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

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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.

**[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]**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**, 10.1137/0904040**[4]**Alston S. Householder,*The theory of matrices in numerical analysis*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR**0175290****[5]**D. G. Luenberger,*Introduction to Linear and Nonlinear Programming*, Addison-Wesley, Reading, Mass., 1965.**[6]**Dianne P. O’Leary,*The block conjugate gradient algorithm and related methods*, Linear Algebra Appl.**29**(1980), 293–322. MR**562766**, 10.1016/0024-3795(80)90247-5**[7]**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**0383715****[8]**Beresford N. Parlett,*The symmetric eigenvalue problem*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. Prentice-Hall Series in Computational Mathematics. MR**570116****[9]**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**, 10.1016/0024-3795(80)90248-7**[10]**Y. Saad,*Krylov subspace methods for solving large unsymmetric linear systems*, Math. Comp.**37**(1981), no. 155, 105–126. MR**616364**, 10.1090/S0025-5718-1981-0616364-6**[11]**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**, 10.1137/0905015**[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]**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**, 10.1090/S0025-5718-1985-0777273-9**[15]**Horst D. Simon,*The Lanczos algorithm with partial reorthogonalization*, Math. Comp.**42**(1984), no. 165, 115–142. MR**725988**, 10.1090/S0025-5718-1984-0725988-X**[16]**H. A. van der Vorst,*An Iterative Method for Solving**Using cg-Information Obtained for the Symmetric Positive Definite Matrix A*, Technical Report 85-32, Delft University of Technology, Mathematics and Informatics, 1985.

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0878697-3

Article copyright:
© Copyright 1987
American Mathematical Society