Using Nonorthogonal Lanczos Vectors in the Computation of Matrix Functions

Abstract

The Lanczos algorithm uses a three-term recurrence to construct an orthonormal basis for the Krylov space corresponding to a symmetric matrix A and a nonzero starting vector $\varphi$. The vectors and recurrence coefficients produced by this algorithm can be used for a number of purposes, including solving linear systems $Au= \varphi$ and computing the matrix exponential $e^{-tA} \varphi$. Although the vectors produced in finite precision arithmetic are not orthogonal, we show why they can still be used effectively for these purposes.