Balancing sparse matrices for computing eigenvalues
Applying a permuted diagonal similarity transform DPAP T D −1 to a matrix A before calculating its eigenvalues can improve the speed and accuracy with which the eigenvalues are computed. This is often called balancing. This paper describes several balancing algorithms for sparse matrices and compare...
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| Vydáno v: | Linear algebra and its applications Ročník 309; číslo 1; s. 261 - 287 |
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| Jazyk: | angličtina |
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15.04.2000
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | Applying a permuted diagonal similarity transform
DPAP
T
D
−1 to a matrix
A before calculating its eigenvalues can improve the speed and accuracy with which the eigenvalues are computed. This is often called
balancing. This paper describes several balancing algorithms for sparse matrices and compares them against each other and the traditional dense algorithm. We first discuss our sparse implementation of the dense algorithm; our code is faster than the dense algorithm when the density of the matrix is no more than approximately .5, and is much faster for large, sparse matrices. We next describe a set of randomized balancing algorithms for matrices that are not given explicitly, i.e. given a vector
x, we can compute only
Ax and perhaps
A
T
x. We motivate these Krylov-based algorithms using Perron–Frobenius theory. Results are given comparing the Krylov-based algorithms to each other and to the sparse and dense direct balancing algorithms, looking at norm reduction, running times, and the accuracy of eigenvalues computed after a matrix is balanced. We conclude that sparse balancing algorithms are efficient preconditioners for eigensolvers. |
|---|---|
| AbstractList | Applying a permuted diagonal similarity transform
DPAP
T
D
−1 to a matrix
A before calculating its eigenvalues can improve the speed and accuracy with which the eigenvalues are computed. This is often called
balancing. This paper describes several balancing algorithms for sparse matrices and compares them against each other and the traditional dense algorithm. We first discuss our sparse implementation of the dense algorithm; our code is faster than the dense algorithm when the density of the matrix is no more than approximately .5, and is much faster for large, sparse matrices. We next describe a set of randomized balancing algorithms for matrices that are not given explicitly, i.e. given a vector
x, we can compute only
Ax and perhaps
A
T
x. We motivate these Krylov-based algorithms using Perron–Frobenius theory. Results are given comparing the Krylov-based algorithms to each other and to the sparse and dense direct balancing algorithms, looking at norm reduction, running times, and the accuracy of eigenvalues computed after a matrix is balanced. We conclude that sparse balancing algorithms are efficient preconditioners for eigensolvers. |
| Author | Demmel, James W. Chen, Tzu-Yi |
| Author_xml | – sequence: 1 givenname: Tzu-Yi surname: Chen fullname: Chen, Tzu-Yi email: tzuyi@cs.berkeley.edu organization: Computer Science Division, University of California, Berkeley, CA 94720, USA – sequence: 2 givenname: James W. surname: Demmel fullname: Demmel, James W. email: demmel@cs.berkeley.edu organization: Computer Science Division and Mathematics Department, University of California, Berkeley, CA 94720, USA |
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| Cites_doi | 10.1007/BFb0121080 10.1287/moor.16.1.208 10.1137/1.9781611971538 10.1287/opre.38.3.439 10.1093/comjnl/14.3.280 10.1145/62038.62043 10.1007/BF02242378 10.1137/0201010 10.1137/S0895479895293247 10.1145/355780.355785 10.1016/B978-044482107-2/50003-0 10.1007/BF02165404 10.1145/321043.321048 10.1090/S0002-9939-1971-0281731-5 |
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| Keywords | Accurate eigenvalues 65F15 Balancing Norm minimization 65F35 Sparse matrix algorithms |
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| References_xml | – volume: 15 start-page: 1 year: 1989 end-page: 14 ident: NEWBIB7 article-title: Sparse matrix test problems publication-title: ACM Trans. Math. Softw. – volume: 30 start-page: 419 year: 1971 end-page: 425 ident: NEWBIB11 article-title: Concerning diagonal similarity of irreducible matrices publication-title: Proc. AMS – reference: D.C. Sorensen, Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations, in Matlab 5.0 distribution, 1995 – reference: R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962 – reference: Z. Bai, D. Day, J. Demmel, J. Dongarra, A test matrix collection for non-Hermitian eigenvalue problems, Technical Report CS-97-355, University of Tennessee, 1997 – reference: T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms, MIT Press, Cambridge, MA, 1990 – volume: 13 start-page: 293 year: 1969 end-page: 304 ident: NEWBIB14 article-title: Balancing a matrix for calculation of eigenvalues and eigenvectors publication-title: Numer. Math. – volume: 1 start-page: 146 year: 1972 end-page: 160 ident: NEWBIB19 article-title: Depth-first search and linear graph algorithms publication-title: SIAM J. Comput. – volume: 10 start-page: 1 year: 1972 end-page: 7 ident: NEWBIB18 article-title: Minimization of norms and logarithmic norms by diagonal similarities publication-title: Computing – reference: R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. 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Mach. doi: 10.1145/321043.321048 – ident: 10.1016/S0024-3795(00)00014-8_NEWBIB5 – ident: 10.1016/S0024-3795(00)00014-8_NEWBIB6 – ident: 10.1016/S0024-3795(00)00014-8_NEWBIB1 – volume: 30 start-page: 419 year: 1971 ident: 10.1016/S0024-3795(00)00014-8_NEWBIB11 article-title: Concerning diagonal similarity of irreducible matrices publication-title: Proc. AMS doi: 10.1090/S0002-9939-1971-0281731-5 – ident: 10.1016/S0024-3795(00)00014-8_NEWBIB2 – ident: 10.1016/S0024-3795(00)00014-8_NEWBIB3 |
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| Snippet | Applying a permuted diagonal similarity transform
DPAP
T
D
−1 to a matrix
A before calculating its eigenvalues can improve the speed and accuracy with which... |
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| SubjectTerms | Accurate eigenvalues Balancing Norm minimization Sparse matrix algorithms |
| Title | Balancing sparse matrices for computing eigenvalues |
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