MONOTONE CONVERGENCE OF THE LANCZOS APPROXIMATIONS TO MATRIX FUNCTIONS OF HERMITIAN MATRICES
Andreas Frommer · 2009
When A is a Hermitian matrix, the action f(A)b of a matrix function f(A) on a vector b can efficiently be approximated via the Lanczos method. In this note we use M-matrix theory to establish that the 2-norm of the error of the sequence of approximations is monotonically decreasing if f is a Stieltjes transform and A is positive definite. We discuss the relation of our approach to a recent, more general monotonicity result of Druskin for Laplace transforms. We also extend the class of functions to certain product type functions. This yields, for example, monotonicity when approximating sign(A)b with A indefinite if the Lanczos method is performed for A2 rather than A.