期刊
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
卷 28, 期 1, 页码 264-284出版社
SIAM PUBLICATIONS
DOI: 10.1137/050630416
关键词
rounding error analysis; backward stability; linear equations; condition numbers; large sparse matrices; iterative solution; Krylov subspace methods; Arnoldi method; generalized minimum residual method; modified Gram-Schmidt; QR factorization; loss of orthogonality; least squares; singular values
The generalized minimum residual method (GMRES) [Y. Saad and M. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869] for solving linear systems Ax = b is implemented as a sequence of least squares problems involving Krylov subspaces of increasing dimensions. The most usual implementation is modified Gram-Schmidt GMRES (MGS-GMRES). Here we show that MGS-GMRES is backward stable. The result depends on a more general result on the backward stability of a variant of the MGS algorithm applied to solving a linear least squares problem, and uses other new results on MGS and its loss of orthogonality, together with an important but neglected condition number, and a relation between residual norms and certain singular values.
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