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class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical tool to algorithmically solve equations</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 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.ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">September 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table><p>In <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, a <b>numerical method</b> is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. </p><meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_method&action=edit&section=1" title="Edit section: Mathematical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=0}"></span> be a <a href="/wiki/Well-posed_problem_(numerical_analysis)" class="mw-redirect" title="Well-posed problem (numerical analysis)">well-posed problem</a>, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:X\times Y\rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:X\times Y\rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8d4c6a5ecddaf5aebf5e9e440b689703c0c0aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.564ex; height:2.176ex;" alt="{\displaystyle F:X\times Y\rightarrow \mathbb {R} }"></span> is a <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_Numbers" class="mw-redirect" title="Complex Numbers">complex</a> functional relationship, defined on the cross-product of an input data set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and an output data set <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, such that exists a <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">locally lipschitz</a> function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:X\rightarrow Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:X\rightarrow Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089b3c3038f9faa5c5a03bdd590ba306d8b99787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.421ex; height:2.509ex;" alt="{\displaystyle g:X\rightarrow Y}"></span> called <a href="/w/index.php?title=Resolvent_(direct_problem)&action=edit&redlink=1" class="new" title="Resolvent (direct problem) (page does not exist)">resolvent</a>, which has the property that for every root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></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 y=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26c08f8fd3471dad5e2c45c2f753ffd7c9aba4ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.509ex; height:2.843ex;" alt="{\displaystyle y=g(x)}"></span>. We define <b>numerical method</b> for the approximation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=0}"></span>, the <a href="/wiki/Sequence" title="Sequence">sequence</a> of problems </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 \left\{M_{n}\right\}_{n\in \mathbb {N} }=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>{</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>{</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{M_{n}\right\}_{n\in \mathbb {N} }=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48d2e4b6af62707e361ceb02633c6fa86c185a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.593ex; height:3.009ex;" alt="{\displaystyle \left\{M_{n}\right\}_{n\in \mathbb {N} }=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} },}"></span></dd></dl> <p>with <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_{n}:X_{n}\times Y_{n}\rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}:X_{n}\times Y_{n}\rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e08c64e69b781b202a61cf7d767dddf4b8a1ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.495ex; height:2.509ex;" alt="{\displaystyle F_{n}:X_{n}\times Y_{n}\rightarrow \mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\in X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}\in X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620c24c43e857091629d53ecbf88855d1f2c7d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.532ex; height:2.509ex;" alt="{\displaystyle x_{n}\in X_{n}}"></span> and <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_{n}\in Y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{n}\in Y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c324821ec401ea96490b2928e935439f6bc06656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.767ex; height:2.509ex;" alt="{\displaystyle y_{n}\in Y_{n}}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>. The problems of which the method consists need not be well-posed. If they are, the method is said to be <i>stable</i> or <i>well-posed</i>.<sup id="cite_ref-quartsaccsal_1-0" class="reference"><a href="#cite_note-quartsaccsal-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Consistency">Consistency</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_method&action=edit&section=2" title="Edit section: Consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Necessary conditions for a numerical method to effectively approximate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=0}"></span> are that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\rightarrow x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}\rightarrow x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9366d8c859fc5925cb1732396229ce0f016de5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.492ex; height:2.176ex;" alt="{\displaystyle x_{n}\rightarrow x}"></span> and that <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> behaves like <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }"></span>. So, a numerical method is called <i>consistent</i> if and only if the sequence of functions <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 \left\{F_{n}\right\}_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>{</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{F_{n}\right\}_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45acbc8a5939c7deb02e0d30d618320fa0fff53c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.539ex; height:3.009ex;" alt="{\displaystyle \left\{F_{n}\right\}_{n\in \mathbb {N} }}"></span> pointwise converges to <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> on the set <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> of its solutions: </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 \lim F_{n}(x,y+t)=F(x,y,t)=0,\quad \quad \forall (x,y,t)\in S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">lim</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim F_{n}(x,y+t)=F(x,y,t)=0,\quad \quad \forall (x,y,t)\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3cfb36e901d5de5f0d59ec08406430595354b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.801ex; height:2.843ex;" alt="{\displaystyle \lim F_{n}(x,y+t)=F(x,y,t)=0,\quad \quad \forall (x,y,t)\in S.