Estimation of uTƒ(A)v for large-scale unsymmetric matrices

Fast algorithms, based on the unsymmetric look‐ahead Lanczos and the Arnoldi process, are developed for the estimation of the functional Φ(ƒ)=uTƒ(A) v for fixed u, v and A, where A∈ℜ︁n×n is a large‐scale unsymmetric matrix. Numerical results are presented which validate the comparable accuracy of bo...

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Published inNumerical linear algebra with applications Vol. 11; no. 1; pp. 75 - 89
Main Authors Guo, Hongbin, Renaut, Rosemary A.
Format Journal Article
LanguageEnglish
Published Chichester, UK John Wiley & Sons, Ltd 01.02.2004
Subjects
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ISSN1070-5325
1099-1506
1099-1506
DOI10.1002/nla.334

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Abstract Fast algorithms, based on the unsymmetric look‐ahead Lanczos and the Arnoldi process, are developed for the estimation of the functional Φ(ƒ)=uTƒ(A) v for fixed u, v and A, where A∈ℜ︁n×n is a large‐scale unsymmetric matrix. Numerical results are presented which validate the comparable accuracy of both approaches. Although the Arnoldi process reaches convergence more quickly in some cases, it has greater memory requirements, and may not be suitable for especially large applications. Copyright © 2003 John Wiley & Sons, Ltd.
AbstractList Fast algorithms, based on the unsymmetric look‐ahead Lanczos and the Arnoldi process, are developed for the estimation of the functional Φ(ƒ)=uTƒ(A) v for fixed u, v and A, where A∈ℜ︁n×n is a large‐scale unsymmetric matrix. Numerical results are presented which validate the comparable accuracy of both approaches. Although the Arnoldi process reaches convergence more quickly in some cases, it has greater memory requirements, and may not be suitable for especially large applications. Copyright © 2003 John Wiley & Sons, Ltd.
Author Guo, Hongbin
Renaut, Rosemary A.
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References_xml – reference: Guo H. Computing trace of function of matrix. Numerical Mathematics (A Journal of Chinese Universities) 2000; 22(2):204-215.
– reference: Saad Y. Iterative Methods for Sparse Linear Systems. PWS Press, 1996.
– reference: Wilkinson JH. Algebraic Eigenvalue Problems. Clarendon Press: Oxford, 1965.
– reference: Guo H. IRR: an algorithm for computing the smallest singular value of large-scale matrices. International Journal of Computer Mathematics 2001; 77(2):89-104.
– reference: Bai Z, Ye Q. Error estimation of the Padé approximation of transfer function via the Lanczos process. Electronic Transactions on Numerical Analysis 1998; 7:1-17.
– reference: Freund RW, Gutknecht M, Nachtigal N. An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices. SIAM Journal of Science and Statistical Computing 1993; 14:137-158.
– reference: Hochbruck M, Lubich C. On Krylov subspace approximations to the matrix exponential operator. SIAM Journal on Numerical Analysis 1997; 34:1911-1925.
– reference: Rinehart RF. The equivalence of definitions of a matrix function. American Mathematical Monthly 1995; 62:395-414.
– reference: Gutknecht MH. A completed theory of the unsymmetric Lanczos process and related algorithm, Part II. SIAM Journal on Matrix Analysis and Applications 1994; 15:15-58.
– reference: Golub GH, van Loan C. Matrix Computations (3rd edn). John Hopkins Press: Baltimore, MD, 1996.
– reference: Bai Z, Fahey M, Golub G. Some large-scale matrix computation problems. Journal of Computational and Applied Mathematics 1996; 74:71-89.
– reference: Higham NJ. The test matrix toolbox for Matlab. No. 237, University of Manchester: England, 1993.
– reference: Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. Journal of Research of the National Bureau of Standards 1950; 45:225-280.
– reference: Gutknecht MH. A completed theory of the unsymmetric Lanczos process and related algorithm, Part I. SIAM Journal on Matrix Analysis and Applications 1992; 13:594-639.
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  article-title: Some large‐scale matrix computation problems
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  year: 2001
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  article-title: IRR: an algorithm for computing the smallest singular value of large‐scale matrices
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  article-title: Some modified matrix eigenvalue problems
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Snippet Fast algorithms, based on the unsymmetric look‐ahead Lanczos and the Arnoldi process, are developed for the estimation of the functional Φ(ƒ)=uTƒ(A) v for...
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SubjectTerms Arnoldi
large-scale unsymmetric matrix
look-ahead Lanczos
matrix function
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Title Estimation of uTƒ(A)v for large-scale unsymmetric matrices
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