2880</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {(b-a)^{5}}{2880}}\,f^{(4)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38ffd6cac3fce4da01d9ded54e22fbdb7602eab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.456ex; height:5.843ex;" alt="{\displaystyle -{\frac {(b-a)^{5}}{2880}}\,f^{(4)}(\xi )}"></span> </td></tr> <tr align="center"> <td>3</td> <td>Méthode de <a href="/wiki/Thomas_Simpson" title="Thomas Simpson">Simpson</a> 3/8   </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{8}}(f_{0}+3f_{1}+3f_{2}+f_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>8</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{8}}(f_{0}+3f_{1}+3f_{2}+f_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/576acf1c9916f415bcdc06ede2876e3687ab8776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.333ex; height:5.343ex;" alt="{\displaystyle {\frac {b-a}{8}}(f_{0}+3f_{1}+3f_{2}+f_{3})}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {(b-a)^{5}}{6480}}\,f^{(4)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>6480</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {(b-a)^{5}}{6480}}\,f^{(4)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713aaa869b1dd59e763743c8ec6f3cf92af97463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.456ex; height:6.009ex;" alt="{\displaystyle -{\frac {(b-a)^{5}}{6480}}\,f^{(4)}(\xi )}"></span> </td></tr> <tr align="center"> <td>4</td> <td>Méthode de <a href="/wiki/George_Boole" title="George Boole">Boole</a>-<a href="/wiki/Yvon_Villarceau" class="mw-redirect" title="Yvon Villarceau">Villarceau</a>   </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{90}}(7f_{0}+32f_{1}+12f_{2}+32f_{3}+7f_{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>90</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>7</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>32</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>12</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>32</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>7</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{90}}(7f_{0}+32f_{1}+12f_{2}+32f_{3}+7f_{4})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c0825d67febc5a0f8529b0cff79c65691119a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.342ex; height:5.343ex;" alt="{\displaystyle {\frac {b-a}{90}}(7f_{0}+32f_{1}+12f_{2}+32f_{3}+7f_{4})}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {(b-a)^{7}}{1935360}}\,f^{(6)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mn>1935360</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {(b-a)^{7}}{1935360}}\,f^{(6)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b326c85505d44cbb36f97f2e58fefa71da8248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.662ex; height:5.843ex;" alt="{\displaystyle -{\frac {(b-a)^{7}}{1935360}}\,f^{(6)}(\xi )}"></span> </td></tr> <tr align="center"> <td>6</td> <td>Méthode de Weddle-Hardy   </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{840}}(41f_{0}+216f_{1}+27f_{2}+272f_{3}+27f_{4}+216f_{5}+41f_{6})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>840</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>41</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>216</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>27</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>272</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>27</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mn>216</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <mn>41</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{840}}(41f_{0}+216f_{1}+27f_{2}+272f_{3}+27f_{4}+216f_{5}+41f_{6})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5260cac3e4ef9e339a6160dbb8fec270d508fc9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:59.872ex; height:5.509ex;" alt="{\displaystyle {\frac {b-a}{840}}(41f_{0}+216f_{1}+27f_{2}+272f_{3}+27f_{4}+216f_{5}+41f_{6})}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {(b-a)^{9}}{1567641600}}\,f^{(8)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mrow> <mn>1567641600</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {(b-a)^{9}}{1567641600}}\,f^{(8)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8a035b317a5e8db5bb8274896ce014b69eb996" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.149ex; height:6.009ex;" alt="{\displaystyle -{\frac {(b-a)^{9}}{1567641600}}\,f^{(8)}(\xi )}"></span> </td></tr> </tbody></table> </center> <p><br /> Les formules relatives aux degrés supérieurs sont donnés dans le tableau suivant : <br /> </p> <center> <table border="" cellpadding="5" cellspacing="0" align="center"> <tbody><tr align="center"> <td><b>Degré</b></td> <td><b>Nombre de points</b></td> <td><b>Formule</b> </td> <td><b>Terme d'erreur</b> </td></tr> <tr align="center"> <td>7</td> <td>Méthode à 8 points<sup id="cite_ref-Wolfram_1-1" class="reference"><a href="#cite_note-Wolfram-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{17280}}(751(f_{0}+f_{7})+3577(f_{1}+f_{6})+1323(f_{2}+f_{5})+2989(f_{3}+f_{4}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>17280</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>751</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3577</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1323</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2989</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{17280}}(751(f_{0}+f_{7})+3577(f_{1}+f_{6})+1323(f_{2}+f_{5})+2989(f_{3}+f_{4}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063f5095d80fae431d0b96e1cc2a47bca8718c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:70.562ex; height:5.509ex;" alt="{\displaystyle {\frac {b-a}{17280}}(751(f_{0}+f_{7})+3577(f_{1}+f_{6})+1323(f_{2}+f_{5})+2989(f_{3}+f_{4}))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {8183}{518400}}{\frac {(b-a)^{9}}{7^{9}}}\,f^{(8)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8183</mn> <mn>518400</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mrow> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {8183}{518400}}{\frac {(b-a)^{9}}{7^{9}}}\,f^{(8)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2edfbc6ec36e3933515384664dd341244b853a00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.267ex; height:6.343ex;" alt="{\displaystyle -{\frac {8183}{518400}}{\frac {(b-a)^{9}}{7^{9}}}\,f^{(8)}(\xi )}"></span> </td></tr> <tr align="center"> <td>8</td> <td>Méthode à 9 points<sup id="cite_ref-Wolfram_1-2" class="reference"><a href="#cite_note-Wolfram-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{28350}}(989(f_{0}+f_{8})+5888(f_{1}+f_{7})-928(f_{2}+f_{6})+10496(f_{3}+f_{5})-4540f_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>28350</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>989</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>5888</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>928</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>10496</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>4540</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{28350}}(989(f_{0}+f_{8})+5888(f_{1}+f_{7})-928(f_{2}+f_{6})+10496(f_{3}+f_{5})-4540f_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635a847c26bb985bbbd6e66657bd1140d2f93b90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:79.341ex; height:5.343ex;" alt="{\displaystyle {\frac {b-a}{28350}}(989(f_{0}+f_{8})+5888(f_{1}+f_{7})-928(f_{2}+f_{6})+10496(f_{3}+f_{5})-4540f_{4}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {2368}{467775}}{\frac {(b-a)^{11}}{8^{11}}}\,f^{(10)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2368</mn> <mn>467775</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mrow> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {2368}{467775}}{\frac {(b-a)^{11}}{8^{11}}}\,f^{(10)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e952672b264328bbfdd10639b59497fda205cf28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.911ex; height:6.343ex;" alt="{\displaystyle -{\frac {2368}{467775}}{\frac {(b-a)^{11}}{8^{11}}}\,f^{(10)}(\xi )}"></span> </td></tr> <tr align="center"> <td>9</td> <td>Méthode à 10 points<sup id="cite_ref-Wolfram_1-3" class="reference"><a