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class="vector-toc-list"> <li id="toc-Convergence_of_Krylov_subspace_methods" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Convergence_of_Krylov_subspace_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Convergence of Krylov subspace methods</span> </div> </a> <ul id="toc-Convergence_of_Krylov_subspace_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Preconditioners" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Preconditioners"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Preconditioners</span> </div> </a> <ul id="toc-Preconditioners-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Methods_of_successive_approximation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Methods_of_successive_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Methods of successive approximation</span> </div> </a> <button aria-controls="toc-Methods_of_successive_approximation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Methods of successive approximation subsection</span> </button> <ul id="toc-Methods_of_successive_approximation-sublist" class="vector-toc-list"> <li id="toc-History" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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mw-first-heading"><span class="mw-page-title-main">Iterative method</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://ar.wikipedia.org/wiki/%D8%B7%D8%B1%D9%8A%D9%82%D8%A9_%D8%AA%D9%83%D8%B1%D8%A7%D8%B1%D9%8A%D8%A9" title="طريقة تكرارية – Arabic" lang="ar" hreflang="ar" data-title="طريقة تكرارية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%81%E0%A6%A8%E0%A6%B0%E0%A6%BE%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A6%BF%E0%A6%AE%E0%A7%82%E0%A6%B2%E0%A6%95_%E0%A6%AA%E0%A6%A6%E0%A7%8D%E0%A6%A7%E0%A6%A4%E0%A6%BF" title="পুনরাবৃত্তিমূলক পদ্ধতি – Bangla" lang="bn" hreflang="bn" data-title="পুনরাবৃত্তিমূলক পদ্ধতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/M%C3%A8tode_iteratiu" title="Mètode iteratiu – Catalan" lang="ca" hreflang="ca" data-title="Mètode iteratiu" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Itera%C4%8Dn%C3%AD_metoda" title="Iterační metoda – Czech" lang="cs" hreflang="cs" data-title="Iterační metoda" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Iterativ_metode" title="Iterativ metode – Danish" lang="da" hreflang="da" data-title="Iterativ metode" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Iteratsioonimeetod" title="Iteratsioonimeetod – Estonian" lang="et" hreflang="et" data-title="Iteratsioonimeetod" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/M%C3%A9todo_iterativo" title="Método iterativo – Spanish" lang="es" hreflang="es" data-title="Método iterativo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D9%88%D8%B4_%D8%AA%DA%A9%D8%B1%D8%A7%D8%B1%D8%B4%D9%88%D9%86%D8%AF%D9%87" title="روش تکرارشونده – Persian" lang="fa" hreflang="fa" data-title="روش تکرارشونده" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9thode_it%C3%A9rative" title="Méthode itérative – French" lang="fr" hreflang="fr" data-title="Méthode itérative" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%98%EB%B3%B5%EB%B2%95" title="반복법 – Korean" lang="ko" hreflang="ko" data-title="반복법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%81%E0%A4%A8%E0%A4%B0%E0%A4%BE%E0%A4%B5%E0%A5%83%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%BF%E0%A4%AE%E0%A5%82%E0%A4%B2%E0%A4%95_%E0%A4%B5%E0%A4%BF%E0%A4%A7%E0%A4%BF" title="पुनरावृत्तिमूलक विधि – Hindi" lang="hi" hreflang="hi" data-title="पुनरावृत्तिमूलक विधि" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Metode_iteratif" title="Metode iteratif – Indonesian" lang="id" hreflang="id" data-title="Metode iteratif" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Metodo_iterativo" title="Metodo iterativo – Italian" lang="it" hreflang="it" data-title="Metodo iterativo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%99%D7%98%D7%94_%D7%90%D7%99%D7%98%D7%A8%D7%98%D7%99%D7%91%D7%99%D7%AA" title="שיטה איטרטיבית – Hebrew" lang="he" hreflang="he" data-title="שיטה איטרטיבית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%8D%E5%BE%A9%E6%B3%95_(%E6%95%B0%E5%80%A4%E8%A8%88%E7%AE%97)" title="反復法 (数値計算) – Japanese" lang="ja" hreflang="ja" data-title="反復法 (数値計算)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/M%C3%A9todo_iterativo" title="Método iterativo – Portuguese" lang="pt" hreflang="pt" data-title="Método iterativo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Itera%C4%8Dn%C3%A1_met%C3%B3da" title="Iteračná metóda – Slovak" lang="sk" hreflang="sk" data-title="Iteračná metóda" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B8%A7%E0%B8%99%E0%B8%8B%E0%B9%89%E0%B8%B3" title="วิธีวนซ้ำ – Thai" lang="th" hreflang="th" data-title="วิธีวนซ้ำ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%BF%AD%E4%BB%A3%E6%B3%95" title="迭代法 – Chinese" lang="zh" hreflang="zh" data-title="迭代法" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algorithm in which each approximation of the solution is derived from prior approximations</div> <p>In <a href="/wiki/Computational_mathematics" title="Computational mathematics">computational mathematics</a>, an <b>iterative method</b> is a <a href="/wiki/Algorithm" title="Algorithm">mathematical procedure</a> that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the <i>i</i>-th approximation (called an "iterate") is derived from the previous ones. </p><p>A