On the Lánczos method for solving symmetric linear systems with several right-hand sides
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- by Youcef Saad PDF
- Math. Comp. 48 (1987), 651-662 Request permission
Abstract:
This paper analyzes a few methods based on the Lanczos algorithm for solving large sparse symmetric linear systems with several right-hand sides. The methods examined are suitable for the situation when the right sides are not too different from one another, as is often the case in time-dependent or parameter-dependent problems. We propose a theoretical error bound for the approximation obtained from a projection process onto a Krylov subspace generated from processing a previous right-hand side.References
-
E. Carnoy & M. Geradin, On the Practical Use of the Lanczos Algorithm in Finite Element Applications to Vibration and Bifurcation Problems, Proc. Conf. on Matrix Pencils, held at Lulea, Sweden, March 1982 (Axel Ruhe, ed.), University of Umea, Springer-Verlag, New York, 1982, pp. 156-176.
R. Chandra, Conjugate Gradient Methods for Partial Differential Equations, Ph.D. Thesis, Computer Science Dept., Yale University, 1978.
- C. W. Gear and Y. Saad, Iterative solution of linear equations in ODE codes, SIAM J. Sci. Statist. Comput. 4 (1983), no. 4, 583–601. MR 725654, DOI 10.1137/0904040
- Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 0175290 D. G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading, Mass., 1965.
- Dianne P. O’Leary, The block conjugate gradient algorithm and related methods, Linear Algebra Appl. 29 (1980), 293–322. MR 562766, DOI 10.1016/0024-3795(80)90247-5
- C. C. Paige and M. A. Saunders, Solutions of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), no. 4, 617–629. MR 383715, DOI 10.1137/0712047
- Beresford N. Parlett, The symmetric eigenvalue problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. MR 570116
- B. N. Parlett, A new look at the Lanczos algorithm for solving symmetric systems of linear equations, Linear Algebra Appl. 29 (1980), 323–346. MR 562767, DOI 10.1016/0024-3795(80)90248-7
- Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp. 37 (1981), no. 155, 105–126. MR 616364, DOI 10.1090/S0025-5718-1981-0616364-6
- Youcef Saad, Practical use of some Krylov subspace methods for solving indefinite and nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 5 (1984), no. 1, 203–228. MR 731892, DOI 10.1137/0905015 Y. Saad & A. Sameh, A Parallel Block Stiefel Method for Solving Positive Definite Systems, Proceedings of the Elliptic Problem Solver Conference (M. H. Schultz, ed.), Los Alamos Scientific Laboratory, Academic Press, New York, 1980, pp. 405-412. Y. Saad & A. Sameh, Iterative Methods for the Solution of Elliptic Differential Equations on Multiprocessors, Proceedings of the CONPAR 81 Conference (Wolfgang Handler, ed.), Springer-Verlag, New York, 1981, pp. 395-411.
- Youcef Saad and Martin H. Schultz, Conjugate gradient-like algorithms for solving nonsymmetric linear systems, Math. Comp. 44 (1985), no. 170, 417–424. MR 777273, DOI 10.1090/S0025-5718-1985-0777273-9
- Horst D. Simon, The Lanczos algorithm with partial reorthogonalization, Math. Comp. 42 (1984), no. 165, 115–142. MR 725988, DOI 10.1090/S0025-5718-1984-0725988-X H. A. van der Vorst, An Iterative Method for Solving $f(A)x = b$ Using cg-Information Obtained for the Symmetric Positive Definite Matrix A, Technical Report 85-32, Delft University of Technology, Mathematics and Informatics, 1985.
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 48 (1987), 651-662
- MSC: Primary 65F10; Secondary 65F50
- DOI: https://doi.org/10.1090/S0025-5718-1987-0878697-3
- MathSciNet review: 878697