Recursion relations for the extended Krylov subspace method

Abstract

The evaluation of matrix functions of the form f (A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small problems. This paper derives short recursion relations for orthonormal bases of extended Krylov subspaces of the type K m,mi+1 (A) = span{A −m+1 v,. .. , A −1 v, v, Av,. .. , A mi v}, m = 1, 2, 3,. .. , with i a positive integer, and describes applications to the evaluation of matrix functions and the computation of rational Gauss quadrature rules.

The structure of matrices in rational Gauss quadrature

“…where i is a positive integer are considered in [21]. An extension in which the relation between the numerator and denominator degrees is more general than in (2.2) has recently been discussed by Díaz-Mendoza et al [8].…”

“…This basis can be expressed with the orthogonal Laurent polynomials (2.4); v j is a multiple of φ j (A)v. Hence, the determination of an orthonormal basis for K m,im+1 (A) is equivalent to the generation of an orthonormal basis of Laurent polynomials for the space L m−1,im . This connection is used in [21] to derive short recursion relations for the vectors (2.6).…”

“…We are in a position to discuss rational Gauss quadrature rules for the approximation of integrals of the form (1.1). The case of i = 1 in (2.2) has been considered in [21], where it is shown that…”

On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations

“…To solve large linear matrix equations, several Krylov subspace projection methods have been proposed (see, e.g., [1,[13][14][15][16][17][18][19][20][21][22][23][24] and the references therein). The main idea developed in these methods is to use a block Krylov subspace or an extended block Krylov subspace and then project the original large matrix equation onto these Krylov subspaces using a Galerkin condition or a minimization property of the obtained residual.…”

“…The main problem is now how to solve the reduced order minimization problem (24). One possibility is the use of the preconditioned global conjugate gradient (PGCG) method.…”