href="#cite_note-Wolfram-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{89600}}(2857(f_{0}+f_{9})+15741(f_{1}+f_{8})+1080(f_{2}+f_{7})+19344(f_{3}+f_{6})+5778(f_{4}+f_{5}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>89600</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>2857</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>15741</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1080</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>19344</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>5778</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{89600}}(2857(f_{0}+f_{9})+15741(f_{1}+f_{8})+1080(f_{2}+f_{7})+19344(f_{3}+f_{6})+5778(f_{4}+f_{5}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ecbabf0aa319ac99b71a6b17ab22c8ca5dfb4ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:90.576ex; height:5.343ex;" alt="{\displaystyle {\frac {b-a}{89600}}(2857(f_{0}+f_{9})+15741(f_{1}+f_{8})+1080(f_{2}+f_{7})+19344(f_{3}+f_{6})+5778(f_{4}+f_{5}))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {519}{394240}}{\frac {(b-a)^{11}}{9^{10}}}\,f^{(10)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>519</mn> <mn>394240</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mrow> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {519}{394240}}{\frac {(b-a)^{11}}{9^{10}}}\,f^{(10)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37652a283f9e02fd993447933dcb80b5c62c77c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.911ex; height:6.343ex;" alt="{\displaystyle -{\frac {519}{394240}}{\frac {(b-a)^{11}}{9^{10}}}\,f^{(10)}(\xi )}"></span> </td></tr> <tr align="center"> <td>10</td> <td>Méthode à 11 points<sup id="cite_ref-Wolfram_1-4" class="reference"><a href="#cite_note-Wolfram-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b-a}{598752}}(16067(f_{0}+f_{10})+106300(f_{1}+f_{9})-48525(f_{2}+f_{8})+272400(f_{3}+f_{7})-260550(f_{4}+f_{6})+427368f_{5})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>598752</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>16067</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>106300</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>48525</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>272400</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>260550</mn> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>427368</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b-a}{598752}}(16067(f_{0}+f_{10})+106300(f_{1}+f_{9})-48525(f_{2}+f_{8})+272400(f_{3}+f_{7})-260550(f_{4}+f_{6})+427368f_{5})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e4128dddd15513b1edb1a289c16da0461680d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:111.544ex; height:5.509ex;" alt="{\displaystyle {\frac {b-a}{598752}}(16067(f_{0}+f_{10})+106300(f_{1}+f_{9})-48525(f_{2}+f_{8})+272400(f_{3}+f_{7})-260550(f_{4}+f_{6})+427368f_{5})}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1346350}{326918592}}{\frac {(b-a)^{13}}{{10}^{13}}}\,f^{(12)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1346350</mn> <mn>326918592</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msup> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1346350}{326918592}}{\frac {(b-a)^{13}}{{10}^{13}}}\,f^{(12)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d59764e63990bee48615afc01f3104ee9a67f78c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.398ex; height:6.343ex;" alt="{\displaystyle -{\frac {1346350}{326918592}}{\frac {(b-a)^{13}}{{10}^{13}}}\,f^{(12)}(\xi )}"></span> </td></tr> </tbody></table> </center> <div class="mw-heading mw-heading2"><h2 id="Ordre_de_la_méthode"><span id="Ordre_de_la_m.C3.A9thode"></span>Ordre de la méthode</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_de_Newton-Cotes&veaction=edit&section=4" title="Modifier la section : Ordre de la méthode" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_de_Newton-Cotes&action=edit&section=4" title="Modifier le code source de la section : Ordre de la méthode"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'<i><a href="/wiki/Calcul_num%C3%A9rique_d%27une_int%C3%A9grale#Ordre_de_la_formule_de_quadrature_et_convergence" title="Calcul numérique d'une intégrale">ordre</a></i> d'une formule de quadrature est définie comme le plus grand