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Residual inverse iteration for the nonlinear eigenvalue problem. (English) Zbl 0594.65026

A variant of inverse iteration is considered for numerical solution of \(A(\lambda)x=0\), where A is a given twice differentiable matrix-valued function and the eigenvector \(x\neq 0\) and eigenvalue \(\lambda\) are sought. To an approximation \(x^{(i)}\) for x, the algorithm adds a correction computed from \(A(\lambda)x^{(i)}\) for suitable trial \(\lambda\). Convergence is proved only for simple eigenvalues, but one of the numerical examples presented, which involves a symmetric definite quadratic eigenvalue problem, has some double eigenvalues and they were found as efficiently as the others. For a discussion of standard inverse iteration and its relation to Newton’s method in the case in which A is a polynomial, see G. Peters and J. H. Wilkinson [SIAM Rev. 21, 339-360 (1979; Zbl 0424.65021)].
Reviewer: A.L.Andrew

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A54 Matrices over function rings in one or more variables
15A18 Eigenvalues, singular values, and eigenvectors

Citations:

Zbl 0424.65021
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