Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the Lánczos method for solving symmetric linear systems with several right-hand sides

Author: Youcef Saad
Journal: Math. Comp. 48 (1987), 651-662
MSC: Primary 65F10; Secondary 65F50
MathSciNet review: 878697
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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.
  • [2] R. Chandra, Conjugate Gradient Methods for Partial Differential Equations, Ph.D. Thesis, Computer Science Dept., Yale University, 1978.
  • [3] W. C. Gear & Y. Saad, "Iterative solution of linear equations in ODE codes," SIAM J. Sci. Statist. Comput., v. 4, 1983, pp. 583-601. MR 725654 (85a:65104)
  • [4] A. S. Householder, Theory of Matrices in Numerical Analysis, Blaisdell, Johnson, Colo., 1964. MR 0175290 (30:5475)
  • [5] D. G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading, Mass., 1965.
  • [6] D. O'Leary, "The block conjugate gradient algorithm and related methods," Linear Algebra Appl., v. 29, 1980, pp. 243-322. MR 562766 (81i:65027)
  • [7] C. C. Paige & M. A. Saunders, "Solution of sparse indefinite systems of linear equations," SIAM J. Numer. Anal., v. 12, 1975, pp. 617-624. MR 0383715 (52:4595)
  • [8] B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, N. J., 1980. MR 570116 (81j:65063)
  • [9] B. N. Parlett, "A new look at the Lanczos algorithm for solving symmetric systems of linear equations," Linear Algebra Appl., v. 29, 1980, pp. 323-346. MR 562767 (83e:65064)
  • [10] Y. Saad, "Krylov subspace methods for solving large unsymmetric linear systems," Math. Comp., v. 37, 1981, pp. 105-126. MR 616364 (83j:65037)
  • [11] Y. Saad, "Practical use of some Krylov subspace methods for solving indefinite and unsymmetric linear systems," SIAM J. Sci. Statist. Comput., v. 5, 1984, pp. 203-228. MR 731892 (85m:65029)
  • [12] 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.
  • [13] 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.
  • [14] Y. Saad & M. H. Schultz, "Conjugate gradient-like algorithms for solving nonsymmetric linear systems," Math. Comp., v. 44, 1985, pp. 417-424. MR 777273 (86d:65047)
  • [15] H. D. Simon, "The Lanczos algorithm with partial reorthogonalization," Math. Comp., v. 42, 1984, pp. 115-142. MR 725988 (85h:65075)
  • [16] 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.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65F10, 65F50

Retrieve articles in all journals with MSC: 65F10, 65F50

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society