Document Zbl 1496.15014

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1496.15014 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">×</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools & Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/1496.15014?"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open — the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span> <a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="an:1496.15014" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span> </span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>  </span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>  </span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>  </span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>  </span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>  </span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>  </span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>  </span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>  </span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>  </span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>  </span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>  </span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>  </span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>  </span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>  </span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>  </span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>  </span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>  </span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a <span class="item">&</span> b </span><span> </span>logical and (default)</a></li> <li><a href="#"><span class="token">a <span class="item">|</span> b </span><span> </span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab   </span><span> </span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>  </span><span> </span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab c<span class="item">"</span></span><span> </span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab c<span class="item">)</span></span><span> </span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki & ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/rakshit.suman" title="Author Profile">Rakshit, Suman</a>; <a href="/authors/khare.swanand-r" title="Author Profile">Khare, S. R.</a></div> <h2 class="title"> <strong>Solution of the linearly structured partial polynomial inverse eigenvalue problem.</strong> <i>(English)</i> <a class="label nowrap" href="/1496.15014">Zbl 1496.15014</a> </h2> <div class="source"> <a href="/serials/482" title="Journal Profile">J. Comput. Appl. Math.</a> <a href="/?q=in%3A482351" title="Articles in this Issue">411, Article ID 114242, 14 p. (2022)</a>. </div> <div class="abstract">This paper deals with the linearly structured partial polynomial inverse eigenvalue problem (LPPIEP) for matrices $A_i \in \mathbb{R}^{n \times n}$, $i=0,1,\ldots,k-1$ of such that the matrix polynomial \[ P(\lambda)=\lambda^k I_n + \sum_{i=0}^{k-1} \lambda^{i} A_i \] has $m$ ($1 \leq m \leq kn$) prescribed eigenvalues and eigenvectors. Note that the coefficients of $P(\lambda)$ may have different linear structures. For instance, some of the matrices can be diagonal matrices, other symmetric tridiagonal, etc. In order to address these situations, the authors study another problem, namely:<br class="zbmathjax-paragraph">Problem (LPPIEP-I). Given two positive integers $k$, $n$ and a set of partial eigenpairs $(\lambda_j, \phi_j)_{j=1}^m$, $1 \leq m \leq kn$, construct a monic matrix polynomial $P(\lambda)$ of degree $k$ in such a way that matrices $A_i \in\mathcal L_i$, $i=0,1,\ldots,k-1$ and $P(\lambda)$ has the specified set $(\lambda_j, \phi_j)_{j=1}^m$, $1 \leq m \leq kn$ as its eigenpairs. <br class="zbmathjax-paragraph">The authors propose an approach for the original LPPIEP. Their results solve some open problems in the theory of polynomial inverse eigenvalue problems. The proposed solution gives an alternative answer to the inverse eigenvalue problems