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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
De 015 789 29X
Author(s):
Tisseur, Francoise
Shorttitle:
Newton's method in floating point arithmetic and iterative refinement of GEP
Source:
SIAM J. Matrix Anal. Appl. 22, No. 4, 1038-1057 (2001).
Classification:
Primary Classification:
65F15Eigenvalues, eigenvectors
Secondary Classification:
65F35Matrix norms, conditioning, scaling
Keywords:
generalized eigenvalue problem; rounding error analysis; iterative improvement of Cholesky method; Newton's method in extended precision; floating point arithmetics
Review:


In the symmetric generalized eigenvalue problem with indefinite A
or B there exists a conflict between efficiency and backward
stability. The paper advocates an iterative refinement of computed
eigenpairs by Newton's method. It is admitted that the linear
solver is unstable and that the Jacobian itself is inaccurate. An
extended precision is only assumed available in computation of the
residual. The key question which is analyzed concerns the
sufficient precision of the residual. It is shown that and when an
improvement of the forward and backward errors may be achieved in
such a situation. Numerical illustrations confirm the theory using
small matrices treated by the usual Cholesky-QR method.
Remarks to the editors:

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