specific implementation with <a href="/wiki/Algorithm#Termination" title="Algorithm">termination</a> criteria for a given iterative method like <a href="/wiki/Gradient_descent" title="Gradient descent">gradient descent</a>, <a href="/wiki/Hill_climbing" title="Hill climbing">hill climbing</a>, <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>, or <a href="/wiki/Quasi-Newton_method" title="Quasi-Newton method">quasi-Newton methods</a> like <a href="/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">BFGS</a>, is an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> of an iterative method or a <b>method of successive approximation</b>. An iterative method is called <i><a href="/wiki/Convergent_series" title="Convergent series">convergent</a></i> if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, <a href="/wiki/Heuristic" title="Heuristic">heuristic</a>-based iterative methods are also common. </p><p>In contrast, <b>direct methods</b> attempt to solve the problem by a finite sequence of operations. In the absence of <a href="/wiki/Rounding_error" class="mw-redirect" title="Rounding error">rounding errors</a>, direct methods would deliver an exact solution (for example, solving a linear system of equations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span> by <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>). Iterative methods are often the only choice for <a href="/wiki/Nonlinear_equation" class="mw-redirect" title="Nonlinear equation">nonlinear equations</a>. However, iterative methods are often useful even for linear problems involving many variables (sometimes on the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Attractive_fixed_points">Attractive fixed points</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=1" title="Edit section: Attractive fixed points"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If an equation can be put into the form <i>f</i>(<i>x</i>) = <i>x</i>, and a solution <b>x</b> is an attractive <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a> of the function <i>f</i>, then one may begin with a point <i>x</i><sub>1</sub> in the <a href="/wiki/Basin_of_attraction" class="mw-redirect" title="Basin of attraction">basin of attraction</a> of <b>x</b>, and let <i>x</i><sub><i>n</i>+1</sub> = <i>f</i>(<i>x</i><sub><i>n</i></sub>) for <i>n</i> ≥ 1, and the sequence {<i>x</i><sub><i>n</i></sub>}<sub><i>n</i> ≥ 1</sub> will converge to the solution <b>x</b>. Here <i>x</i><sub><i>n</i></sub> is the <i>n</i>th approximation or iteration of <i>x</i> and <i>x</i><sub><i>n</i>+1</sub> is the next or <i>n</i> + 1 iteration of <i>x</i>. Alternately, superscripts in parentheses are often used in numerical methods, so as not to interfere with subscripts with other meanings. (For example, <i>x</i><sup>(<i>n</i>+1)</sup> = <i>f</i>(<i>x</i><sup>(<i>n</i>)</sup>).) If the function <i>f</i> is <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a>, a sufficient condition for convergence is that the <a href="/wiki/Spectral_radius" title="Spectral radius">spectral radius</a> of the derivative is strictly bounded by one in a neighborhood of the fixed point. If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist. </p> <div class="mw-heading mw-heading2"><h2 id="Linear_systems">Linear systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=2" title="Edit section: Linear systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the case of a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a>, the two main classes of iterative methods are the <b>stationary iterative methods</b>, and the more general <a href="/wiki/Krylov_subspace" title="Krylov subspace">Krylov subspace</a> methods. </p> <div class="mw-heading mw-heading3"><h3 id="Stationary_iterative_methods">Stationary iterative methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=3" title="Edit section: Stationary iterative methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Introduction">Introduction</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=4" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stationary iterative methods solve a linear system with an <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a> approximating the original one; and based on a measurement of the error in the result (<a href="/wiki/Residual_(numerical_analysis)" title="Residual (numerical analysis)">the residual</a>), form a "correction equation" for which this process is repeated. While these methods are simple to derive, implement, and analyze, convergence is only guaranteed for a limited class of matrices. </p> <div class="mw-heading mw-heading4"><h4 id="Definition">Definition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=5" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <i>iterative method</i> is defined 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 \mathbf {x} ^{k+1}:=\Psi (\mathbf {x} ^{k})\,,\quad k\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>:=</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ^{k+1}:=\Psi (\mathbf {x} ^{k})\,,\quad k\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8af8d4bf79d5c843ddee71b03886a4e2453fd034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.678ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} ^{k+1}:=\Psi (\mathbf {x} ^{k})\,,\quad k\geq 0}"></span></dd></dl> <p>and for a given linear system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span> with