Publication Date

12-3-1979

Abstract

This paper is concerned with the efficient numerical solution of the matrix equation AX2 + BX + C =0, where A,B,C and X are all square matrices. Such a matrix X is called a solvent. This matrix equation is very closely related to the problem of finding scalars lambda and nonzero vectors x such that (lambda2A + lambdaB + C)x=0. The latter equation represents a quadratic eigenvalue problem with each lambda and x called an eigenvalue and eigenvector, respectively. Such equations have many important physical applications which we survey.

By presenting an algorithm to calculate solvents, we show how the eigenvalue problem can be solved as a by-product. Some comparisons are made between our algorithm and other methods currently available for solving both the solvent and eigenvalue problems. In addition, we present some new theory on the existence of a real solvent as well as a perturbation analysis of quadratic matrix equations. We also study the effects of rounding errors on the presented algorithm, and give some numerical examples.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Cleve Barry Moler

Second Committee Member

Steven Arthur Pruess

Third Committee Member

Clifford Ray Qualls

Fourth Committee Member

Henry Cosad Harpending

Language

English

Document Type

Dissertation

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