Using nonorthogonal Lanczos vectors in the computation of matrix functions

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 phi. The vectors and recurrence coefficients produced by this algorithm can be used for a number of purposes, including solving linear systems Au = phi and computing the matrix exponential e(-tA)phi. Although the vectors produced in finite precision arithmetic are not orthogonal, we show why they can still be used effectively for these purposes.