exact solution <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 \mathbf {x} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7854cd1cbbc521a6d45d17d621a9e4286ced0ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.465ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} ^{*}}"></span> the <i>error</i> 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 \mathbf {e} ^{k}:=\mathbf {x} ^{k}-\mathbf {x} ^{*}\,,\quad k\geq 0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>:=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} ^{k}:=\mathbf {x} ^{k}-\mathbf {x} ^{*}\,,\quad k\geq 0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2026c72f2288e498972120f31a61c2e658a20910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.114ex; height:3.009ex;" alt="{\displaystyle \mathbf {e} ^{k}:=\mathbf {x} ^{k}-\mathbf {x} ^{*}\,,\quad k\geq 0\,.}"></span></dd></dl> <p>An iterative method is called <i>linear</i> if there exists a matrix <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\in \mathbb {R} ^{n\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\in \mathbb {R} ^{n\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bb86023ba714c65dd2f31f3961152ad5d32872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.768ex; height:2.343ex;" alt="{\displaystyle C\in \mathbb {R} ^{n\times n}}"></span> such that </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 \mathbf {e} ^{k+1}=C\mathbf {e} ^{k}\quad \forall \,k\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} ^{k+1}=C\mathbf {e} ^{k}\quad \forall \,k\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3723c41b08310913c6122095a1c905af9882ab06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.067ex; height:2.843ex;" alt="{\displaystyle \mathbf {e} ^{k+1}=C\mathbf {e} ^{k}\quad \forall \,k\geq 0}"></span></dd></dl> <p>and this matrix is called the <i>iteration matrix</i>. An iterative method with a given iteration matrix <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> is called <i>convergent</i> if the following holds </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 _{k\rightarrow \infty }C^{k}=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\rightarrow \infty }C^{k}=0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b20e99201f1efae30670b03d4b800ff455fe53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.711ex; height:4.343ex;" alt="{\displaystyle \lim _{k\rightarrow \infty }C^{k}=0\,.}"></span></dd></dl> <p>An important theorem states that for a given iterative method and its iteration matrix <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> it is convergent if and only if its <a href="/wiki/Spectral_radius" title="Spectral radius">spectral radius</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3502886d455ab1f5536b4a3444bd7af7fddaf12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.778ex; height:2.843ex;" alt="{\displaystyle \rho (C)}"></span> is smaller than unity, that is, </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 \rho (C)<1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (C)<1\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b918cadc635aa15a35a6e7f9571834890b8b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.072ex; height:2.843ex;" alt="{\displaystyle \rho (C)<1\,.}"></span></dd></dl> <p>The basic iterative methods work by <a href="/wiki/Matrix_splitting" title="Matrix splitting">splitting</a> the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> into </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 A=M-N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>M</mi> <mo>−</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=M-N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c26fb458b97dfa0014f6a57d59be1b4cd59194" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.188ex; height:2.343ex;" alt="{\displaystyle A=M-N}"></span></dd></dl> <p>and here the matrix <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> should be easily <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a>. The iterative methods are now defined 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 M\mathbf {x} ^{k+1}=N\mathbf {x} ^{k}+b\,,\quad k\geq 0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>N</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\mathbf {x} ^{k+1}=N\mathbf {x} ^{k}+b\,,\quad k\geq 0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d97348cde84efcadb299cd95d021bee7473e481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.792ex; height:3.009ex;" alt="{\displaystyle M\mathbf {x} ^{k+1}=N\mathbf {x} ^{k}+b\,,\quad k\geq 0\,.}"></span></dd></dl> <p>From this follows that the iteration matrix is given 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 C=I-M^{-1}A=M^{-1}N\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>I</mi> <mo>−</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> <mo>=</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>N</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=I-M^{-1}A=M^{-1}N\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/141a76165486e97317fbbecf39adf8d2d8a2602e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.479ex; height:2.843ex;" alt="{\displaystyle C=I-M^{-1}A=M^{-1}N\,.