entier <span class="texhtml"><i>m</i></span> pour lequel la valeur calculée par la formule vaut exactement l'intégrale recherchée pour tout polynôme de degré inférieur ou égal à <span class="texhtml"><i>m</i></span>. </p><p>L'ordre de la formule de Newton-Cotes de degré <span class="texhtml"><i>n</i></span> est supérieur ou égal à <span class="texhtml"><i>n</i></span>, car on a alors <span class="texhtml"><i>L</i>=<i>f</i></span> pour tout <span class="texhtml"><i>f</i></span> polynôme de degré inférieur ou égal à <span class="texhtml"><i>n</i></span>. </p><p>On peut en fait montrer le résultat suivant<sup id="cite_ref-Demailly_2-0" class="reference"><a href="#cite_note-Demailly-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>: </p> <blockquote style="width:90%; border-left: solid #D0D0D0 1px; padding-left:1em;"> <p>Si <span class="texhtml"><i>n</i></span> est impair, alors la méthode de Newton-Cotes de degré <span class="texhtml"><i>n</i></span> est d'ordre <span class="texhtml"><i>n</i></span>. </p><p>Si <span class="texhtml"><i>n</i></span> est pair, alors la méthode de Newton-Cotes de degré <span class="texhtml"><i>n</i></span> est d'ordre <span class="texhtml"><i>n</i>+1</span>. </p> </blockquote> <p>L'ordre donne une indication de l'efficacité d'une formule de quadrature. Les formules de Newton-Cotes sont donc généralement utilisées pour des degrés pairs. </p> <div class="mw-heading mw-heading2"><h2 id="Convergence">Convergence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_de_Newton-Cotes&veaction=edit&section=5" title="Modifier la section : Convergence" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_de_Newton-Cotes&action=edit&section=5" title="Modifier le code source de la section : Convergence"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bien qu'une formule de Newton-Cotes puisse être établie pour n'importe quel degré, l'utilisation de degrés supérieurs peut causer des erreurs d'arrondi<sup id="cite_ref-Demailly_2-1" class="reference"><a href="#cite_note-Demailly-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>, et la convergence n’est pas assurée lorsque le degré augmente à cause du <a href="/wiki/Ph%C3%A9nom%C3%A8ne_de_Runge" title="Phénomène de Runge">phénomène de Runge</a>. Pour cette raison, il est généralement préférable de se restreindre aux premiers degrés, et d'utiliser des <a href="/wiki/Calcul_num%C3%A9rique_d%27une_int%C3%A9grale#Formules_composites" title="Calcul numérique d'une intégrale">formules composites</a> pour améliorer la précision de la formule de quadrature. Toutefois, la méthode de Newton-Cotes d'ordre 8 est employée dans le livre <i>Computer Methods for Mathematical Computations</i>, de Forsythe, Malcolm et Moler, qui a joui d'un succès certain dans les années 70 et 80. Elle y apparaît sous la forme d'une méthode adaptative : QUANC8<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_de_Newton-Cotes&veaction=edit&section=6" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_de_Newton-Cotes&action=edit&section=6" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Wolfram-1"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Wolfram_1-0">a</a> <a href="#cite_ref-Wolfram_1-1">b</a> <a href="#cite_ref-Wolfram_1-2">c</a> <a href="#cite_ref-Wolfram_1-3">d</a> et <a href="#cite_ref-Wolfram_1-4">e</a></sup> </span><span class="reference-text"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Newton-CotesFormulas.html">Weisstein, Eric W. "Newton-Cotes Formulas." From MathWorld--A Wolfram Web Resource</a></span> </li> <li id="cite_note-Demailly-2"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Demailly_2-0">a</a> et <a href="#cite_ref-Demailly_2-1">b</a></sup> </span><span class="reference-text"><span class="ouvrage" id="Demailly2006"><span class="ouvrage" id="Jean-Pierre_Demailly2006"><a href="/wiki/Jean-Pierre_Demailly" title="Jean-Pierre Demailly">Jean-Pierre <span class="nom_auteur">Demailly</span></a>, <cite class="italique">Analyse numérique et équations différentielles</cite>, <a href="/wiki/EDP_Sciences" title="EDP Sciences">EDP Sciences</a>, <abbr class="abbr" title="collection">coll.