Publication Date
5-1-1974
Abstract
This paper studies the Generalized Eigenvalue Problem Ax=λBx for real, rectangular matrices A and B. Several current algorithms for solving this problem are examined, most of which involve a determination of the rank of B, or one of its submatrices. An example is given in which such a decision cannot be made without introducing unnecessary error into the problem. The QZR Algorithm is then introduced as an algorithm which uses unitary transformations to reduce the matrices to a prescribed canonical form. This canonical form provides the information necessary for making decisions about rank. Since A and B can be non square, additional considerations come into the concept of a solution to the Generalized Rectangular Eigenvalue Problem. Two main results are given. One relates the size of the residual back to the original matrices. This result comes from considering the minimum perturbation necessary in the original matrices to change a given "close" solution into an "exact" solution. The second result comes from a backward error analysis of the algorithm. It proves that the resulting matrices are unitarily equivalent to the original matrices plus a small perturbation. A listing of the QZR Algorithm is included, and several examples are discussed.
Degree Name
Mathematics
Level of Degree
Doctoral
Department Name
Mathematics & Statistics
First Committee Member (Chair)
Cleve Barry Moler
Second Committee Member
Alfred Samuel Carasso
Third Committee Member
Steven Arthur Pruess
Language
English
Document Type
Dissertation
Recommended Citation
Burris, Charles Henry Jr.. "The Numerical Solution of the Generalized Eigenvalue Problem for Rectangular Matrices." (1974). https://digitalrepository.unm.edu/math_etds/223