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=6" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Basic examples of stationary iterative methods use a splitting of the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> such 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 A=D+L+U\,,\quad D:={\text{diag}}((a_{ii})_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>D</mi> <mo>+</mo> <mi>L</mi> <mo>+</mo> <mi>U</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>D</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>diag</mtext> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=D+L+U\,,\quad D:={\text{diag}}((a_{ii})_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679d0c484114c5f7ee92b86af8c4a748fc1653db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.505ex; height:2.843ex;" alt="{\displaystyle A=D+L+U\,,\quad D:={\text{diag}}((a_{ii})_{i})}"></span></dd></dl> <p>where <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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is only the diagonal part 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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is the strict lower <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular part</a> 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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. Respectively, <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is the strict upper triangular part 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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. </p> <ul><li><a href="/wiki/Modified_Richardson_iteration" title="Modified Richardson iteration">Richardson method</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M:={\frac {1}{\omega }}I\quad (\omega \neq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ω</mi> </mfrac> </mrow> <mi>I</mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M:={\frac {1}{\omega }}I\quad (\omega \neq 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162ad40e0c6264f72d40e3d2174c3fc57f112519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.48ex; height:5.176ex;" alt="{\displaystyle M:={\frac {1}{\omega }}I\quad (\omega \neq 0)}"></span></li> <li><a href="/wiki/Jacobi_method" title="Jacobi method">Jacobi method</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M:=D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:=</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M:=D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67cb4b92e2d7c65ebab9da539b2f75e385d7dea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.112ex; height:2.176ex;" alt="{\displaystyle M:=D}"></span></li> <li><a href="/wiki/Jacobi_method#Weighted_Jacobi_method" title="Jacobi method">Damped Jacobi method</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M:={\frac {1}{\omega }}D\quad (\omega \neq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ω</mi> </mfrac> </mrow> <mi>D</mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M:={\frac {1}{\omega }}D\quad (\omega \neq 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e06315def635749dfc0835c20afec2a6260c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.232ex; height:5.176ex;" alt="{\displaystyle M:={\frac {1}{\omega }}D\quad (\omega \neq 0)}"></span></li> <li><a href="/wiki/Gauss%E2%80%93Seidel_method" title="Gauss–Seidel method">Gauss–Seidel method</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M:=D+L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:=</mo> <mi>D</mi> <mo>+</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M:=D+L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696ef4d1b03304481bdcae995cc0621f7e1e2bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.535ex; height:2.343ex;" alt="{\displaystyle M:=D+L}"></span></li> <li><a href="/wiki/Successive_over-relaxation" title="Successive over-relaxation">Successive over-relaxation method</a> (SOR): <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:={\frac {1}{\omega }}D+L\quad (\omega \neq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ω</mi> </mfrac> </mrow> <mi>D</mi> <mo>+</mo> <mi>L</mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M:={\frac {1}{\omega }}D+L\quad (\omega \neq 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d02c8164f54bbab1557f52e801ee6d31e1df506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.656ex; height:5.176ex;" alt="{\displaystyle M:={\frac {1}{\omega }}D+L\quad (\omega \neq 0)}"></span></li> <li><a href="/wiki/Symmetric_successive_over-relaxation" title="Symmetric successive over-relaxation">Symmetric successive over-relaxation</a> (SSOR): <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:={\frac {1}{\omega (2-\omega )}}(D+\omega L)D^{-1}(D+\omega U)\quad (\omega \not \in \{0,2\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>+</mo> <mi>ω</mi> <mi>L</mi> <mo stretchy="false">)</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>D</mi> <mo>+</mo> <mi>ω</mi> <mi>U</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>∉</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M:={\frac {1}{\omega (2-\omega )}}(D+\omega L)D^{-1}(D+\omega U)\quad (\omega \not \in \{0,2\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f42e194a99d71741a39de3132f785a1bf18ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.492ex; height:6.009ex;" alt="{\displaystyle M:={\frac {1}{\omega (2-\omega )}}(D+\omega L)D^{-1}(D+\omega U)\quad (\omega \not \in \{0,2\})}"></span></li></ul> <p>Linear stationary iterative methods are also called <a href="/wiki/Relaxation_(iterative_method)" title="Relaxation (iterative method)">relaxation