Year of Publication

2015

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department

Mathematics

First Advisor

Dr. Qiang Ye

Abstract

The problem of computing the matrix exponential etA arises in many theoretical and practical problems. Many methods have been developed to accurately and efficiently compute this matrix function or its product with a vector, i.e., etAv. In the past few decades, with the increasing need of the computation for large sparse matrices, iterative methods such as the Krylov subspace methods have proved to be a powerful class of methods in dealing with many linear algebra problems. The Krylov subspace methods have been introduced for computing matrix exponentials by Gallopoulos and Saad, and the corresponding error bounds that aim at explaining the convergence properties have been extensively studied. Many of those bounds show that the speed of convergence depends on the norm of the matrix, while some others emphasize the important role played by the spectral distribution. For example, it is shown in a recent work by Ye that the speed of convergence is also determined by the condition number for a symmetric negative definite matrix. Namely the convergence is fast for a well-conditioned matrix no matter how large the norm is.

In this dissertation, we derive new error bounds for computing etAv for non-symmetric A, using the spectral information of A. Our result is based on the assumption that A is negative definite, i.e., the field of values of A lies entirely in the left half of the complex plane, such that the underlying dynamic system is stable. The new bounds show that the speed of convergence is related to the size and shape of the rectangle containing the field of values, and they agree with the existing results when A is symmetric. Furthermore, we also derive a simpler error bound for the special case when A is skew-Hermitian. This bound explains an observed convergence behavior where the approximation error initially stagnates for certain number of iterations before it starts to converge. In deriving our new error bounds, we use sharper estimates of the decay property of exponentials of Hessenberg matrices, by constructing Faber polynomial approximating exponential function in the region containing the field of values. The Jacobi elliptic functions are used to construct the conformal mappings and generate the Faber polynomials. We also present numerical tests to demonstrate the behavior of the new error bounds.

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