This book presents a general class of numerical algorithms for solving systems of linear equations and systems of nonlinear equations. The algorithms belong to a class of projection methods. The contributions of this well-written monograph are twofold. First, it presents a unified approach to and generalizes classical algorithms for solving linear and nonlinear systems. The authors show these classical algorithms to belong to a general class, the ABS class (the authors, Abaffy and Spedicato, have named the class ABS to acknowledge the contributions of Abaffy, Charles Broyden, and Spedicato in introducing parameters associated with the class of algorithms). A basic concept defining the class is that for a linear system Ax=b of m equations in n unknowns, the ABS method computes a solution x as the (m+1)th iterate of a sequence of approximations x(i) to x having the following property: the approximation x(i+1) obtained at the ith iteration is a solution to the first i equations. In the linear case, the ABS class contains essentially all possible algorithms with the property that “they can solve, in exact arithmetic, a linear system starting from an arbitrary point and in a number of iterations no greater than the number of equations.” In the nonlinear case the ABS class contains Newton’s method and many of its generalizations.

A second contribution of the book may lie in the evidence it provides on how well the ABS algorithms can perform in terms of accuracy, storage requirements, and overhead costs. The authors provide theoretical comparisons between the ABS and some classical algorithms. In some ways, however, the results presented here might be considered preliminary. For the most part, this monograph omits software implementation, numerical computations, and performance on vector and parallel computers. The authors postpone these topics for further consideration.