methods</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Krylov_subspace_methods">Krylov subspace methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=7" title="Edit section: Krylov subspace methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Krylov subspace methods<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> work by forming a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of the sequence of successive matrix powers times the initial residual (the <b>Krylov sequence</b>). The approximations to the solution are then formed by minimizing the residual over the subspace formed. The prototypical method in this class is the <a href="/wiki/Conjugate_gradient_method" title="Conjugate gradient method">conjugate gradient method</a> (CG) which assumes that the system matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite</a>. For symmetric (and possibly indefinite) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> one works with the <a href="/wiki/Minimal_residual_method" title="Minimal residual method">minimal residual method</a> (MINRES). In the case of non-symmetric matrices, methods such as the <a href="/wiki/Generalized_minimal_residual_method" title="Generalized minimal residual method">generalized minimal residual method</a> (GMRES) and the <a href="/wiki/Biconjugate_gradient_method" title="Biconjugate gradient method">biconjugate gradient method</a> (BiCG) have been derived. </p> <div class="mw-heading mw-heading4"><h4 id="Convergence_of_Krylov_subspace_methods">Convergence of Krylov subspace methods</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=8" title="Edit section: Convergence of Krylov subspace methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since these methods form a basis, it is evident that the method converges in <i>N</i> iterations, where <i>N</i> is the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practice <i>N</i> can be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of the <a href="/wiki/Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">spectrum</a> of the operator. </p> <div class="mw-heading mw-heading3"><h3 id="Preconditioners">Preconditioners</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=9" title="Edit section: Preconditioners"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such as <a href="/wiki/GMRES" class="mw-redirect" title="GMRES">GMRES</a> (alternatively, <a href="/wiki/Preconditioning" class="mw-redirect" title="Preconditioning">preconditioned</a> Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area. </p> <div class="mw-heading mw-heading2"><h2 id="Methods_of_successive_approximation">Methods of successive approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=10" title="Edit section: Methods of successive approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical methods relating to successive approximation include: </p> <ul><li><a href="/wiki/Babylonian_method" class="mw-redirect" title="Babylonian method">Babylonian method</a>, for finding square roots of numbers<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Fixed-point_iteration" title="Fixed-point iteration">Fixed-point iteration</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>Means of finding zeros of functions: <ul><li><a href="/wiki/Halley%27s_method" title="Halley's method">Halley's method</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li></ul></li> <li>Differential-equation matters: <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a>, on existence of solutions of differential equations</li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a>, for numerical solution of differential equations</li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="History">History</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=11" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a> used iterative methods to calculate the sine of 1° and <span class="texhtml mvar" style="font-style:italic;">π</span> in <i>The Treatise of Chord and Sine</i> to high precision. An early iterative method for <a href="/wiki/Gauss%E2%80%93Seidel_method" title="Gauss–Seidel method">solving a linear system</a> appeared in a letter of <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2019)">citation needed</span></a></i>]</sup>. </p><p>The theory of stationary iterative methods was solidly established with the work of <a href="/wiki/D.M._Young" class="mw-redirect" title="D.M. Young">D.M. Young</a> starting in the 1950s. The conjugate gradient method was also invented in the 1950s, with independent developments by <a href="/wiki/Cornelius_Lanczos" title="Cornelius Lanczos">Cornelius Lanczos</a>, <a href="/wiki/Magnus_Hestenes" title="Magnus Hestenes">Magnus Hestenes</a> and <a href="/wiki/Eduard_Stiefel" title="Eduard Stiefel">Eduard Stiefel</a>, but its nature and applicability were misunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>, especially the elliptic type. </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=Iterative_method&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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analysis</a></li> <li><a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">Root-finding algorithm</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=Iterative_method&action=edit&section=13" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAmritkarde_SturlerŚwirydowiczTafti2015" class="citation journal cs1">Amritkar, Amit; de Sturler, Eric; Świrydowicz, Katarzyna; Tafti, Danesh; Ahuja, Kapil (2015). "Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver". <i>Journal of Computational Physics</i>. <b>303</b>: 222. