A Jacobi--Davidson Type SVD Method
Abstract
We discuss a new method for the iterative computation of a portion of the singular values and vectors of a large sparse matrix. Similar to the Jacobi--Davidson method for the eigenvalue problem, we compute in each step a correction by (approximately) solving a correction equation. We give a few variants of this Jacobi--Davidson SVD (JDSVD) method with their theoretical properties. It is shown that the JDSVD can be seen as an accelerated (inexact) Newton scheme. We experimentally compare the method with some other iterative SVD methods.
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References
1.
Jane Cullum, Ralph Willoughby, Mark Lake, A Lánczos algorithm for computing singular values and vectors of large matrices, SIAM J. Sci. Statist. Comput., 4 (1983), 197–215
2.
E. R. Davidson, The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real‐symmetric matrices, J. Comput. Phys., 17 (1975), pp. 87–94.
3.
J. Dongarra, Improving the accuracy of computed singular values, SIAM J. Sci. Statist. Comput., 4 (1983), 712–719
4.
Diederik Fokkema, Gerard Sleijpen, Henk Van der Vorst, Accelerated inexact Newton schemes for large systems of nonlinear equations, SIAM J. Sci. Comput., 19 (1998), 657–674
5.
G. Golub, W. Kahan, Calculating the singular values and pseudo‐inverse of a matrix, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 205–224
6.
Gene Golub, Franklin Luk, Michael Overton, A block Lánczos method for computing the singular values of corresponding singular vectors of a matrix, ACM Trans. Math. Software, 7 (1981), 149–169
7.
G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The John Hopkins University Press, Baltimore, London, 1996.
8.
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
9.
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
10.
Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards, 45 (1950), 255–282
11.
Chris Paige, Beresford Parlett, Henk van der Vorst, Approximate solutions and eigenvalue bounds from Krylov subspaces, Numer. Linear Algebra Appl., 2 (1995), 115–133
12.
B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, 1997 (corrected reprint of the 1980 original).
13.
B. Philippe, M. Sadkane, Computation of the fundamental singular subspace of a large matrix, Linear Algebra Appl., 257 (1997), 77–104
14.
Gerard Sleijpen, Albert Booten, Diederik Fokkema, Henk Van der Vorst, Jacobi‐Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), 595–633, International Linear Algebra Year (Toulouse, 1995)
15.
Gerard Sleijpen, Henk Van der Vorst, A Jacobi‐Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17 (1996), 401–425
16.
G. L. G. Sleijpen and H. A. van der Vorst, The Jacobi‐Davidson method for eigenvalue problems and its relation with accelerated inexact Newton schemes, in Iterative Methods in Linear Algebra II, IMACS Series in Computational and Applied Mathematics 3, S. D. Margenov and P. S. Vassilevski, eds., IMACS, New Brunswick, NJ, 1996, pp. 377–389.
17.
Gerard Sleijpen, Henk van der Vorst, Ellen Meijerink, Efficient expansion of subspaces in the Jacobi‐Davidson method for standard and generalized eigenproblems, Electron. Trans. Numer. Anal., 7 (1998), 75–89, Large scale eigenvalue problems (Argonne, IL, 1997)
18.
G. Stewart, Ji Sun, Matrix perturbation theory, Computer Science and Scientific Computing, Academic Press Inc., 1990xvi+365
19.
Sabine Van Huffel, Iterative algorithms for computing the singular subspace of a matrix associated with its smallest singular values, Linear Algebra Appl., 154/156 (1991), 675–709
20.
Sowmini Varadhan, Michael Berry, Gene Golub, Approximating dominant singular triplets of large sparse matrices via modified moments, Numer. Algorithms, 13 (1996), 123–152
21.
J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, UK, 1965.
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SIAM Journal on Scientific Computing
Pages: 606 - 628
ISSN (online): 1095-7197
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Copyright © 2001 Society for Industrial and Applied Mathematics.
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Published online: 4 August 2006
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