{"title":"移位和缩放吉尼布雷矩阵的 GMRES、伪谱和 Crouzeix 猜想","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":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2024-03-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/mcom/3963\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3963","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
摘要
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。
GMRES, pseudospectra, and Crouzeix’s conjecture for shifted and scaled Ginibre matrices
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.
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