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 in | Numerical linear algebra with applications Vol. 11; no. 1; pp. 75 - 89 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Chichester, UK
John Wiley & Sons, Ltd
01.02.2004
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1070-5325 1099-1506 1099-1506 |
| DOI | 10.1002/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. |
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| 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 | 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. 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. 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. Guo H. Computing trace of function of matrix. Numerical Mathematics (A Journal of Chinese Universities) 2000; 22(2):204-215. Rinehart RF. The equivalence of definitions of a matrix function. American Mathematical Monthly 1995; 62:395-414. Golub GH. Some modified matrix eigenvalue problems. SIAM Review 1973; 15:318-334. Wilkinson JH. Algebraic Eigenvalue Problems. Clarendon Press: Oxford, 1965. 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. 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. Saad Y. Iterative Methods for Sparse Linear Systems. PWS Press, 1996. Higham NJ. The test matrix toolbox for Matlab. No. 237, University of Manchester: England, 1993. 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. Hochbruck M, Lubich C. On Krylov subspace approximations to the matrix exponential operator. SIAM Journal on Numerical Analysis 1997; 34:1911-1925. Bai Z, Fahey M, Golub G. Some large-scale matrix computation problems. Journal of Computational and Applied Mathematics 1996; 74:71-89. Draux A. PolynÂmes Orthogonaux formels-Applications. Lecture Notes in Mathematics, vol. 974. Springer: Berlin, 1983. Golub GH, van Loan C. Matrix Computations (3rd edn). John Hopkins Press: Baltimore, MD, 1996. 1993; 14 1995; 62 1950; 45 2000 1973; 15 2000; 22 1997; 34 1998 1965 1996 1996; 74 1974 1992; 13 1983 1994; 15 1993 2001; 77 1998; 7 |
| 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. – reference: Draux A. PolynÂmes Orthogonaux formels-Applications. Lecture Notes in Mathematics, vol. 974. Springer: Berlin, 1983. – reference: Golub GH. Some modified matrix eigenvalue problems. SIAM Review 1973; 15:318-334. – volume: 34 start-page: 1911 year: 1997 end-page: 1925 article-title: On Krylov subspace approximations to the matrix exponential operator publication-title: SIAM Journal on Numerical Analysis – volume: 7 start-page: 1 year: 1998 end-page: 17 article-title: Error estimation of the Padé approximation of transfer function via the Lanczos process publication-title: Electronic Transactions on Numerical Analysis – year: 1983 – volume: 62 start-page: 395 year: 1995 end-page: 414 article-title: The equivalence of definitions of a matrix function publication-title: American Mathematical Monthly – volume: 74 start-page: 71 year: 1996 end-page: 89 article-title: Some large‐scale matrix computation problems publication-title: Journal of Computational and Applied Mathematics – volume: 45 start-page: 225 year: 1950 end-page: 280 article-title: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators publication-title: Journal of Research of the National Bureau of Standards – volume: 13 start-page: 594 year: 1992 end-page: 639 article-title: A completed theory of the unsymmetric Lanczos process and related algorithm, Part I publication-title: SIAM Journal on Matrix Analysis and Applications – year: 1965 – volume: 15 start-page: 15 year: 1994 end-page: 58 article-title: A completed theory of the unsymmetric Lanczos process and related algorithm, Part II publication-title: SIAM Journal on Matrix Analysis and Applications – volume: 77 start-page: 89 issue: 2 year: 2001 end-page: 104 article-title: IRR: an algorithm for computing the smallest singular value of large‐scale matrices publication-title: International Journal of Computer Mathematics – year: 1996 – year: 2000 – year: 1974 – volume: 22 start-page: 204 issue: 2 year: 2000 end-page: 215 article-title: Computing trace of function of matrix publication-title: Numerical Mathematics – volume: 14 start-page: 137 year: 1993 end-page: 158 article-title: An implementation of the look‐ahead Lanczos algorithm for non‐Hermitian matrices publication-title: SIAM Journal of Science and Statistical Computing – volume: 15 start-page: 318 year: 1973 end-page: 334 article-title: Some modified matrix eigenvalue problems publication-title: SIAM Review – year: 1993 – year: 1998 |
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| Title | Estimation of uTƒ(A)v for large-scale unsymmetric matrices |
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