Collocation method

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</nav> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <p>In mathematics, a <b>collocation method</b> is a method for the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Numerical_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Numerical analysis">numerical</a> solution of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ordinary_differential_equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ordinary differential equation">ordinary differential equations</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Partial_differential_equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Partial differential equation">partial differential equations</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Integral_equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Integral equation">integral equations</a>. The idea is to choose a finite-dimensional space of candidate solutions (usually <a href="https://en-m-wikipedia-org.translate.goog/wiki/Polynomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Polynomial">polynomials</a> up to a certain degree) and a number of points in the domain (called <i>collocation points</i>), and to select that solution which satisfies the given equation at the collocation points.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ordinary_differential_equations"><span class="tocnumber">1</span> <span class="toctext">Ordinary differential equations</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Example:_The_trapezoidal_rule"><span class="tocnumber">1.1</span> <span class="toctext">Example: The trapezoidal rule</span></a></li> <li class="toclevel-2 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Other_examples"><span class="tocnumber">1.2</span> <span class="toctext">Other examples</span></a></li> </ul></li> <li class="toclevel-1 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Orthogonal_collocation_method"><span class="tocnumber">2</span> <span class="toctext">Orthogonal collocation method</span></a></li> <li class="toclevel-1 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notes"><span class="tocnumber">3</span> <span class="toctext">Notes</span></a></li> <li class="toclevel-1 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References"><span class="tocnumber">4</span> <span class="toctext">References</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Ordinary_differential_equations">Ordinary differential equations</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Collocation_method&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Ordinary differential equations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>Suppose that the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ordinary_differential_equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ordinary differential equation">ordinary differential equation</a></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 y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> y </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> , </mo> <mspace width="1em"></mspace> <mi> y </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d7dc94cd047fc8f2ea0d20976dbf2e33f30eac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.512ex; height:3.009ex;" alt="{\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},}"> </noscript><span class="lazy-image-placeholder" style="width: 30.512ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d7dc94cd047fc8f2ea0d20976dbf2e33f30eac" data-alt="{\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>is to be solved over the interval <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 [t_{0},t_{0}+h]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> h </mi> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [t_{0},t_{0}+h]} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67dfc6adc51170b9c42340d31fdcb4e095250c18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.295ex; height:2.843ex;" alt="{\displaystyle [t_{0},t_{0}+h]}"> </noscript><span class="lazy-image-placeholder" style="width: 10.295ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67dfc6adc51170b9c42340d31fdcb4e095250c18" data-alt="{\displaystyle [t_{0},t_{0}+h]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Choose <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 c_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle c_{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2f8052630e67b00d04e3487e1d68ed7070470b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.096ex; height:2.009ex;" alt="{\displaystyle c_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.096ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2f8052630e67b00d04e3487e1d68ed7070470b" data-alt="{\displaystyle c_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> from 0 ≤ <i>c</i><sub>1</sub>< <i>c</i><sub>2</sub>< ... < <i>c</i><sub><i>n</i></sub> ≤ 1.</p> <p>The corresponding (polynomial) collocation method approximates the solution <i>y</i> by the polynomial <i>p</i> of degree <i>n</i> which satisfies the initial condition <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 p(t_{0})=y_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p(t_{0})=y_{0}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae777564830ce254a2afe7c6f7aa4d4a0b0c813" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.254ex; height:2.843ex;" alt="{\displaystyle p(t_{0})=y_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.254ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae777564830ce254a2afe7c6f7aa4d4a0b0c813" data-alt="{\displaystyle p(t_{0})=y_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, and the differential 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 p'(t_{k})=f(t_{k},p(t_{k}))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> p </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> , </mo> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p'(t_{k})=f(t_{k},p(t_{k}))} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d9e3016b2abe8464e68f8264f844d319d7abbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:19.737ex; height:3.009ex;" alt="{\displaystyle