Estimates of the trace of the inverse of a symmetric matrix using the modified Chebyshev algorithm

In this paper we study how to compute an estimate of the trace of the inverse of a symmetric matrix by using Gauss quadrature and the modified Chebyshev algorithm. As auxiliary polynomials we use the shifted Chebyshev polynomials. Since this can be too costly in computer storage for large matrices w...

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Vydáno v:	Numerical algorithms Ročník 51; číslo 3; s. 309 - 318
Hlavní autor:	Meurant, Gérard
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Boston Springer US 01.07.2009 Springer Nature B.V
Témata:	Algebra Algorithms Chebyshev approximation Computer Science Computer simulation Estimates Inverse Mathematical analysis Matrices (mathematics) Numeric Computing Numerical Analysis Original Paper Polynomials Quadratures Symmetry Theory of Computation Trace Inverse Chebyshev algorithm
ISSN:	1017-1398, 1572-9265
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Abstract	In this paper we study how to compute an estimate of the trace of the inverse of a symmetric matrix by using Gauss quadrature and the modified Chebyshev algorithm. As auxiliary polynomials we use the shifted Chebyshev polynomials. Since this can be too costly in computer storage for large matrices we also propose to compute the modified moments with a stochastic approach due to Hutchinson (Commun Stat Simul 18:1059–1076, 1989 ).
AbstractList	In this paper we study how to compute an estimate of the trace of the inverse of a symmetric matrix by using Gauss quadrature and the modified Chebyshev algorithm. As auxiliary polynomials we use the shifted Chebyshev polynomials. Since this can be too costly in computer storage for large matrices we also propose to compute the modified moments with a stochastic approach due to Hutchinson (Commun Stat Simul 18:1059--1076, 1989). In this paper we study how to compute an estimate of the trace of the inverse of a symmetric matrix by using Gauss quadrature and the modified Chebyshev algorithm. As auxiliary polynomials we use the shifted Chebyshev polynomials. Since this can be too costly in computer storage for large matrices we also propose to compute the modified moments with a stochastic approach due to Hutchinson (Commun Stat Simul 18:1059–1076, 1989 ).
Author	Meurant, Gérard
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Cites_doi	10.1080/03610918908812806 10.1016/0377-0427(96)00018-0 10.1007/BF01406683 10.2307/2004418 10.1093/oso/9780198506720.001.0001 10.1090/S0025-5718-69-99647-1 10.1515/9781400833887 10.1007/10703040_2
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References	CR2 Golub, Welsch (CR8) 1969; 23 Golub, Meurant (CR7) 2009 Bai, Fahey, Golub (CR1) 1996; 74 Bai, Golub, Palma, Dongarra, Hernandez (CR4) 1999 Gautschi (CR5) 2004 Sack, Donovan (CR10) 1972; 18 Golub, Meurant, Griffiths, Watson (CR6) 1994 Wheeler (CR11) 1974; 4 Bai, Golub (CR3) 1997; 4 Hutchinson (CR9) 1989; 18 Z. Bai (9246_CR1) 1996; 74 9246_CR2 G.H. Golub (9246_CR8) 1969; 23 Z. Bai (9246_CR3) 1997; 4 M. Hutchinson (9246_CR9) 1989; 18 Z. Bai (9246_CR4) 1999 G.H. Golub (9246_CR7) 2009 J.C. Wheeler (9246_CR11) 1974; 4 G.H. Golub (9246_CR6) 1994 R.A. Sack (9246_CR10) 1972; 18 W. Gautschi (9246_CR5) 2004
References_xml	– volume: 18 start-page: 1059 year: 1989 end-page: 1076 ident: CR9 article-title: A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines publication-title: Commun. Stat. Simul. doi: 10.1080/03610918908812806 – volume: 74 start-page: 71 year: 1996 end-page: 89 ident: CR1 article-title: Some large scale matrix computation problems publication-title: Comp. J. Appl. Math. doi: 10.1016/0377-0427(96)00018-0 – volume: 18 start-page: 465 issue: 5 year: 1972 end-page: 478 ident: CR10 article-title: An algorithm for Gaussian quadrature given modified moments publication-title: Numer. Math. doi: 10.1007/BF01406683 – year: 2009 ident: CR7 publication-title: Matrices, Moments and Quadrature with Applications – volume: 23 start-page: 221 year: 1969 end-page: 230 ident: CR8 article-title: Calculation of Gauss quadrature rules publication-title: Math. Comp. doi: 10.2307/2004418 – volume: 4 start-page: 287 issue: 2 year: 1974 end-page: 296 ident: CR11 article-title: Modified moments and Gaussian quadrature. In: Proceedings of the international conference on Padé approximants, continued fractions and related topics, Univ. Colorado, Boulder. Rocky Mt. publication-title: J. Math. – ident: CR2 – start-page: 4 year: 1999 end-page: 19 ident: CR4 article-title: Some unusual matrix eigenvalue problems publication-title: Proceedings of Vecpar’98—Third International Conference for Vector and Parallel Processing – start-page: 105 year: 1994 end-page: 156 ident: CR6 article-title: Matrices, moments and quadrature publication-title: Numerical Analysis 1993 – volume: 4 start-page: 29 year: 1997 end-page: 38 ident: CR3 article-title: Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices publication-title: Ann. Numer. Math. – year: 2004 ident: CR5 publication-title: Orthogonal Polynomials: Computation and Approximation – volume-title: Orthogonal Polynomials: Computation and Approximation year: 2004 ident: 9246_CR5 doi: 10.1093/oso/9780198506720.001.0001 – volume: 4 start-page: 29 year: 1997 ident: 9246_CR3 publication-title: Ann. Numer. Math. – volume: 23 start-page: 221 year: 1969 ident: 9246_CR8 publication-title: Math. Comp. doi: 10.1090/S0025-5718-69-99647-1 – volume: 4 start-page: 287 issue: 2 year: 1974 ident: 9246_CR11 publication-title: J. Math. – volume: 18 start-page: 1059 year: 1989 ident: 9246_CR9 publication-title: Commun. Stat. Simul. doi: 10.1080/03610918908812806 – volume: 74 start-page: 71 year: 1996 ident: 9246_CR1 publication-title: Comp. J. Appl. Math. doi: 10.1016/0377-0427(96)00018-0 – start-page: 105 volume-title: Numerical Analysis 1993 year: 1994 ident: 9246_CR6 – volume: 18 start-page: 465 issue: 5 year: 1972 ident: 9246_CR10 publication-title: Numer. Math. doi: 10.1007/BF01406683 – ident: 9246_CR2 – volume-title: Matrices, Moments and Quadrature with Applications year: 2009 ident: 9246_CR7 doi: 10.1515/9781400833887 – start-page: 4 volume-title: Proceedings of Vecpar’98—Third International Conference for Vector and Parallel Processing year: 1999 ident: 9246_CR4 doi: 10.1007/10703040_2
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SubjectTerms	Algebra Algorithms Chebyshev approximation Computer Science Computer simulation Estimates Inverse Mathematical analysis Matrices (mathematics) Numeric Computing Numerical Analysis Original Paper Polynomials Quadratures Symmetry Theory of Computation
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Title	Estimates of the trace of the inverse of a symmetric matrix using the modified Chebyshev algorithm
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