Enhanced matrix function approximation

Abstract

Matrix functions of the form f (A) v, where A is a large symmetric matrix, f is a function, and v not equal 0 is a vector, are commonly approximated by first applying a few, say n, steps of the symmetric Lanczos process to A with the initial vector v in order to determine an orthogonal section of A. The latter is represented by a (small) n x n tridiagonal matrix to which f is applied. This approach uses the n first Lanczos vectors provided by the Lanczos process. However, n steps of the Lanczos process yield n + 1 Lanczos vectors. This paper discusses how the (n + 1) st Lanczos vector can be used to improve the quality of the computed approximation of f (A) v. Also the approximation of expressions of the form v(T) f (A) v is considered.