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Journal Articles Journal of Computational and Applied Mathematics Year : 2021

The extended global Lanczos method, Gauss–Radau quadrature, and matrix function approximation

Carl Jagels
  • Function : Author
Lothar Reichel

Abstract

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.

Dates and versions

hal-04413526 , version 1 (23-01-2024)

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Carl Jagels, Khalide Jbilou, Lothar Reichel. The extended global Lanczos method, Gauss–Radau quadrature, and matrix function approximation. Journal of Computational and Applied Mathematics, 2021, 381, pp.113027. ⟨10.1016/j.cam.2020.113027⟩. ⟨hal-04413526⟩
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