M.E. Hochstenbach

A Jacobi--Davidson Type SVD Method

We discuss a new method for the iterative computation of a portion of the singular values and vectors of a large sparse matrix. Similar to the Jacobi--Davidson method for the eigenvalue problem, we compute in each step a correction by (approximately) solving a correction equation. We give a few variants of this Jacobi--Davidson SVD (JDSVD) method with their theoretical properties. It is shown that the JDSVD can be seen as an accelerated (inexact) Newton scheme. We experimentally compare the method with some other iterative SVD methods.

A Jacobi--Davidson Method for Solving Complex Symmetric Eigenvalue Problems

We discuss variants of the Jacobi--Davidson method for solving the generalized complex symmetric eigenvalue problem. The Jacobi--Davidson algorithm can be considered as an accelerated inexact Rayleigh quotient iteration. We show that it is appropriate to replace the Euclidean inner product in

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

{\mathbb C}^n

Controlling Inner Iterations in the Jacobi-Davidson Method

with an indefinite inner product. The Rayleigh quotient based on this indefinite inner product leads to an asymptotically cubically convergent Rayleigh quotient iteration. Advantages of the method are illustrated by numerical examples. We deal with problems from electromagnetics that require the computation of interior eigenvalues. The main drawback that we experience in these particular examples is the lack of efficient preconditioners.

Alternatives to the Rayleigh Quotient for the Quadratic 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.

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

The Jacobi-Davidson method is an eigenvalue solver which uses an inner-outer scheme. In the outer iteration one tries to approximate an eigenpair while in the inner iteration a linear system has to be solved, often iteratively, with the ultimate goal to make progress for the outer loop. In this paper we prove a relation between the residual norm of the inner linear system and the residual norm of the eigenvalue problem. We show that the latter may be estimated inexpensively during the inner iterations. On this basis, we propose a stopping strategy for the inner iterations to maximally exploit the strengths of the method. These results extend previous results obtained for the special case of Hermitian eigenproblems with the conjugate gradient or the symmetric QMR method as inner solver. The present analysis applies to both standard and generalized eigenproblems, does not require symmetry, and is compatible with most iterative methods for the inner systems. It can also be extended to other types of inner-outer eigenvalue solvers, such as inexact inverse iteration or inexact Rayleigh quotient iteration. The effectiveness of our approach is illustrated by a few numerical experiments, including the comparison of a standard Jacobi-Davidson code with the same code enhanced by our stopping strategy.

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

We consider the quadratic eigenvalue problem

Probabilistic Upper Bounds for the Matrix Two-Norm

\lambda^2 Ax + \lambda Bx + Cx = 0.

Spectral collocation solutions to multiparameter Mathieu’s system

Suppose that

A Golub---Kahan-Type Reduction Method for Matrix Pairs

u