Bor Plestenjak

A Jacobi--Davidson Type Method for the Two-Parameter Eigenvalue Problem

We present a new numerical method for computing selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. The method does not require good initial approximations and is able to tackle large problems that are too expensive for methods that compute all eigenvalues. The new method uses a two-sided approach and is a generalization of the Jacobi--Davidson type method for right definite two-parameter eigenvalue problems [M. E. Hochstenbach and B. Plestenjak, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 392--410]. Here we consider the much wider class of nonsingular problems. In each step we first compute Petrov triples of a small projected two-parameter eigenvalue problem and then expand the left and right search spaces using approximate solutions to appropriate correction equations. Using a selection technique, it is possible to compute more than one eigenpair. Some numerical examples are presented.

ON THE SINGULAR TWO-PARAMETER EIGENVALUE PROBLEM ∗

In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. In this paper, the above relation to singular two-parameter eigenvalue problems is extended, and it is shown that the simple finite regular eigenvalues of a two-parameter eigenvalue problem and the associated system of generalized eigenvalue problems agree. This enables one to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems.

Backward error, condition numbers, and pseudospectra for the multiparameter eigenvalue problem

We define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem (MEP). The pseudospectrum for the MEP is defined and characterized. We show that the distance from a right definite MEP to the closest non right definite MEP is related to the smallest unbounded pseudospectrum. Some numerical results are given.

A Continuation Method for a Right Definite Two-Parameter Eigenvalue Problem

The continuation method has been successfully applied to the classical

A Jacobi--Davidson Type Method for a Right Definite Two-Parameter Eigenvalue Problem

Ax=\lambda x

Spectral collocation solutions to multiparameter Mathieu’s system

and to the generalized

Numerical Methods for the Tridiagonal Hyperbolic Quadratic Eigenvalue Problem

Ax=\lambda Bx

An algorithm for drawing planar graphs

eigenvalue problems. Shimasaki applied the continuation method to the right definite two-parameter problem, which resulted in a discretization of a two-parameter Sturm--Liouville problem. We show that the continuation method can be used for a general right definite two-parameter problem and we give a sketch of the algorithm. For a local convergent method we use the tensor Rayleigh quotient iteration (TRQI), which is a generalization of the Rayleigh iterative method to two-parameter problems. We show its convergence and compare it with Newtons method and with the generalized Rayleigh quotient iteration (GRQI), studied by Ji, Jiang, and Lee.

A Sylvester-Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants

We present a new numerical iterative method for computing selected eigenpairs of a right definite two-parameter eigenvalue problem. The method works even without good initial approximations and is able to tackle large problems that are too expensive for existing methods. The new method is similar to the Jacobi--Davidson method for the eigenvalue problem. In each step, we first compute Ritz pairs of a small projected right definite two-parameter eigenvalue problem and then expand the search spaces using approximate solutions of appropriate correction equations. We present two alternatives for the correction equations, introduce a selection technique that makes it possible to compute more than one eigenpair, and give some numerical results.

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Our main aim is the accurate computation of a large number of specified eigenvalues and eigenvectors of Mathieu’s system as a multiparameter eigenvalue problem (MEP). The reduced wave equation, for small deflections, is solved directly without approximations introduced by the classical Mathieu functions. We show how for moderate values of the cut-off collocation parameter the QR algorithm and the Arnoldi method may be applied successfully, while for larger values the Jacobi–Davidson method is the method of choice with respect to convergence, accuracy and memory usage.