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.

Controlling inner iterations in the Jacobi-Davidson method

The correction equation in the Jacobi-Davidson method is effective in a subspace orthogonal to the current eigenvector approximation, while for the continuation of the process only vectors orthogonal to the search subspace are of importance. Such a vector is obtained by orthogonalizing the (approximate) solution of the correction equation against the search subspace. As an alternative, a variant of the correction equation can be formulated that is restricted to the subspace orthogonal to the current search subspace. In this paper, we discuss the effectivity of this variant. Our investigation is also motivated by the fact that the restricted correction equation can be used for avoiding stagnation in case of defective eigenvalues. Moreover, this equation plays a key role in the inexact TRQ method cite{SYa98.

Alternative correction equations in the Jacobi-Davidson method

The methods used for extracting an approximate eigenpair are crucial in sparse iterative eigensolvers. Using least squares and line search techniques this paper devises a method for an approximate eigenpair extraction. Numerical comparison of the Jacobi–Davidson method using the suggested method of eigenpair extraction, Rayleigh–Ritz, and refined Ritz projections shows that the suggested method is a viable alternative.

The Least squares and line search in extracting eigenpairs in Jacobi–Davidson method

In this paper, a canonical transformation is proposed to solve the eigenvalue problem related to the dynamics of rotor-bearing systems. In this problem, all matrices are real, but they may not be s...

A Numerical Method to Solve Nonsymmetric Eigensystems Applied to Dynamics of Turbomachinery

In this paper, we describe a new algorithm to compute the extreme eigenvalue/eigenvector pairs of a symmetric matrix. The proposed algorithm can be viewed as an extension of the Jacobi transformation method for symmetric matrix diagonalization to the case where we want to compute just a few eigenvalues/eigenvectors. The method is also particularly well suited for the computation of sparse eigenspaces. We show the effectiveness of the method for sparse low-rank approximations and show applications to random symmetric matrices, graph Fourier transforms, and with the sparse principal component analysis in image classification experiments.

An iterative Jacobi-like algorithm to compute a few sparse eigenvalue-eigenvector pairs.

A new variant of Arnoldi method for approximation of eigenpairs

It is generally believed that the Ritz vectors do not coincide with the refined Ritz vectors in the Arnoldi method for computing eigenvalues of matrices. We show that this coincidence is theoretically possible. We provide a necessary and sufficient condition for this coincidence to happen and give examples to illustrate the same. Using Lanczos polynomials, we give a polynomial characterization of refined Ritz vectors of symmetric matrices that is different from the one available in the literature.

Mashetti Ravibabu

Papers

A new variant of Arnoldi method for approximation of eigenpairs

On refined Ritz vectors and polynomial characterization

The Least squares and line search in extracting eigenpairs in Jacobi–Davidson method