considered in the literature for a specific structure such as symmetric, skew-symmetric, etc. However, in this paper the framework is generalized to an arbitrary linear structure. In particular, for any given specific linear structure, this paper presents a way that systematically generates a system of linear equations and obtain the solution of the corresponding inverse eigenvalue problem. <br class="zbmathjax-paragraph">The contributions made in this paper can be summarized as follows: <dl class="zbmathjax-itemize"><dt class="zbmathjax-item1">(i)</dt><dd class="zbmathjax-item1"> the proposed methods are capable to solve LPPIEP and LPPIEP-I using a set of $m$, $1 \leq m \leq kn$, eigenpairs without imposing any restrictions on it;</dd><dt class="zbmathjax-item1">(ii)</dt><dd class="zbmathjax-item1"> it is possible to study the partial inverse eigenvalue problem for different linear structures at once;</dd><dt class="zbmathjax-item1">(iii)</dt><dd class="zbmathjax-item1"> the authors derive some necessary and sufficient conditions on the eigendata for the existence of solution to these problems;</dd><dt class="zbmathjax-item1">(iv)</dt><dd class="zbmathjax-item1"> explicit analytical expressions for the family of matrix polynomials are provided;</dd><dt class="zbmathjax-item1">(v)</dt><dd class="zbmathjax-item1"> sensitivity of the solutions to the perturbations of the eigendata is also discussed;</dd><dt class="zbmathjax-item1">(vi)</dt><dd class="zbmathjax-item1"> the minimum norm solutions of LPPIEP and LPPIEP-I are obtained.</dd></dl><div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10330">Juan Ramón Torregrosa Sánchez (Valencia)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7531707">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A15A29" title="MSC2020">15A29</a> </td> <td class="space"> Inverse problems in linear algebra </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15A18" title="MSC2020">15A18</a> </td> <td class="space"> Eigenvalues, singular values, and eigenvectors </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15A06" title="MSC2020">15A06</a> </td> <td class="space"> Linear equations (linear algebraic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15A83" title="MSC2020">15A83</a> </td> <td class="space"> Matrix completion problems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amatrix+polynomial">matrix polynomial</a>; <a href="/?q=ut%3Alinearly+structure+matrix">linearly structure matrix</a>; <a href="/?q=ut%3Apolynomial+inverse+eigenvalue+problem">polynomial inverse eigenvalue problem</a>; <a href="/?q=ut%3Apolynomial+eigenvalue+problem">polynomial eigenvalue problem</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 1496.15014" data-ciurl="/ci/07531707" data-biburl="/bibtex/07531707.bib" data-amsurl="/amsrefs/07531707.bib" data-xmlurl="/xml/07531707.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07531707.pdf" title="Zbl 1496.15014 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/10.1016/j.cam.2022.114242" aria-label="DOI for “Solution of the linearly structured partial polynomial inverse eigenvalue problem”" title="10.1016/j.cam.2022.114242">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1904.10339"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Gohberg, I.; Lancaster, P.; Rodman, L., Matrix Polynomials (2005), Springer</td> </tr><tr> <td>[2]</td> <td class="space">Datta, B. N., Numerical Linear Algebra and Applications (2010), SIAM · <a href="/1187.65027" class="nowrap">Zbl 1187.65027</a></td> </tr><tr> <td>[3]</td> <td class="space">Tisseur, F.; Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev., 43, 2, 235-286 (2001) · <a href="/0985.65028" class="nowrap">Zbl 0985.65028</a></td> </tr><tr> <td>[4]</td> <td class="space">Cai, Y. F.; Kuo, Y. C.; Lin, W. W.; Xu, S. F., Solutions to a quadratic inverse eigenvalue problem, Linear Algebra Appl., 430, 5, 1590-1606 (2009) · <a href="/1165.65017" class="nowrap">Zbl 1165.65017</a></td> </tr><tr> <td>[5]</td> <td class="space">Dmytryshyn, A., Skew-Symmetric Matrix Pencils: Stratification Theory and Tools (2014), Umeå