Neumaier, A. Residual inverse iteration for the nonlinear eigenvalue problem. (English) Zbl 0594.65026 SIAM J. Numer. Anal. 22, 914-923 (1985). 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 Cited in 52 Documents 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 Keywords:inverse iteration; eigenvector; eigenvalue; Convergence; numerical examples Citations:Zbl 0424.65021 PDFBibTeX XMLCite \textit{A. Neumaier}, SIAM J. Numer. Anal. 22, 914--923 (1985; Zbl 0594.65026) Full Text: DOI