The extended global Lanczos method, Gauss–Radau quadrature, and matrix function approximation
Résumé
The need to evaluate expressions of the form I(f ) := trace (W T f (A)W ), where the matrix
A ∈ Rn×n is symmetric, W ∈ Rn×k with 1 ≤ k ≪ n, and f is a function defined on
the convex hull of the spectrum of A, arises in many applications including network
analysis and machine learning. When the matrix A is large, the evaluation of I(f ) by
first computing f (A) may be prohibitively expensive. In this situation it is attractive to
compute an approximation of I(f ) by first applying a few steps of a global Lanczos-type
method to reduce A to a small matrix and then evaluating f at this reduced matrix. The
computed approximation can be interpreted as a quadrature rule. The present paper
generalizes the extended global Lanczos method introduced in Bentbib et al. (2018)
and discusses the computation of error-bounds and error estimates. Numerical examples
illustrate the performance of the techniques described.