universitet, (Ph.D. thesis)</td> </tr><tr> <td>[6]</td> <td class="space">Mehrmann, V.; Watkins, D., Polynomial eigenvalue problems with Hamiltonian structure, Electron. Trans. Numer. Anal., 13, 106-118 (2002) · <a href="/1065.65054" class="nowrap">Zbl 1065.65054</a></td> </tr><tr> <td>[7]</td> <td class="space">Bai, Z. J., Symmetric tridiagonal inverse quadratic eigenvalue problems with partial eigendata, Inverse Problems, 24, 1, 1-29 (2007)</td> </tr><tr> <td>[8]</td> <td class="space">Gladwell, G. M.L., Inverse problems in vibration, Appl. Mech. Rev., 39, 7, 1013-1018 (1986) · <a href="/0588.73110" class="nowrap">Zbl 0588.73110</a></td> </tr><tr> <td>[9]</td> <td class="space">Barcilon, V., On the solution of inverse eigenvalue problems of high orders, Geophys. J. Int., 39, 1, 143-154 (1974) · <a href="/0288.34019" class="nowrap">Zbl 0288.34019</a></td> </tr><tr> <td>[10]</td> <td class="space">Chu, M. T., Inverse eigenvalue problems, SIAM Rev., 40, 1, 1-39 (1998) · <a href="/0915.15008" class="nowrap">Zbl 0915.15008</a></td> </tr><tr> <td>[11]</td> <td class="space">Datta, B. N.; Sokolov, V., Quadratic inverse eigenvalue problems, active vibration control and model updating, Appl. Comput. Math., 8, 2, 170-191 (2009) · <a href="/1194.15008" class="nowrap">Zbl 1194.15008</a></td> </tr><tr> <td>[12]</td> <td class="space">De Terán, F.; Dopico, F. M.; Van Dooren, P., Matrix polynomials with completely prescribed eigenstructure, SIAM J. Matrix Anal. Appl., 36, 1, 302-328 (2015) · <a href="/1315.15021" class="nowrap">Zbl 1315.15021</a></td> </tr><tr> <td>[13]</td> <td class="space">Cai, J.; Chen, J., Iterative solutions of generalized inverse eigenvalue problem for partially bisymmetric matrices, Linear Multilinear Algebra, 65, 8, 1643-1654 (2017) · <a href="/1369.15011" class="nowrap">Zbl 1369.15011</a></td> </tr><tr> <td>[14]</td> <td class="space">Gigola, S.; Lebtahi, L.; Thome, N., The inverse eigenvalue problem for a hermitian reflexive matrix and the optimization problem, J. Comput. Appl. Math., 291, 449-457 (2016) · <a href="/1329.15047" class="nowrap">Zbl 1329.15047</a></td> </tr><tr> <td>[15]</td> <td class="space">Laurie, D. P., Solving the inverse eigenvalue problem via the eigenvector matrix, J. Comput. Appl. Math., 35, 1-3, 277-289 (1991) · <a href="/0727.65029" class="nowrap">Zbl 0727.65029</a></td> </tr><tr> <td>[16]</td> <td class="space">Liu, Z.; Zhang, Y.; Ferreira, C.; Ralha, R., On inverse eigenvalue problems for block Toeplitz matrices with toeplitz blocks, Appl. Math. Comput., 216, 6, 1819-1830 (2010) · <a href="/1192.65040" class="nowrap">Zbl 1192.65040</a></td> </tr><tr> <td>[17]</td> <td class="space">Moghaddam, M. R.; Mirzaei, H.; Ghanbari, K., On the generalized inverse eigenvalue problem of constructing symmetric pentadiagonal matrices from three mixed eigendata, Linear Multilinear Algebra, 63, 6, 1154-1166 (2015) · <a href="/1315.15009" class="nowrap">Zbl 1315.15009</a></td> </tr><tr> <td>[18]</td> <td class="space">Mourad, B.; Abbas, H.; Moslehian, M. S., A note on the inverse spectral problem for symmetric doubly stochastic matrices, Linear Multilinear Algebra, 63, 12, 2537-2545 (2015) · <a href="/1333.15030" class="nowrap">Zbl 1333.15030</a></td> </tr><tr> <td>[19]</td> <td class="space">Šmigoc*, H., Construction of nonnegative matrices and the inverse eigenvalue problem, Linear Multilinear Algebra, 53, 2, 85-96 (2005) · <a href="/1080.15007" class="nowrap">Zbl 1080.15007</a></td> </tr><tr> <td>[20]</td> <td class="space">Wei, Y.; Dai, H., An inverse eigenvalue problem for Jacobi matrix, Appl. Math. Comput., 251, 633-642 (2015) · <a href="/1328.15030" class="nowrap">Zbl 1328.15030</a></td> </tr><tr> <td>[21]</td> <td class="space">Xu, S., On inverse spectrum problems for normal nonnegative matrices, Linear Multilinear Algebra, 34, 3-4, 353-364 (1993) · <a href="/0815.15006" class="nowrap">Zbl 0815.15006</a></td> </tr><tr> <td>[22]</td> <td class="space">Yuan, Y. X.; Dai, H., A generalized inverse eigenvalue problem in structural dynamic model updating, J. Comput. Appl. Math., 226, 1, 42-49 (2009) · <a