</abbr> « Grenoble Sciences », <time>2006</time>, 344 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-7598-0112-1" title="Spécial:Ouvrages de référence/978-2-7598-0112-1"><span class="nowrap">978-2-7598-0112-1</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=KwRJ5Vc5O-8C&pg=PA63">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 63<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analyse+num%C3%A9rique+et+%C3%A9quations+diff%C3%A9rentielles&rft.pub=EDP+Sciences&rft.aulast=Demailly&rft.aufirst=Jean-Pierre&rft.date=2006&rft.pages=63&rft.tpages=344&rft.isbn=978-2-7598-0112-1&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+de+Newton-Cotes"></span></span></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><a rel="nofollow" class="external text" href="http://www.netlib.org/fmm/">Code source de QUANC8</a></span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Liens_externes">Liens externes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_de_Newton-Cotes&veaction=edit&section=7" title="Modifier la section : Liens externes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_de_Newton-Cotes&action=edit&section=7" title="Modifier le code source de la section : Liens externes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.math-linux.com/spip.php?article57">Formules de Newton-Cotes</a> sur Math-Linux.com</li> <li><span class="ouvrage" id="Weisstein"><span class="ouvrage" id="Eric_W._Weisstein"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, « <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Newton-CotesFormulas.html"><cite style="font-style:normal;" lang="en"><span class="lang-en" lang="en">Newton-Cotes Formulas</span></cite></a> », sur <span class="italique"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span></span></span></li></ul> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Analyse_num%C3%A9rique" title="Modèle:Palette Analyse numérique"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Analyse_num%C3%A9rique&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Analyse_num%C3%A9rique" title="Analyse numérique">Analyse numérique</a></div></th> </tr> <tr> <th class="navbox-group" style="width: 17em"><a href="/wiki/Algorithme_de_recherche_d%27un_z%C3%A9ro_d%27une_fonction" title="Algorithme de recherche d'un zéro d'une fonction">Recherche de zéro</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Algorithme_de_Josephy-Newton" title="Algorithme de Josephy-Newton">Méthode de Josephy-Newton</a></li> <li><a href="/wiki/M%C3%A9thode_de_la_s%C3%A9cante" title="Méthode de la sécante">Méthode de la sécante</a></li> <li><a href="/wiki/M%C3%A9thode_de_Newton" title="Méthode de Newton">Méthode de Newton</a></li> <li><a href="/wiki/Point_fixe" title="Point fixe">Point fixe</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width: 17em"><a href="/wiki/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques)">Transformations de matrice</a></th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Matrice_de_Hessenberg" title="Matrice de Hessenberg">Matrice de Hessenberg</a></li> <li><a href="/wiki/D%C3%A9composition_LU" title="Décomposition LU">Décomposition LU</a></li> <li><a href="/wiki/Factorisation_de_Cholesky" title="Factorisation de Cholesky">Factorisation de Cholesky</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width: 17em">Résolutions de systèmes</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/M%C3%A9thode_de_Gauss-Seidel" title="Méthode de Gauss-Seidel">Méthode de Gauss-Seidel</a></li> <li><a href="/wiki/M%C3%A9thode_de_surrelaxation_successive" title="Méthode de surrelaxation successive">Méthode de surrelaxation successive (SOR)</a></li> <li><a href="/wiki/M%C3%A9thode_de_Jacobi" title="Méthode de Jacobi">Méthode de Jacobi</a></li> <li><a href="/wiki/D%C3%A9composition_QR" title="Décomposition QR">Décomposition QR</a></li> <li><a href="/wiki/D%C3%A9composition_LU" title="Décomposition LU">Décomposition LU</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width: 17em"><a href="/wiki/Calcul_num%C3%A9rique_d%27une_int%C3%A9grale" title="Calcul numérique d'une