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Extended block Hessenberg process for the evaluation of matrix functions

A. H. Bentbib, M. EL Ghomari, K. Jbilou, EL. M. Sadek

TL;DR

This work addresses efficient computation of matrix functions applied to a block $V$ and the solution of shifted linear systems for large sparse matrices $A$ by projecting onto an extended block Krylov subspace ${\rm K}^e_m(A,V)$. It introduces the extended block Hessenberg process (EBHA) that builds a concise basis ${\rm V}_{2m}$ and a small projected matrix ${\rm \nT}_{2m}$, enabling the approximation ${\rm I}_{2m}(f)= {\rm \nV}_{2m} f({\rm \nT}_{2m}) E_1 \boldsymbol{\nabla}_{1,1}$. The paper proves exactness for Laurent polynomials up to certain degrees and provides an exponential error bound, along with a practical MF-EBH scheme and a restarted approach for multiple shifted systems. Numerical experiments on large-scale matrices demonstrate that MF-EBH often achieves lower CPU time than the extended block Arnoldi method while maintaining accuracy. The methods have broad applicability in scientific computing where large-scale matrix functions and multi-RHS shifted systems arise.

Abstract

In the present paper, we propose a block variant of the extended Hessenberg process for computing approximations of matrix functions and other problems producing large-scale matrices. Applications to the computation of a matrix function such as f(A)V, where A is an nxn large sparse matrix, V is an nxp block with p<<n, and f is a function are presented. Solving shifted linear systems with multiple right hand sides are also given. Computing approximations of these matrix problems appear in many scientific and engineering applications. Different numerical experiments are provided to show the effectiveness of the proposed method for these problems.

Extended block Hessenberg process for the evaluation of matrix functions

TL;DR

This work addresses efficient computation of matrix functions applied to a block and the solution of shifted linear systems for large sparse matrices by projecting onto an extended block Krylov subspace . It introduces the extended block Hessenberg process (EBHA) that builds a concise basis and a small projected matrix , enabling the approximation . The paper proves exactness for Laurent polynomials up to certain degrees and provides an exponential error bound, along with a practical MF-EBH scheme and a restarted approach for multiple shifted systems. Numerical experiments on large-scale matrices demonstrate that MF-EBH often achieves lower CPU time than the extended block Arnoldi method while maintaining accuracy. The methods have broad applicability in scientific computing where large-scale matrix functions and multi-RHS shifted systems arise.

Abstract

In the present paper, we propose a block variant of the extended Hessenberg process for computing approximations of matrix functions and other problems producing large-scale matrices. Applications to the computation of a matrix function such as f(A)V, where A is an nxn large sparse matrix, V is an nxp block with p<<n, and f is a function are presented. Solving shifted linear systems with multiple right hand sides are also given. Computing approximations of these matrix problems appear in many scientific and engineering applications. Different numerical experiments are provided to show the effectiveness of the proposed method for these problems.
Paper Structure (9 sections, 6 theorems, 75 equations, 1 figure, 6 tables, 3 algorithms)

This paper contains 9 sections, 6 theorems, 75 equations, 1 figure, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The extended block Hessenberg process
  3. Application to the approximation of matrix functions
  4. Approximations for $\exp(A)V$
  5. Shifted block linear systems
  6. Numerical experiments
  7. Examples for the approximation of matrix functions $f(A)V$
  8. Examples of the shifted linear systems
  9. Conclusion

Key Result

Proposition 2.1

Assume that $m$ steps of Algorithm algorithm:EBHA have been run and let $\widetilde{\mathbb{T}}_{2m}=\mathbb{V}_{2m+1}^LA\mathbb{V}_{2m}\in{\mathbb R}^{(2m+1)p\times 2mp}$, with $\mathbb{V}_{2m+1}=[V_1,\ldots,V_{2m},V_{2m+1}]\in\mathbb{R}^{n\times (2m+1)p}$, then we have the following relation where the matrix $E_m=[e_{2p(m-1)+1},\ldots,e_{2mp}]\in{\mathbb R}^{2mp\times 2p}$ is made up of the las

Figures (1)

  • Figure 5.1: Evolution of the absolute error determined by the MF-EBH method when approximating $\sqrt{A}V$ (left plot) and $\log{(A)}V$ (right plot)

Theorems & Definitions (12)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 2 more