This book is written as an introduction to polynomial matrix computa­ tions. It is a companion volume to an earlier book on Methods and Applications of Error-Free Computation by R. T. Gregory and myself, published by Springer-Verlag, New York, 1984. This book is intended for seniors and graduate students in computer and system sciences, and mathematics, and for researchers in the fields of computer science, numerical analysis, systems theory, and computer algebra. Chapter I introduces the basic concepts of abstract algebra, including power series and polynomials. This chapter is essentially meant for bridging the gap between the abstract algebra and polynomial matrix computations. Chapter II is concerned with the evaluation and interpolation of polynomials. The use of these techniques for exact inversion of poly­ nomial matrices is explained in the light of currently available error-free computation methods. In Chapter III, the principles and practice of Fourier evaluation and interpolation are described. In particular, the application of error-free discrete Fourier transforms for polynomial matrix computations is consi­ dered.

I Algebraic Concepts.- 1 Introduction.- 2 Groups, Rings, Integral Domains, and Fields.- 3 Power Series and Polynomials.- 4 Chinese Remainder Theorem and Interpolation.- 5 Polynomials in Several Variables.- II Polynomial Matrix-Evaluation, Interpolation, Inversion.- 1 Introduction.- 2 Results from Matrix Theory.- 3 Matrix Method-Evaluation and Interpolation of Single Variable Polynomials.- 4 Tensor Product Method-Evaluation and Interpolation of Multi-variable Polynomials.- III Fourier Evaluation and Interpolation.- 1 Introduction.- 2 Discrete Fourier Transform over a Ring.- 3 Convolution.- 4 Error-Free DFT.- 5 Polynomial Evaluation-Interpolation-Multiplication.- 6 Multivariable Polynomial Interpolation.- IV Polynomial Hensel Codes.- 1 Introduction.- 2 Hensel Fields.- 3 Isomorphic Algebras.- 4 Hensel Codes for Rational Polynomials.- 5 Arithmetic of Hensel Codes.- 6 Forward and Inverse Mapping Algorithms.- 7 Direct Solution of Linear Systems and Matrix Inversion.- 8 Hensel-Newton-Schultz Iterative Matrix Inversion.- V Matrix Computations-Euclidean and Non-Euclidean Domains.- 1 Introduction.- 2 Matrices over Euclidean Domains.- 3 Matrices over Non-Euclidean Domains.- 4 Multivariable Polynomial Hensel Codes.