href="/1175.65049" class="nowrap">Zbl 1175.65049</a></td> </tr><tr> <td>[23]</td> <td class="space">Zhao, L.; Hu, X.; Zhang, L., Inverse eigenvalue problems for bisymmetric matrices under a central principal submatrix constraint, Linear Multilinear Algebra, 59, 2, 117-128 (2011) · <a href="/1211.15013" class="nowrap">Zbl 1211.15013</a></td> </tr><tr> <td>[24]</td> <td class="space">Zhao, M.; Jia, Z.; Wei, M., Symmetric inverse generalized eigenvalue problem with submatrix constraints in structural dynamic model updating, Electron. J. Linear Algebra, 22, 729-742 (2011) · <a href="/1254.15018" class="nowrap">Zbl 1254.15018</a></td> </tr><tr> <td>[25]</td> <td class="space">De Sa, E. M., Imbedding conditions for $\lambda $-matrices, Linear Algebra Appl., 24, 33-50 (1979) · <a href="/0395.15009" class="nowrap">Zbl 0395.15009</a></td> </tr><tr> <td>[26]</td> <td class="space">Elhay, S.; Ram, Y. M., An affine inverse eigenvalue problem, Inverse Problems, 18, 2, 455-466 (2002) · <a href="/0999.65020" class="nowrap">Zbl 0999.65020</a></td> </tr><tr> <td>[27]</td> <td class="space">Rakshit, S.; Khare, S. R., Symmetric band structure preserving finite element model updating with no spillover, Mech. Syst. Signal Process., 116, 415-431 (2019)</td> </tr><tr> <td>[28]</td> <td class="space">Ram, Y. M., Inverse eigenvalue problem for a modified vibrating system, SIAM J. Appl. Math., 53, 6, 1762-1775 (1993) · <a href="/0802.15011" class="nowrap">Zbl 0802.15011</a></td> </tr><tr> <td>[29]</td> <td class="space">Datta, B. N.; Sarkissian, D., Theory and computations of some inverse eigenvalue problems for the quadratic pencil, Contemp. Math., 280, 221-240 (2001) · <a href="/1008.34075" class="nowrap">Zbl 1008.34075</a></td> </tr><tr> <td>[30]</td> <td class="space">Datta, B. N.; Sokolov, V., A solution of the affine quadratic inverse eigenvalue problem, Linear Algebra Appl., 434, 7, 1745-1760 (2011) · <a href="/1211.65047" class="nowrap">Zbl 1211.65047</a></td> </tr><tr> <td>[31]</td> <td class="space">Kuo, Y. C.; Lin, W. W.; Xu, S. F., On a general solution of partially described inverse quadratic eigenvalue problems and its applications, SIAM J. Matrix Anal. Appl. (2005)</td> </tr><tr> <td>[32]</td> <td class="space">Lancaster, P.; Zaballa, I., On the inverse symmetric quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 35, 1, 254-278 (2014) · <a href="/1320.47007" class="nowrap">Zbl 1320.47007</a></td> </tr><tr> <td>[33]</td> <td class="space">Lin, M. M.; Dong, B.; Chu, M. T., Inverse mode problems for real and symmetric quadratic models, Inverse Problems, 26, 6, 1-20 (2010) · <a href="/1206.37027" class="nowrap">Zbl 1206.37027</a></td> </tr><tr> <td>[34]</td> <td class="space">Qian, J.; Cheng, M., Quadratic inverse eigenvalue problem for damped gyroscopic systems, J. Comput. Appl. Math., 255, 306-312 (2014) · <a href="/1291.65235" class="nowrap">Zbl 1291.65235</a></td> </tr><tr> <td>[35]</td> <td class="space">Rakshit, S.; Khare, S. R.; Datta, B. N., Symmetric tridiagonal partial quadratic inverse eigenvalue problem (SPQIEP), (2018 International Conference on Power, Signals, Control and Computation. 2018 International Conference on Power, Signals, Control and Computation, EPSCICON (2018), IEEE), 1-6</td> </tr><tr> <td>[36]</td> <td class="space">Ram, Y. M.; Elba, S., An inverse eigenvalue problem for the symmetric tridiagonal quadratic pencil with application to damped oscillatory systems, SIAM J. Appl. Math., 56, 1, 232-244 (1996) · <a href="/0853.15005" class="nowrap">Zbl 0853.15005</a></td> </tr><tr> <td>[37]</td> <td class="space">Yuan, Y.; Zhou, L.; Chen, J., An inverse quadratic eigenvalue problem for damped gyroscopic systems, Linear Multilinear Algebra, 67, 1-14 (2019)</td> </tr><tr> <td>[38]</td> <td class="space">Yuan, Y.; Dai, H., On a class of inverse quadratic eigenvalue problem, J. Comput. Appl. Math., 235, 8, 2662-2669 (2011) · <a href="/1209.15017" class="nowrap">Zbl 1209.15017</a></td> </tr><tr> <td>[39]</td> <td class="space">Zhao, K.; Yao, G., Application of the alternating direction method for an inverse monic quadratic eigenvalue problem, Appl. Math. Comput., 