p'(t_{k})=f(t_{k},p(t_{k}))}"> </noscript><span class="lazy-image-placeholder" style="width: 19.737ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d9e3016b2abe8464e68f8264f844d319d7abbb" data-alt="{\displaystyle p'(t_{k})=f(t_{k},p(t_{k}))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> at all <i>collocation points</i> <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 t_{k}=t_{0}+c_{k}h}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mi> h </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t_{k}=t_{0}+c_{k}h} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2321534852ad99197bd27b94b3a4519312caeea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.196ex; height:2.509ex;" alt="{\displaystyle t_{k}=t_{0}+c_{k}h}"> </noscript><span class="lazy-image-placeholder" style="width: 13.196ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2321534852ad99197bd27b94b3a4519312caeea0" data-alt="{\displaystyle t_{k}=t_{0}+c_{k}h}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\ldots ,n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k=1,\ldots ,n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02703686f808b37fedb436806fa72ca3522e22de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.045ex; height:2.509ex;" alt="{\displaystyle k=1,\ldots ,n}"> </noscript><span class="lazy-image-placeholder" style="width: 12.045ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02703686f808b37fedb436806fa72ca3522e22de" data-alt="{\displaystyle k=1,\ldots ,n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. This gives <i>n</i> + 1 conditions, which matches the <i>n</i> + 1 parameters needed to specify a polynomial of degree <i>n</i>.</p> <p>All these collocation methods are in fact implicit <a href="https://en-m-wikipedia-org.translate.goog/wiki/Runge%E2%80%93Kutta_methods?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Runge–Kutta methods">Runge–Kutta methods</a>. The coefficients <i>c</i><sub><i>k</i></sub> in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods. <sup id="cite_ref-1" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></p> <div class="mw-heading mw-heading3"> <h3 id="Example:_The_trapezoidal_rule">Example: The trapezoidal rule</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Collocation_method&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Example: The trapezoidal rule" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Pick, as an example, the two collocation points <i>c</i><sub>1</sub> = 0 and <i>c</i><sub>2</sub> = 1 (so <i>n</i> = 2). The collocation conditions are</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 p(t_{0})=y_{0},\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p(t_{0})=y_{0},\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/856aee4a12855be6f1717345dd71d5e85e0db3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.288ex; height:2.843ex;" alt="{\displaystyle p(t_{0})=y_{0},\,}"> </noscript><span class="lazy-image-placeholder" style="width: 11.288ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/856aee4a12855be6f1717345dd71d5e85e0db3e4" data-alt="{\displaystyle p(t_{0})=y_{0},\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <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 p'(t_{0})=f(t_{0},p(t_{0})),\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> p </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> , </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p'(t_{0})=f(t_{0},p(t_{0})),\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ee02a8324317dd055d3f35d9671faeb5c0cbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.667ex; height:3.009ex;" alt="{\displaystyle p'(t_{0})=f(t_{0},p(t_{0})),\,}"> </noscript><span class="lazy-image-placeholder" style="width: 20.667ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ee02a8324317dd055d3f35d9671faeb5c0cbae" data-alt="{\displaystyle p'(t_{0})=f(t_{0},p(t_{0})),\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <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 p'(t_{0}+h)=f(t_{0}+h,p(t_{0}+h)).\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> p </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> h </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> h </mi> <mo> , </mo> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> h </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> . </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p'(t_{0}+h)=f(t_{0}+h,p(t_{0}+h)).\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f8be76eb3762b9a65182779c40fb8aa4892406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:33.205ex; height:3.009ex;" alt="{\displaystyle p'(t_{0}+h)=f(t_{0}+h,p(t_{0}+h)).\,}"> </noscript><span class="lazy-image-placeholder" style="width: 33.205ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f8be76eb3762b9a65182779c40fb8aa4892406" data-alt="{\displaystyle p'(t_{0}+h)=f(t_{0}+h,p(t_{0}+h)).\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>There are three conditions, so <i>p</i> should be a polynomial of degree 2. Write <i>p</i> in the form</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 p(t)=\alpha (t-t_{0})^{2}+\beta (t-t_{0})+\gamma \,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> α </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo> − </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> β </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo> − </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> γ </mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p(t)=\alpha (t-t_{0})^{2}+\beta (t-t_{0})+\gamma \,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df388b1a8e71c861e3e6b1c7c9db7fa117b2e97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:32.977ex; height:3.176ex;" alt="{\displaystyle p(t)=\alpha (t-t_{0})^{2}+\beta (t-t_{0})+\gamma \,}"> </noscript><span class="lazy-image-placeholder" style="width: 32.977ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df388b1a8e71c861e3e6b1c7c9db7fa117b2e97" data-alt="{\displaystyle p(t)=\alpha (t-t_{0})^{2}+\beta (t-t_{0})+\gamma \,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>to simplify the computations. Then the collocation conditions can be solved to give the coefficients</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}\alpha &={\frac {1}{2h}}{\Big (}f(t_{0}+h,p(t_{0}+h))-f(t_{0},p(t_{0})){\Big )},\\\beta &=f(t_{0},p(t_{0})),\\\gamma &=y_{0}.