Document Zbl 1091.65041

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Comput.</a> <a href="/?q=in%3A124338" title="Articles in this Issue">75, No. 254, 791-815 (2006)</a>. </div> <div class="abstract">Author’s abstract: For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. <br class="zbmathjax-paragraph">Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as generalized minimal residual (GMRES) methods and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A3165">Dietrich Braess (Bochum)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5015493">Cited in <strong>204</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F35" title="MSC2020">65F35</a> </td> <td class="space"> Numerical computation of matrix norms, conditioning, scaling </td> </tr><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><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F50" title="MSC2020">65F50</a> </td> <td class="space"> Computational methods for sparse matrices </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Apreconditioner">preconditioner</a>; <a href="/?q=ut%3Ageneralized+minimal+residual+%28GMRES%29+methods">generalized minimal residual (GMRES) methods</a>; <a href="/?q=ut%3Alarge+sparse+systems">large sparse systems</a>; <a href="/?q=ut%3Apreconditioning">preconditioning</a>; <a href="/?q=ut%3Ablock+Jacobi-type+preconditioner">block Jacobi-type preconditioner</a>; <a href="/?q=ut%3Ablock+Gauss-Seidel+type+preconditioner">block Gauss-Seidel type preconditioner</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 1091.65041" data-ciurl="/ci/05015493" data-biburl="/bibtex/05015493.bib" data-amsurl="/amsrefs/05015493.bib" data-xmlurl="/xml/05015493.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05015493.pdf" title="Zbl 1091.65041 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/10.1090/S0025-5718-05-01801-6" aria-label="DOI for “Structured preconditioners for nonsingular matrices of block two-by-two structures”" title="10.1090/S0025-5718-05-01801-6">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">Habib Ammari, Gang Bao, and Aihua W. 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