A Generalized Rayleigh Quotient Iteration for Computing Simple Eigenvalues of Nonnormal Matrices
Hubert Schwetlick, Ralf Lösche · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · 2000
A Newton type method for approximating eigenpairs of nonnormal matrices is proposed. It takes full advantage of known approximations and circumvents shortcomings of Newton's method applied to the nonlinear system Ax—xλ = 0, (xTx—1)/2 = 0 caused by a possibly ill-conditioned Jacobian. The eigenvalue problem is considered as a nonlinear parameter dependent system of equations F(x, λ) = Ax—xλ = 0 without a normalization condition. If λ is an eigenvalue of A then (0, λ) is a bifurcation point on the solution manifold of the system. In the paper, a bifurcation point algorithm proposed by Griewank/Reddien and modified by Allgower/Schwetlick is adapted to the case of computing a simple eigenvalue and the corresponding right and left eigenvector of a nonnormal matrix. The extended algorithm with udapted bordering vectors can be viewed as generalized Rayleigh quotient iteration. Surprisingly it turns out that, unlike Newton's method and standard Rayleigh quotient iteration, these algorithms coming from nonlinear analysis, automatically give the right bordering directions, and the linear systems to be solved per step have uniformly bounded condition numbers. The algorithms are analyzed, and numerical comparisons with other algorithms of Rayleigh quotient iteration type are given.