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi> α </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mi> h </mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em"> ( </mo> </mrow> </mrow> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> h </mi> <mo> , </mo> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> h </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em"> ) </mo> </mrow> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mi> β </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mi> γ </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\alpha &={\frac {1}{2h}}{\Big (}f(t_{0}+h,p(t_{0}+h))-f(t_{0},p(t_{0})){\Big )},\\\beta &=f(t_{0},p(t_{0})),\\\gamma &=y_{0}.\end{aligned}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4ffaa970f5eea5a73379ab090bf7ea49315caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.921ex; margin-bottom: -0.25ex; width:45.073ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\alpha &={\frac {1}{2h}}{\Big (}f(t_{0}+h,p(t_{0}+h))-f(t_{0},p(t_{0})){\Big )},\\\beta &=f(t_{0},p(t_{0})),\\\gamma &=y_{0}.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 45.073ex;height: 11.509ex;vertical-align: -4.921ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4ffaa970f5eea5a73379ab090bf7ea49315caa" data-alt="{\displaystyle {\begin{aligned}\alpha &={\frac {1}{2h}}{\Big (}f(t_{0}+h,p(t_{0}+h))-f(t_{0},p(t_{0})){\Big )},\\\beta &=f(t_{0},p(t_{0})),\\\gamma &=y_{0}.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The collocation method is now given (implicitly) by</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 y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Big (}f(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mi> p </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> h </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mi> h </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em"> ( </mo> </mrow> </mrow> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mi> h </mi> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em"> ) </mo> </mrow> </mrow> <mo> , </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Big (}f(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b55a4b5fb1a70e54e57b79852182f5c2ea3b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.061ex; height:5.176ex;" alt="{\displaystyle y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Big (}f(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,}"> </noscript><span class="lazy-image-placeholder" style="width: 53.061ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b55a4b5fb1a70e54e57b79852182f5c2ea3b50" data-alt="{\displaystyle y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Big (}f(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>where <i>y</i><sub>1</sub> = <i>p</i>(<i>t</i><sub>0</sub> + <i>h</i>) is the approximate solution at <i>t</i> = <i>t</i><sub>1</sub> = <i>t</i><sub>0</sub> + <i>h</i>.</p> <p>This method is known as the "<a href="https://en-m-wikipedia-org.translate.goog/wiki/Trapezoidal_rule_(differential_equations)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Trapezoidal rule (differential equations)">trapezoidal rule</a>" for differential equations. Indeed, this method can also be derived by rewriting the differential equation as</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 y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau ,y(\tau ))\,{\textrm {d}}\tau ,\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> t </mi> </mrow> </msubsup> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> τ </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> τ </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> d </mtext> </mrow> </mrow> <mi> τ </mi> <mo> , </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau ,y(\tau ))\,{\textrm {d}}\tau ,\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded285da945408d355b58163ce0c3d2b6acfa2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.685ex; height:6.509ex;" alt="{\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau ,y(\tau ))\,{\textrm {d}}\tau ,\,}"> </noscript><span class="lazy-image-placeholder" style="width: 31.685ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded285da945408d355b58163ce0c3d2b6acfa2d3" data-alt="{\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau ,y(\tau ))\,{\textrm {d}}\tau ,\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>and approximating the integral on the right-hand side by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Trapezoidal_rule?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Trapezoidal rule">trapezoidal rule</a> for integrals.</p> <div class="mw-heading mw-heading3"> <h3 id="Other_examples">Other examples</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Collocation_method&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Other examples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gauss%E2%80%93Legendre_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gauss–Legendre method">Gauss–Legendre methods</a> use the points of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gauss%E2%80%93Legendre_quadrature?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gauss–Legendre quadrature">Gauss–Legendre quadrature</a> as collocation points. The Gauss–Legendre method based on <i>s</i> points has order 2<i>s</i>.<sup id="cite_ref-2" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> All Gauss–Legendre methods are <a href="https://en-m-wikipedia-org.translate.goog/wiki/A-stability?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="A-stability">A-stable</a>.