Linear approximation (LA) is one of the most used computational tools in engineering, as well as in a wide range of scientific modeling. As the authors illustrate in chapter 1, the spectrum of applications for LA includes business, human physiology, mortality rate and survival analysis, fuel consumption, wind energy, chemical processes, and digital image restoration and pattern classification, to mention only a few.

This impressive work is organized into five parts, with a total of 23 chapters and four appendices. It presents a plethora of methods used in discrete LA. The first part of the book includes four chapters that introduce the reader, in tutorial style, to the universe of approximation, linear algebra and linear programming, and elementary computational functional analysis. The second chapter explains the connection between discrete LA and overdetermined linear equations, provides comparisons among different norms used in approximation, and gives readers practical information about computing precision, data interpretation, and representation of matrices and vectors in computers for use in C programming. Dynamic allocation is used to support variable size problems. The third chapter presents the simplex algorithm, and also motivates the use of linear programming to solve LA models. Chapter 4, “Efficient Solutions of Linear Equations,” provides a clear presentation of vector and matrix norms, ill-conditioned matrices, Gauss LU factorization, orthogonal factorization methods, and the Gauss-Jordan method, in addition to a complete presentation of rounding errors in arithmetic operations.

In the second part of the book, the L₁ approximation is investigated. Chapters 5 through 9 cover all aspects of L₁ approximation: basic facts, one-sided approximation, bounded variables algorithm, polygonal approximation of place curves, and piecewise L₁ approximation of plane curves. The third part of the book, chapters 10 to 16, presents theoretical aspects and algorithms of Chebyshev approximation: basic facts, one-sided approach, bounded variables algorithm, constrained approximation, strict approximation, piecewise methodology, and linear inequalities. The fourth section, in three chapters, covers least squares approximation, describing the pseudoinverse approach for solving linear least squares problems. Chapter 17 addresses factorization methods, explicit formulas for pseudoinverse, the singular value decomposition method, and practical computational aspects related to methods of decomposition such as those of Cholesky, Gauss, and Householder. The pseudoinverse in linear spaces, multicollinearity, collinearity, and ill-conditioned aspects are discussed in the last two sections of the chapter. Piecewise linear least squares approximation is analyzed in chapter 18, and chapter 19 gives readers methods to find solutions of ill-posed linear systems.

The last section is dedicated to underdetermined systems of linear equations. Chapters 20 and 21 discuss the L₁ solution of underdetermined linear systems--not only the fundamental case, but also the bounded case (each element of the solution would be bounded between -1 and +1). The case when the L₁ norm of the solution vector is as small as possible is also analyzed for the bounded case. The algorithms for Chebyshev approximation in the case of an underdetermined case is analyzed in chapter 22, while the bounded least square solution of such systems can be obtained as shown in chapter 23. The appendices offer information about C function prototypes, utilities, and C common functions.

C source code is provided for the algorithms described in the book. The code is distributed among the chapters and a main program is presented in Appendix B. The reader will benefit from the included CD-ROM containing the source code, test drivers, and common functions. The code is portable with minimum intervention, as explained in the text.

The first of the appendices lists all of the references used in the book. However, a chapter-by-chapter bibliography is also provided. The references were selected to be relevant to the book’s subject. I also appreciated the well-designed index.

I recommend this book to all people working in scientific computing. The entire work represents a valuable reference in numerical linear approximation, and could be used by students (undergraduates or graduates), scientists, and engineers.