{"title":"GMRES, pseudospectra, and Crouzeix’s conjecture for shifted and scaled Ginibre matrices","authors":"Tyler Chen, Anne Greenbaum, Thomas Trogdon","doi":"10.1090/mcom/3963","DOIUrl":null,"url":null,"abstract":"<p>We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N times upper N\"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">N\\times N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N right-arrow normal infinity\"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">N\\to \\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix’s conjecture.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"42 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3963","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted N×NN\times N matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the N→∞N\to \infty behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix’s conjecture.
我们研究了应用于线性方程组的 GMRES 算法，该方程组涉及一个经过缩放和移位的 N × N N 次矩阵，该矩阵的条目是独立的复高斯。当这个线性方程组的右边独立于这个随机矩阵时，GMRES残余误差的 N → ∞ Nto \infty 行为就可以精确地确定。为了处理右手侧依赖于随机矩阵的情况，我们研究了吉尼布雷矩阵的伪谱和数值范围，并证明了克鲁齐猜想的限制版本。
期刊介绍：
All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are.
This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.
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