Recursion Relations for the Extended Krylov Subspace Method
نویسندگان
چکیده
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(A) = span{A−m+1v, . . . , Av, v, Av, . . . , Amiv}, 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.
منابع مشابه
The Extended Krylov Subspace Method and Orthogonal Laurent Polynomials
Abstract. The need to evaluate expressions of the form f(A)v, where A is a large sparse or structured symmetric matrix, v is a vector, and f is a nonlinear function, arises in many applications. The extended Krylov subspace method can be an attractive scheme for computing approximations of such expressions. This method projects the approximation problem onto an extended Krylov subspace K(A) = s...
متن کاملSolving Ill-Posed Cauchy Problems by a Krylov Subspace Method
We study the numerical solution of a Cauchy problem for a self-adjoint elliptic partial differential equation uzz − Lu = 0 in three space dimensions (x, y, z) , where the domain is cylindrical in z. Cauchy data are given on the lower boundary and the boundary values on the upper boundary is sought. The problem is severely illposed. The formal solution is written as a hyperbolic cosine function ...
متن کاملRecursive Krylov-based multigrid cycles
We consider multigrid cycles based on the recursive use of a two–grid method, in which the coarse–grid system is solved by μ ≥ 1 steps of a Krylov subspace iterative method. The approach is further extended by allowing such inner iterations only at levels of given multiplicity, a V–cycle formulation being used at all other levels. For symmetric positive definite systems and symmetric multigrid ...
متن کاملShort recurrences for computing extended Krylov bases for Hermitian and unitary matrices
It is well known that the projection of a matrix A onto a Krylov subspace span { h, Ah, Ah, . . . , Ak−1h } results in a matrix of Hessenberg form. We show that the projection of the same matrix A onto an extended Krylov subspace, which is of the form span { A−krh, . . . , A−2h, A−1h,h, Ah, Ah . . . , A`h } , is a matrix of so-called extended Hessenberg form which can be characterized uniquely ...
متن کاملWedderburn rank reduction and Krylov subspace method for tensor approximation. Part 1: Tucker case
We propose algorithms for Tucker approximation of 3-tensor, that use it only through tensor-by-vector-by-vector multiplication subroutine. In matrix case, Krylov methods are methods of choice to approximate dominant column and row subspace of sparse or structured matrix given through matrix-by-vector subroutine. However, direct generalization to tensor case, proposed recently by Eldén and Savas...
متن کامل