Computational methods in linear algebra and related problems

Abstract

The importance of linear algebra can be summarized by the single statement: "Linear Algebra pervades and enriches almost all areas in numerical computation." Ever so many methods are available in literature for the computer solution of problems in linear algebra. Yet, this field is ever-expanding, with more and more new concepts and algorithms being developed almost every day. The reason for this rapid growth in this area is partly due to the advent of very-high-speed, large-memory computers and partly due to the fact that no one definite computational method in linear algebra can be said to be best suited for all types of a particular problem. The objective of this dissertation is to present the new studies which the author has carried out during the past few years on the following general and varied aspects of computational problems in linear algebra, as well as the closely related problem of solving for the zeros of a polynomial. i) Properties of matrices under certain logical operations. ii) Construction of a class of matrices called simply-invertible matrices. iii) Triangular partitioning scheme for matrix inversion. iv) Power series method for inverting matrices. v) Preventing failure of LU-decomposition by a correction procedure. vi) Methods for obtaining generalized inverse of rectangular and singular square matrices over real, complex, and finite fields. vii) A study that a combined Newton–McAuley method is better suited for solving polynomials with repeated roots than either of these methods independently. All these aspects are organized in this dissertation in nine chapters.