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Reviewer: Gabriel Constantin Barzescu

Ever-increasing numbers of computer specialists and non-specialists face computational problems whose solution requires numerical analysis methods. Consequently, a new book about these methods and their implementation on microcomputers is of real interest for many people even if it is only a practical introduction that addresses students and practicing engineers, rather than a book for scientists or research engineers with serious backgrounds in the field. This book is organized into 11 chapters. Chapter 1 provides an elementary introduction to some aspects of computers and computing. It discusses error reduction in numerical analysis and gives some history of computational complexity and numerical analysis. Chapter 2 covers both simple and complex methods of solving nonlinear algebraic equations, including the quotient-difference method and the methods of Bairstow, Laguerre, Moore, and Broyden. Chapter 3, on solving linear algebraic equations, presents the well-known methods of Gauss and Gauss-Jordan, the triangular decomposition method, and the iteration methods of Jacobi and Gauss-Seidel. It gives some refinements and discusses convergence criteria and other aspects of these methods. Chapter 4, “Numerical Integration,” contains the methods that are used most in the field, including Romberg's, Gauss's, and Patterson's. Chapter 5 deals with numerical differentiation, and Chapter 6 is a comprehensive overview of methods to solve differential equations (Euler's, Runge-Kutta, predictor-corrector, finite-difference, etc.). Chapter 7 contains some methods for approximating functions (e.g., methods based on Chebyshev polynomials). Chapter 8, “Fitting Functions to Data,” contains such classical methods as Lagrange's interpolation formula and the least squares method together with some aspects of Fourier analysis and piecewise curve fitting with cubic splines. Chapter 9 presents Monte Carlo methods, and Chapter 10 discusses the algebraic eigenvalue problem. The last chapter gives some information about the testing of microcomputer performance and about some software libraries for numerical analysis. Every chapter contains BASIC programs that illustrate most of the methods described. The choice of BASIC is questionable, of course, but it is not a serious impediment. The book addresses nonspecialists in numerical analysis but leaves some topics insufficiently developed for this audience. For example, the cubic spline approach given in Chapter 8 is computationally inefficient because of the instabilities induced by the monomial representation. For computational purposes, a cubic spline is better represented as a linear combination of uniform B-splines. Moreover, this approach has the advantage of local control over the shape of the curve. The book has a general bibliography of over 60 titles, but the lack of a consistent list of references for each chapter or method will impede those who want detailed information on specific topics. Despite these shortcomings, the book is a good compromise between an introductory theoretical course and a practical guide to simple implementations of the numerical analysis methods widely used on microcomputers.