Alternative correction equations in the Jacobi-Davidson method

Abstract

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 [19].

Iterative methods for calculations of extreme eigenvalues of large symmetric matrices

“…To improve the Davidson method it was proposed to use the approach introduced by Jacobi that implies the calculation of a correction vector in an orthogonal space. Following this way, the Jacobi-Davidson method has been developed [51,52,53]. In this method a new correction vector U is calculated from the equation…”

Behavior of the Correction Equations in the Jacobi–Davidson Method

“…Generally speaking, the solutions of the correction equations in (3) and (4) do not lead to the same expansion of the projection subspace, and, as emphasized in [9], it is unclear which of them is more effective in actual computations.…”

On the correction equation of the Jacobi–Davidson method

“…In the first variant, we want the columns of Q and P to be bi-orthogonal, i.e., Q H P is a diagonal matrix. Then the right correction equation reads 4) and the left correction equation reads…”

“…In [4], Genseberger and Sleijpen proposed an alternative correction equation for the Jacobi-Davidson method. Consider some subspace W such that u k ∈ W ⊆ span{V k }.…”

“…Consider some subspace W such that u k ∈ W ⊆ span{V k }. Let W be an orthonormal basis for W, the alternative correction equation reads [4]…”