Document Zbl 1361.65016

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Algorithms</a> <a href="/?q=in%3A361537" title="Articles in this Issue">73, No. 4, 927-955 (2016)</a>. </div> <div class="abstract">To avoid complex arithmetic, the authors consider an equivalent $2\times 2$ real block system (with diagonal blocks $W$ and $T$) to a given complex linear system and propose a one-parameter iterative method called scaled-splitted iteration (SCSP). The eigenvalues of the iteration matrix of this method are investigated for the case of $W,T$ being symmetric positive semi-definite, its spectral radius is shown to be less than $1$ when the iteration parameter obeys certain inequalities depending on the extreme generalized eigenvalues of the pair $W,T$. Similarly, the optimal iteration parameter is obtained in dependence of those extreme eigenvalues. The splitting also gives rise to a preconditioner, and the preconditioned matrix is shown to have eigenvalues on a straight line orthogonal to the real axis and with center $(1,0)$. Similar results are obtained for an iteration (called PGSOR) proposed by the authors in [Numer. Linear Algebra Appl. 22, No. 4, 761–776 (2015; <a href="/1363.65049">Zbl 1363.65049</a>)]. Here, the eigenvalues of the preconditioned matrix are contained in a circle around the same centre $(1,0)$.<br class="zbmathjax-paragraph">They show also results of extensive numerical experiments comparing their methods to GMRES and its restarted or ILU-preconditioned variants on 4 example systems (including a discrete 3D Helmholtz equation). Whereas for small systems the Matlab backslash operator is faster, for large systems the preconditioned forms of GMRES and also a further method proposed by <span class="zbmathjax-textit">Z.-Z. Bai</span> [Sci. China, Math. 56, No. 12, 2523–2538 (2013; <a href="/1305.65114">Zbl 1305.65114</a>)], perform well and produce computing times not too far from each other.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A4361">Gisbert Stoyan (Budapest)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6673277">Cited in <strong>36</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F10" title="MSC2020">65F10</a> </td> <td class="space"> Iterative numerical methods for linear systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acomplex+symmetric+linear+systems">complex symmetric linear systems</a>; <a href="/?q=ut%3Anew+iterative+solution+approach">new iterative solution approach</a>; <a href="/?q=ut%3Asymmetric+positive+definite+preconditioner">symmetric positive definite preconditioner</a>; <a href="/?q=ut%3Ascaled-splitted+iteration">scaled-splitted iteration</a>; <a href="/?q=ut%3Aeigenvalues+of+the+iteration+matrix">eigenvalues of the iteration matrix</a>; <a href="/?q=ut%3Anumerical+experiment">numerical experiment</a>; <a href="/?q=ut%3AHelmholtz+equation">Helmholtz equation</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1305.65114">Zbl 1305.65114</a>; <a href="/1363.65049">Zbl 1363.65049</a></div> <div class="software"> <h3>Software:</h3><a href="/software/558">Matlab</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1361.65016" data-ciurl="/ci/06673277" data-biburl="/bibtex/06673277.bib" data-amsurl="/amsrefs/06673277.bib" data-xmlurl="/xml/06673277.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06673277.pdf" title="Zbl 1361.65016 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s11075-016-0123-x" aria-label="DOI for “A new iterative method for solving a class of complex symmetric system of linear equations”" title="10.1007/s11075-016-0123-x">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Hiptmair, R.: Finite elements in computational electromagnetism. 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