A numerical solver for general bordered tridiagonal matrix equations

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• 作者：J. Abderramá ; n Marrero p>jc.abderraman@upm.es" class="auth_mail" title="E-mail the corresponding authorp>
• 关键词：Accuracy ; Bordered tridiagonal matrix ; Givens rotation ; QR factorization
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：72
• 期：11
• 页码：2731-2740
• 全文大小：567 K

文摘

To overcome several limitations of symbolic algorithms introduced recently for matrices of large order, a fast numerical solver is proposed for the matrix linear equation pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116305375&_mathId=si3.gif&_user=111111111&_pii=S0898122116305375&_rdoc=1&_issn=08981221&md5=8f51171635a94802f432bf7ab1b72cac" title="Click to view the MathML source">AX=Bpan>pan class="mathContainer hidden">pan class="mathCode">$AX=B$pan>pan>pan>, where the pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116305375&_mathId=si4.gif&_user=111111111&_pii=S0898122116305375&_rdoc=1&_issn=08981221&md5=5b60ddb0e8b34fee5009210172559487" title="Click to view the MathML source">n×npan>pan class="mathContainer hidden">pan class="mathCode">$n\times n$pan>pan>pan> coefficient matrix pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116305375&_mathId=si5.gif&_user=111111111&_pii=S0898122116305375&_rdoc=1&_issn=08981221&md5=c8ca09719105cd2746a6c6f7ea558fce" title="Click to view the MathML source">Apan>pan class="mathContainer hidden">pan class="mathCode">$A$pan>pan>pan> is a general nonsingular bordered tridiagonal matrix. Its sparse structure is preserved through partial Givens reduction. In particular, the matrix inverse of pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116305375&_mathId=si5.gif&_user=111111111&_pii=S0898122116305375&_rdoc=1&_issn=08981221&md5=c8ca09719105cd2746a6c6f7ea558fce" title="Click to view the MathML source">Apan>pan class="mathContainer hidden">pan class="mathCode">$A$pan>pan>pan> can be computed. For a wide range of bordered tridiagonal linear systems pan id="mmlsi7" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116305375&_mathId=si7.gif&_user=111111111&_pii=S0898122116305375&_rdoc=1&_issn=08981221&md5=210c988862e2e1b5aa082a3eecbd2700" title="Click to view the MathML source">Ax=bpan>pan class="mathContainer hidden">pan class="mathCode">$Ax=b$pan>pan>pan>, the solution is computed in linear time using back substitution and Sherman–Morrison’s formula. Numerical comparisons illustrate the results.