<sup id="cite_ref-3" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></p> <p>In fact, one can show that the order of a collocation method corresponds to the order of the quadrature rule that one would get using the collocation points as weights.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Orthogonal_collocation_method">Orthogonal collocation method</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Collocation_method&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Orthogonal collocation method" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>In direct collocation method, we are essentially performing variational calculus with the finite-dimensional subspace of piecewise linear functions (as in trapezoidal rule), or cubic functions, or other piecewise polynomial functions. In orthogonal collocation method, we instead use the finite-dimensional subspace spanned by the first N vectors in some <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthogonal polynomials">orthogonal polynomial</a> basis, such as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Legendre_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Legendre polynomials">Legendre polynomials</a>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Notes">Notes</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Collocation_method&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <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 class="mw-references-wrap"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFAscherPetzold1998">Ascher & Petzold 1998</a>; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFIserles1996">Iserles 1996</a>, pp. 43–44</span></li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFIserles1996">Iserles 1996</a>, pp. 47</span></li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Collocation_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFIserles1996">Iserles 1996</a>, pp. 63</span></li> </ol> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Collocation_method&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <ul> <li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAscherPetzold1998" class="citation cs2">Ascher, Uri M.; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Linda_Petzold?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Linda Petzold">Petzold, Linda R.</a> (1998), <i>Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations</i>, Philadelphia: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Society_for_Industrial_and_Applied_Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-89871-412-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-89871-412-8"><bdi>978-0-89871-412-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Methods+for+Ordinary+Differential+Equations+and+Differential-Algebraic+Equations&rft.place=Philadelphia&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1998&rft.isbn=978-0-89871-412-8&rft.aulast=Ascher&rft.aufirst=Uri+M.&rft.au=Petzold%2C+Linda+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollocation+method" class="Z3988"></span>.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHairerNørsettWanner1993" class="citation cs2">Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), <i>Solving ordinary differential equations I: Nonstiff problems</i>, Berlin, New York: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Springer-Verlag?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-3-540-56670-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-3-540-56670-0"><bdi>978-3-540-56670-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solving+ordinary+differential+equations+I%3A+Nonstiff+problems&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=978-3-540-56670-0&rft.aulast=Hairer&rft.aufirst=Ernst&rft.au=N%C3%B8rsett%2C+Syvert+Paul&rft.au=Wanner%2C+Gerhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollocation+method" class="Z3988"></span>.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIserles1996" class="citation cs2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Arieh_Iserles?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Arieh Iserles">Iserles, Arieh</a> (1996), <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3D7Zofw3SFTWIC%26q%3D%2522Collocation%2Bmethod%2522"><i>A First Course in the Numerical Analysis of Differential Equations</i></a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cambridge_University_Press?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cambridge University Press">Cambridge University Press</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-521-55655-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-521-55655-2"><bdi>978-0-521-55655-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+the+Numerical+Analysis+of+Differential+Equations&rft.pub=Cambridge+University+Press&rft.date=1996&rft.isbn=978-0-521-55655-2&rft.aulast=Iserles&rft.aufirst=Arieh&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7Zofw3SFTWIC%26q%3D%2522Collocation%2Bmethod%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollocation+method" class="Z3988"></span>.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangChenWu2009" class="citation cs2">Wang, Yingwei; Chen, Suqin; Wu, Xionghua (2009), "A rational spectral collocation method for solving a class of parameterized singular perturbation problems", <i>Journal of Computational and Applied Mathematics</i>, <b>233</b> (10): 2652–2660, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1016%252Fj.cam.2009.11.011">10.1016/j.cam.2009.11.011</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+and+Applied+Mathematics&rft.atitle=A+rational+spectral+collocation+method+for+solving+a+class+of+parameterized+singular+perturbation+problems&rft.volume=233&rft.issue=10&rft.pages=2652-2660&rft.date=2009&rft_id=info%3Adoi%2F10.1016%2Fj.cam.2009.11.011&rft.aulast=Wang&rft.aufirst=Yingwei&rft.au=Chen%2C+Suqin&rft.au=Wu%2C+Xionghua&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollocation+method" class="Z3988"></span>.</li> </ul> <p><br></p> <div 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Collocation method - Wikipedia