Newton's method in floating point arithmetic and iterative refinement of generalized eigenvalue problems (Q2706305)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Newton's method in floating point arithmetic and iterative refinement of generalized eigenvalue problems |
scientific article |
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19 March 2001
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symmetric generalized eigenvalue problem
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rounding error analysis
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Cholesky method
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Newton's method
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floating point arithmetics
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extended precision
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numerical examples
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backward stability
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iterative refinement
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Newton's method in floating point arithmetic and iterative refinement of generalized eigenvalue problems (English)
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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 the 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 the 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.
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