244, 32-41 (2014) · <a href="/1335.15020" class="nowrap">Zbl 1335.15020</a></td> </tr><tr> <td>[40]</td> <td class="space">Kuo, Y. C.; Lin, W. W.; Xu, S. F., Solutions of the partially described inverse quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 29, 1, 33-53 (2006) · <a href="/1137.65020" class="nowrap">Zbl 1137.65020</a></td> </tr><tr> <td>[41]</td> <td class="space">Kuo, Y. C.; Datta, B. N., Quadratic model updating with no spill-over and incomplete measured data: Existence and computation of solution, Linear Algebra Appl., 436, 7, 2480-2493 (2012) · <a href="/1236.65037" class="nowrap">Zbl 1236.65037</a></td> </tr><tr> <td>[42]</td> <td class="space">Rakshit, S.; Khare, S. R.; Datta, B. N., Symmetric tridiagonal structure preserving finite element model updating problem for the quadratic model, Mech. Syst. Signal Process., 107, 278-290 (2018)</td> </tr><tr> <td>[43]</td> <td class="space">Batzke, L.; Mehl, C., On the inverse eigenvalue problem for T-alternating and T-palindromic matrix polynomials, Linear Algebra Appl., 452, 172-191 (2014) · <a href="/1288.65045" class="nowrap">Zbl 1288.65045</a></td> </tr><tr> <td>[44]</td> <td class="space">McLaughlin, J. R., An inverse eigenvalue problem of order four, SIAM J. Math. Anal., 7, 5, 646-661 (1976) · <a href="/0338.34016" class="nowrap">Zbl 0338.34016</a></td> </tr><tr> <td>[45]</td> <td class="space">Cai, Y. F.; Qian, J.; Xu, S. F., Robust partial pole assignment problem for high order control systems, Automatica, 48, 7, 1462-1466 (2012) · <a href="/1246.93056" class="nowrap">Zbl 1246.93056</a></td> </tr><tr> <td>[46]</td> <td class="space">Mao, X.; Dai, H., Minimum norm partial eigenvalue assignment of high order linear system with no spill-over, Linear Algebra Appl., 438, 5, 2136-2154 (2013) · <a href="/1257.93047" class="nowrap">Zbl 1257.93047</a></td> </tr><tr> <td>[47]</td> <td class="space">Datta, B. N.; Deng, S.; Sokolov, V.; Sarkissian, D., An optimization technique for damped model updating with measured data satisfying quadratic orthogonality constraint, Mech. Syst. Signal Process., 23, 6, 1759-1772 (2009)</td> </tr><tr> <td>[48]</td> <td class="space">Landau, H. J., The inverse eigenvalue problem for real symmetric Toeplitz matrices, J. Amer. Math. Soc., 7, 3, 749-767 (1994) · <a href="/0813.15006" class="nowrap">Zbl 0813.15006</a></td> </tr><tr> <td>[49]</td> <td class="space">Pickmann, H.; Soto, R. L.; Egana, J.; Salas, M., An inverse eigenvalue problem for symmetrical tridiagonal matrices, Comput. Math. Appl., 54, 5, 699-708 (2007) · <a href="/1155.65329" class="nowrap">Zbl 1155.65329</a></td> </tr><tr> <td>[50]</td> <td class="space">Lancaster, P., Inverse spectral problems for semisimple damped vibrating systems, SIAM J. Matrix Anal. Appl., 29, 1, 279-301 (2007) · <a href="/1132.74017" class="nowrap">Zbl 1132.74017</a></td> </tr><tr> <td>[51]</td> <td class="space">Wang, P.; Yuan, S.; Xie, X., Least-squares Hermitian problem of complex matrix equation $( A X B , C X D ) = ( E , F )$, J. Inequal. Appl., 2016, 1, 1-13 (2016) · <a href="/1354.65080" class="nowrap">Zbl 1354.65080</a></td> </tr><tr> <td>[52]</td> <td class="space">Ben-Israel, A.; Greville, T. N.E., Generalized Inverses: Theory and Applications (2003), Springer Science & Business Media · <a href="/1026.15004" class="nowrap">Zbl 1026.15004</a></td> </tr><tr> <td>[53]</td> <td class="space">Ben-Israel, A., On error bounds for generalized inverses, SIAM J. Numer. Anal., 3, 4, 585-592 (1966) · <a href="/0147.13201" class="nowrap">Zbl 0147.13201</a></td> </tr><tr> <td>[54]</td> <td class="space">Gladwell, G. M.L.; Jones, T. H.; Willms, N. B., A test matrix for an inverse eigenvalue problem, J. Appl. Math., 2014, 1-6 (2014) · <a href="/1442.15012" class="nowrap">Zbl 1442.15012</a></td> </tr><tr> <td>[55]</td> <td class="space">Yuan, Y.; Dai, H., Solutions to an inverse monic quadratic eigenvalue problem, Linear Algebra Appl., 434, 11, 2367-2381 (2011) · <a href="/1213.65068" class="nowrap">Zbl 1213.65068</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

CINXE.COM

Document Zbl 1496.15014 - zbMATH Open