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1501.03358">1501.03358</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015JCoPh.303..222A">2015JCoPh.303..222A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jcp.2015.09.040">10.1016/j.jcp.2015.09.040</a>.</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+Physics&rft.atitle=Recycling+Krylov+subspaces+for+CFD+applications+and+a+new+hybrid+recycling+solver&rft.volume=303&rft.pages=222&rft.date=2015&rft_id=info%3Aarxiv%2F1501.03358&rft_id=info%3Adoi%2F10.1016%2Fj.jcp.2015.09.040&rft_id=info%3Abibcode%2F2015JCoPh.303..222A&rft.aulast=Amritkar&rft.aufirst=Amit&rft.au=de+Sturler%2C+Eric&rft.au=%C5%9Awirydowicz%2C+Katarzyna&rft.au=Tafti%2C+Danesh&rft.au=Ahuja%2C+Kapil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIterative+method" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Charles George Broyden and Maria Terasa Vespucci: <i>Krylov Solvers for Linear Algebraic Systems: Krylov Solvers</i>, Elsevier, ISBN 0-444-51474-0, (2004).</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_mathematics/">"Babylonian mathematics"</a>. <i>Babylonian mathematics</i>. December 1, 2000.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Babylonian+mathematics&rft.atitle=Babylonian+mathematics&rft.date=2000-12-01&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FBabylonian_mathematics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIterative+method" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFday1960" class="citation book cs1">day, Mahlon (November 2, 1960). <i>Fixed-point theorems for compact convex sets</i>. Mahlon M day.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fixed-point+theorems+for+compact+convex+sets&rft.pub=Mahlon+M+day&rft.date=1960-11-02&rft.aulast=day&rft.aufirst=Mahlon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIterative+method" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iterative_method&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Iterative_methods" class="extiw" title="commons:Category:Iterative methods">Iterative methods</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.netlib.org/linalg/html_templates/Templates.html">Templates for the Solution of Linear Systems</a></li> <li><a rel="nofollow" class="external text" href="http://www-users.cs.umn.edu/~saad/books.html">Y. 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optimization">Optimization</a>: <a href="/wiki/Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a class="mw-selflink selflink">methods</a>, and <a href="/wiki/Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Unconstrained_nonlinear" style="font-size:114%;margin:0 4em"><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Unconstrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Golden-section_search" title="Golden-section search">Golden-section search</a></li> <li><a href="/wiki/Powell%27s_method" title="Powell's method">Powell's method</a></li> <li><a href="/wiki/Line_search" title="Line search">Line search</a></li> <li><a href="/wiki/Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li> <li><a href="/wiki/Successive_parabolic_interpolation" title="Successive parabolic interpolation">Successive parabolic interpolation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Gradient" title="Gradient">Gradients</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Local_convergence" title="Local convergence">Convergence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Trust_region" title="Trust region">Trust region</a></li> <li><a href="/wiki/Wolfe_conditions" title="Wolfe conditions">Wolfe conditions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quasi-Newton_method" title="Quasi-Newton method">Quasi–Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Berndt%E2%80%93Hall%E2%80%93Hall%E2%80%93Hausman_algorithm" title="Berndt–Hall–Hall–Hausman algorithm">Berndt–Hall–Hall–Hausman</a></li> <li><a href="/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">Broyden–Fletcher–Goldfarb–Shanno</a> and <a href="/wiki/Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li> <li><a href="/wiki/Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li> <li><a href="/wiki/Symmetric_rank-one" title="Symmetric rank-one">Symmetric rank-one (SR1)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Other methods</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li> <li><a href="/wiki/Gradient_descent" title="Gradient descent">Gradient</a></li> <li><a href="/wiki/Mirror_descent" title="Mirror descent">Mirror</a></li> <li><a href="/wiki/Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li> <li><a href="/wiki/Powell%27s_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">Truncated Newton</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Max_paraboloid.svg" class="mw-file-description" title="Optimization computes maxima and minima."><img alt="Graph of a strictly concave quadratic function with unique maximum." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/150px-Max_paraboloid.svg.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/225px-Max_paraboloid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/300px-Max_paraboloid.svg.png 2x" data-file-width="700" data-file-height="560" /></a><figcaption>Optimization computes maxima and minima.