The Extended Krylov Subspace Method and Orthogonal Laurent Polynomials
نویسندگان
چکیده
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) = span{Av, . . . , Av,v, Av, . . . , Av} of fairly small dimension, and then solves the small approximation problem so obtained. We review available results for the extended Krylov subspace method and relate them to properties of Laurent polynomials. The structure of the projected problem receives particular attention. We are concerned with the situations when m = l and m = 2l.
منابع مشابه
COMPUTING exp(−τA)b WITH LAGUERRE POLYNOMIALS
Abstract. This paper discusses a method based on Laguerre polynomials combined with a Filtered Conjugate Residual (FCR) framework to compute the product of the exponential of a matrix by a vector. The method implicitly uses an expansion of the exponential function in a series of orthogonal Laguerre polynomials, much like existing methods based on Chebyshev polynomials do. Owing to the fact that...
متن کاملConvergence analysis of the extended Krylov subspace method for the Lyapunov equation
The Extended Krylov Subspace Method has recently arisen as a competitive method for solving large-scale Lyapunov equations. Using the theoretical framework of orthogonal rational functions, in this paper we provide a general a-priori error estimate when the known term has rankone. Special cases, such as symmetric coefficient matrix, are also treated. Numerical experiments confirm the proved the...
متن کاملFormal orthogonal polynomials for an arbitrary moment matrix and Lanczos type methods∗
We give a framework for formal orthogonal polynomials with respect to an arbitrary moment matrix. When the moment matrix is Hankel, this simplifies to the classical framework. The relation with Padé approximation and with Krylov subspace methods is given. 1 Formal block orthogonal polynomials We consider a linear functional defined on the space of polynomials in two variables, defined by the mo...
متن کاملA Preconditioned Recycling GMRES Solver for Stochastic Helmholtz Problems
We present a parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace method for the numerical solution of stochastic indefinite elliptic equations with two random coefficients. Karhunen-Loève expansions are used to represent the stochastic variables and the stochastic Galerkin method with double orthogonal polynomials is used to derive a sequence of uncoupled determi...
متن کاملComputing Approximate Extended Krylov Subspaces without Explicit Inversion
It will be shown that extended Krylov subspaces –under some assumptions– can be retrieved without any explicit inversion or system solves involved. Instead we do the necessary computations of A−1v in an implicit way using the information from an enlarged standard Krylov subspace. It is well-known that both for classical and extended Krylov spaces, direct unitary similarity transformations exist...
متن کامل