intégrale">Intégration numérique</a></th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/M%C3%A9thode_du_point_m%C3%A9dian" title="Méthode du point médian">Méthode du point médian</a></li> <li><a href="/wiki/M%C3%A9thode_des_trap%C3%A8zes" title="Méthode des trapèzes">Méthode des trapèzes</a></li> <li><a href="/wiki/M%C3%A9thode_de_Simpson" title="Méthode de Simpson">Méthode de Simpson</a></li> <li><a href="/wiki/M%C3%A9thodes_de_quadrature_de_Gauss" title="Méthodes de quadrature de Gauss">Méthodes de quadrature de Gauss</a></li> <li><a class="mw-selflink selflink">Formule de Newton-Cotes</a></li> <li><a href="/wiki/M%C3%A9thode_de_Romberg" title="Méthode de Romberg">Méthode de Romberg</a></li> <li><a href="/wiki/M%C3%A9thode_de_Monte-Carlo" title="Méthode de Monte-Carlo">Méthode de Monte-Carlo</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width: 17em"><a href="/wiki/R%C3%A9solution_num%C3%A9rique_des_%C3%A9quations_diff%C3%A9rentielles" title="Résolution numérique des équations différentielles">Équations différentielles</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/M%C3%A9thode_d%27Euler" title="Méthode d'Euler">Méthode d'Euler</a> (et <a href="/wiki/M%C3%A9thode_d%27Euler_semi-implicite" title="Méthode d'Euler semi-implicite">semi-implicite</a>)</li> <li><a href="/wiki/M%C3%A9thode_de_Heun" title="Méthode de Heun">Méthode de Heun</a></li> <li><a href="/wiki/M%C3%A9thodes_de_Runge-Kutta" title="Méthodes de Runge-Kutta">Méthodes de Runge-Kutta</a></li> <li><a href="/wiki/Int%C3%A9gration_de_Verlet" title="Intégration de Verlet">Intégration de Verlet</a></li> <li><a href="/wiki/M%C3%A9thode_saute-mouton" title="Méthode saute-mouton">Leapfrog</a></li> <li><a href="/wiki/M%C3%A9thode_lin%C3%A9aire_%C3%A0_pas_multiples" title="Méthode linéaire à pas multiples">Méthode linéaire à pas multiples</a> (<a href="/wiki/M%C3%A9thodes_d%27Adams-Bashforth" title="Méthodes d'Adams-Bashforth">Adams-Bashforth</a>, <a href="/w/index.php?title=Backward_differentiation_formula&action=edit&redlink=1" class="new" title="Backward differentiation formula (page inexistante)">backward differentiation formula</a> <a href="https://en.wikipedia.org/wiki/backward_differentiation_formula" class="extiw" title="en:backward differentiation formula"><span class="indicateur-langue" title="Article en anglais : « backward differentiation formula »">(en)</span></a>)</li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width: 17em"><a href="/wiki/Interpolation_num%C3%A9rique" title="Interpolation numérique">Interpolation numérique</a></th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Courbe_de_B%C3%A9zier" title="Courbe de Bézier">Courbe de Bézier</a></li> <li><a href="/wiki/Surface_de_B%C3%A9zier" title="Surface de Bézier">Surface de Bézier</a></li> <li><a href="/wiki/Spline" title="Spline">Spline</a></li> <li><a href="/wiki/B-spline" title="B-spline">B-spline</a></li> <li><a href="/wiki/NURBS" title="NURBS">NURBS</a></li> <li><a href="/wiki/Interpolation_polynomiale" title="Interpolation polynomiale">Interpolation polynomiale</a></li> <li><a href="/wiki/Interpolation_lagrangienne" title="Interpolation lagrangienne">Interpolation lagrangienne</a></li> <li><a href="/wiki/Interpolation_newtonienne" title="Interpolation newtonienne">Interpolation newtonienne</a></li> <li><a href="/wiki/Interpolation_d%27Hermite" title="Interpolation d'Hermite">Interpolation d'Hermite</a></li> <li><a href="/wiki/Suite_de_polyn%C3%B4mes_orthogonaux" title="Suite de polynômes orthogonaux">Suite de polynômes orthogonaux</a></li> <li><a href="/wiki/Interpolation_bilin%C3%A9aire" title="Interpolation bilinéaire">Interpolation bilinéaire</a></li> <li><a href="/wiki/Interpolation_bicubique" title="Interpolation bicubique">Interpolation bicubique</a></li></ul> </div></td> </tr> </tbody></table> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Analyse" title="Portail de l'analyse"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/24px-Nuvola_apps_kmplot.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/36px-Nuvola_apps_kmplot.svg.png 1.5x, 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