}"></span></dd></dl> <p>When <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_{n}=F,\forall n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>F</mi> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F,\forall n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a00d8d70099d6268f457b21e73826700c6635f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.792ex; height:2.509ex;" alt="{\displaystyle F_{n}=F,\forall n\in \mathbb {N} }"></span> on <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> the method is said to be <i>strictly consistent</i>.<sup id="cite_ref-quartsaccsal_1-1" class="reference"><a href="#cite_note-quartsaccsal-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </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=Numerical_method&action=edit&section=3" title="Edit section: Convergence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Denote by <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 \ell _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/044b64cebd71c71a2eb9d2658f219fc16705c80d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.188ex; height:2.509ex;" alt="{\displaystyle \ell _{n}}"></span> a sequence of <i>admissible perturbations</i> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span> for some numerical method <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> (i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+\ell _{n}\in X_{n}\forall n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+\ell _{n}\in X_{n}\forall n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4eaae702440177ef614af88f62e6fa03b164168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.548ex; height:2.509ex;" alt="{\displaystyle x+\ell _{n}\in X_{n}\forall n\in \mathbb {N} }"></span>) and with <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_{n}(x+\ell _{n})\in Y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{n}(x+\ell _{n})\in Y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b1465a4e25f2fd4e92494c7fe33a61228d94d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.935ex; height:2.843ex;" alt="{\displaystyle y_{n}(x+\ell _{n})\in Y_{n}}"></span> the value such that <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_{n}(x+\ell _{n},y_{n}(x+\ell _{n}))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(x+\ell _{n},y_{n}(x+\ell _{n}))=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53cc83dd047edb46356366e0c60f7753799430f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.701ex; height:2.843ex;" alt="{\displaystyle F_{n}(x+\ell _{n},y_{n}(x+\ell _{n}))=0}"></span>. A condition which the method has to satisfy to be a meaningful tool for solving the problem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c910dd64922e23f212bc46e970af0540811c48f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=0}"></span> is <i>convergence</i>: </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}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall \ell _{n}:\|\ell _{n}\|<\delta _{\varepsilon ,n_{0}}\Rightarrow \|y_{n}(x+\ell _{n})-y\|\leq \varepsilon .\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 /> <mtd> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∃</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∃</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> such that</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">∀</mi> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">⇒</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mi>ε</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall \ell _{n}:\|\ell _{n}\|<\delta _{\varepsilon ,n_{0}}\Rightarrow \|y_{n}(x+\ell _{n})-y\|\leq \varepsilon .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea0e99df283bc0ad41e30da628607e9e7cb8474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.644ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall \ell _{n}:\|\ell _{n}\|<\delta _{\varepsilon ,n_{0}}\Rightarrow \|y_{n}(x+\ell _{n})-y\|\leq \varepsilon .\end{aligned}}}"></span></dd></dl> <p>One can easily prove that the point-wise convergence of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{n}\}_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{y_{n}\}_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f41a4b09535776a0ad688223a27e0a5dbbb6003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.184ex; height:2.843ex;" alt="{\displaystyle \{y_{n}\}_{n\in \mathbb {N} }}"></span> to <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> implies the convergence of the associated method.<sup id="cite_ref-quartsaccsal_1-2" class="reference"><a href="#cite_note-quartsaccsal-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_method&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li> <li><a href="/wiki/Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numerical_method&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-quartsaccsal-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-quartsaccsal_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-quartsaccsal_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-quartsaccsal_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output 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.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="CITEREFQuarteroni,_Sacco,_Saleri2000" class="citation book cs1">Quarteroni, Sacco, Saleri (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171114040621/http://www.techmat.vgtu.lt/~inga/Files/Quarteroni-SkaitMetod.pdf"><i>Numerical Mathematics</i></a> <span class="cs1-format">(PDF)</span>. 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