Error Bounds for the Krylov Subspace Methods for Computations of Matrix Exponentials

2016/03/23 by Hao Wang, Wang, Hao, Qiang Ye +1
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1603.07358

openalex publication_date 2016/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing e^-τAv for a given τ>0 and v ∈ Cⁿ, where A is a large sparse non-Hermitian matrix. The \em a priori error bounds relate the convergence to λ_min((A+A^*)/(2)), λ_max((A+A^*)/(2)) (the smallest and the largest eigenvalue of the Hermitian part of A) and |λ_max((A-A^*)/(2))| (the largest eigenvalue in absolute value of the skew-Hermitian part of A), which define a rectangular region enclosing the field of values of A. In particular, our bounds explain an observed superlinear convergence behavior where the error may first stagnate for certain iterations before it starts to converge. The special case that A is skew-Hermitian is also considered. Numerical examples are given to demonstrate the theoretical bounds.

Citations

• Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later
• The logarithmic norm. History and modern theory

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