</figcaption></figure></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Constrained_nonlinear" style="font-size:114%;margin:0 4em"><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Constrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrier_function" title="Barrier function">Barrier methods</a></li> <li><a href="/wiki/Penalty_method" title="Penalty method">Penalty methods</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Differentiable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li> <li><a href="/wiki/Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li> <li><a href="/wiki/Successive_linear_programming" title="Successive linear programming">Successive linear programming</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_optimization" style="font-size:114%;margin:0 4em"><a href="/wiki/Convex_optimization" title="Convex optimization">Convex optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Convex_minimization" class="mw-redirect" title="Convex minimization">Convex<br /> minimization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li> <li><a href="/wiki/Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li> <li><a href="/wiki/Subgradient_method" title="Subgradient method">Subgradient method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_programming" title="Linear programming">Linear</a> and<br /><a href="/wiki/Quadratic_programming" title="Quadratic programming">quadratic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_programming#Interior_point" title="Linear programming">Interior point</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_scaling" title="Affine scaling">Affine scaling</a></li> <li><a href="/wiki/Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li> <li><a href="/wiki/Karmarkar%27s_algorithm" title="Karmarkar's algorithm">Projective algorithm of Karmarkar</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Matroid" title="Matroid">Basis-</a><a href="/wiki/Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li> <li><a href="/wiki/Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li> <li><a href="/wiki/Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li> <li><a href="/wiki/Lemke%27s_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</a></li> <li><a href="/wiki/Active-set_method" title="Active-set method">Active-set method</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial" style="font-size:114%;margin:0 4em"><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Combinatorial</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li> <li><a href="/wiki/Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li> <li><a href="/wiki/Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li> <li><a href="/wiki/Integer_programming" title="Integer programming">Integer programming</a> <ul><li><a href="/wiki/Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="/wiki/Branch_and_cut" title="Branch and cut">cut</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph<br /> algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Minimum_spanning_tree" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Minimum_spanning_tree" title="Minimum spanning tree">Minimum<br /> spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bor%C5%AFvka%27s_algorithm" title="Borůvka's algorithm">Borůvka</a></li> <li><a href="/wiki/Prim%27s_algorithm" title="Prim's algorithm">Prim</a></li> <li><a href="/wiki/Kruskal%27s_algorithm" title="Kruskal's algorithm">Kruskal</a></li></ul> </div></td></tr></tbody></table><div> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Shortest_path" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a> <ul><li><a href="/wiki/Shortest_Path_Faster_Algorithm" class="mw-redirect" title="Shortest Path Faster Algorithm">SPFA</a></li></ul></li> <li><a href="/wiki/Dijkstra%27s_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li> <li><a href="/wiki/Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Flow_network" title="Flow network">Network flows</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dinic%27s_algorithm" title="Dinic's algorithm">Dinic</a></li> <li><a href="/wiki/Edmonds%E2%80%93Karp_algorithm" title="Edmonds–Karp algorithm">Edmonds–Karp</a></li> <li><a href="/wiki/Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li> <li><a href="/wiki/Push%E2%80%93relabel_maximum_flow_algorithm" title="Push–relabel maximum flow algorithm">Push–relabel maximum flow</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Metaheuristics" style="font-size:114%;margin:0 4em"><a href="/wiki/Metaheuristic" title="Metaheuristic">Metaheuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li> <li><a href="/wiki/Hill_climbing" title="Hill climbing">Hill climbing</a></li> <li><a href="/wiki/Local_search_(optimization)" title="Local search (optimization)">Local search</a></li> <li><a href="/wiki/Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li> <li><a href="/wiki/Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li> <li><a href="/wiki/Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li> <li><a href="/wiki/Tabu_search" title="Tabu search">Tabu search</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Comparison_of_optimization_software" title="Comparison of optimization software">Software</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" 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