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<title>Matrix-free methods</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Matrix-free methods</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Computational_mathematics" title="Computational mathematics">computational mathematics</a>, a <b>matrix-free method</b> is an algorithm for solving a <a href="Linear_system_of_equations" class="mw-redirect" title="Linear system of equations">linear system of equations</a> or an <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> problem that does not store the coefficient <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> explicitly, but accesses the matrix by evaluating matrix-vector products.<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> Such methods can be preferable when the matrix is so big that storing and manipulating it would cost a lot of memory and computing time, even with the use of methods for <a href="Sparse_matrix" title="Sparse matrix">sparse matrices</a>. Many <a href="Iterative_method" title="Iterative method">iterative methods</a> allow for a matrix-free implementation, including:

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<ul><li>the <a href="Power_method" class="mw-redirect" title="Power method">power method</a>,</li>

<li>the <a href="Lanczos_algorithm" title="Lanczos algorithm">Lanczos algorithm</a>,<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></li>

<li>Locally Optimal Block Preconditioned Conjugate Gradient Method (<a href="LOBPCG" title="LOBPCG">LOBPCG</a>),<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>Wiedemann's coordinate recurrence algorithm,<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>the <a href="Conjugate_gradient_method" title="Conjugate gradient method">conjugate gradient method</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Krylov_subspace" title="Krylov subspace">Krylov subspace methods</a>.</li></ul>

<p>Distributed solutions have also been explored using coarse-grain parallel software systems to achieve homogeneous solutions of linear systems.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>It is generally used in solving non-linear equations like Euler's equations in <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a>. Matrix-free conjugate gradient method has been applied in the non-linear elasto-plastic finite element solver.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Solving these equations requires the calculation of the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> which is costly in terms of CPU time and storage. To avoid this expense, matrix-free methods are employed. In order to remove the need to calculate the Jacobian, the Jacobian vector product is formed instead, which is in fact a vector itself. Manipulating and calculating this vector is easier than working with a large matrix or linear system.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFLangvilleMeyer2006" class="citation cs2"><a href="Amy_Langville" title="Amy Langville">Langville, Amy N.</a>; Meyer, Carl D. (2006), <i>Google's PageRank and beyond: the science of search engine rankings</i>, <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, p. 40, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-12202-1</bdi></cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFCoppersmith1993" class="citation cs2">Coppersmith, Don (1993), "Solving linear equations over GF(2): Block Lanczos algorithm", <i>Linear Algebra and Its Applications</i>, <b>192</b>: <span class="nowrap">33–</span>60, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0024-3795%2893%2990235-G">10.1016/0024-3795(93)90235-G</a></span></cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFKnyazev2001" class="citation journal cs1">Knyazev, Andrew V. (2001). "Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method". <i><a href="SIAM_Journal_on_Scientific_Computing" title="SIAM Journal on Scientific Computing">SIAM Journal on Scientific Computing</a></i>. <b>23</b> (2): <span class="nowrap">517–</span>541. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001SJSC...23..517K">2001SJSC...23..517K</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.2862">10.1.1.34.2862</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS1064827500366124">10.1137/S1064827500366124</a>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFWiedemann1986" class="citation cs2">Wiedemann, D. (1986), <a rel="nofollow" class="external text" href="http://www.enseignement.polytechnique.fr/profs/informatique/Francois.Morain/Master1/Crypto/projects/Wiedemann86.pdf">"Solving sparse linear equations over finite fields"</a> <span class="cs1-format">(PDF)</span>, <i>IEEE Transactions on Information Theory</i>, <b>32</b>: <span class="nowrap">54–</span>62, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIT.1986.1057137">10.1109/TIT.1986.1057137</a></cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFLamacchiaOdlyzko1991" class="citation cs2">Lamacchia, B. A.; Odlyzko, A. M. (1991), "Solving Large Sparse Linear Systems Over Finite Fields", <i>Advances in Cryptology – CRYPT0' 90</i>, Lecture Notes in Computer Science, vol. 537, p. 109, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-38424-3_8">10.1007/3-540-38424-3_8</a></span>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-54508-8</bdi></cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFKaltofenLobo1996" class="citation cs2">Kaltofen, E.; Lobo, A. (1996), "Distributed Matrix-Free Solution of Large Sparse Linear Systems over Finite Fields", <i>Algorithmica</i>, vol. 24, no. <span class="nowrap">3–</span>4, pp. <span class="nowrap">311–</span>348, <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.7470">10.1.1.17.7470</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FPL00008266">10.1007/PL00008266</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13305650">13305650</a></cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFPrabhuneKrishnan2020" class="citation journal cs1">Prabhune, Bhagyashree C.; Krishnan, Suresh (4 March 2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cad.2020.102829">"A fast matrix-free elasto-plastic solver for predicting residual stresses in additive manufacturing"</a>. <i>Computer Aided Design</i>. <b>123</b>: 102829. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cad.2020.102829">10.1016/j.cad.2020.102829</a></span>.</cite></span>

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</style><div id="Numerical_linear_algebra64" style="font-size:114%;margin:0 4em"><a href="Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Floating_point" class="mw-redirect" title="Floating point">Floating point</a></li>

<li><a href="Numerical_stability" title="Numerical stability">Numerical stability</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Problems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="System_of_linear_equations" title="System of linear equations">System of linear equations</a></li>

<li><a href="Matrix_decomposition" title="Matrix decomposition">Matrix decompositions</a></li>

<li><a href="Matrix_multiplication" title="Matrix multiplication">Matrix multiplication</a> (<a href="Matrix_multiplication_algorithm" title="Matrix multiplication algorithm">algorithms</a>)</li>

<li><a href="Matrix_splitting" title="Matrix splitting">Matrix splitting</a></li>

<li><a href="Sparse_matrix" title="Sparse matrix">Sparse problems</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Hardware</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="CPU_cache" title="CPU cache">CPU cache</a></li>

<li><a href="Cache-oblivious_algorithm" title="Cache-oblivious algorithm">Cache-oblivious algorithm</a></li>

<li><a href="Multiprocessing" title="Multiprocessing">Multiprocessing</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Automatically_Tuned_Linear_Algebra_Software" title="Automatically Tuned Linear Algebra Software">ATLAS</a></li>

<li><a href="MATLAB" title="MATLAB">MATLAB</a></li>

<li><a href="Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">Basic Linear Algebra Subprograms (BLAS)</a></li>

<li><a href="LAPACK" title="LAPACK">LAPACK</a></li>

<li><a href="Comparison_of_linear_algebra_libraries" title="Comparison of linear algebra libraries">Specialized libraries</a></li>

<li><a href="Comparison_of_numerical-analysis_software" title="Comparison of numerical-analysis software">General purpose software</a></li></ul>

</div></td></tr></tbody></table></div>

<p><br>

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