Fike C.T. Computer Evaluation of Mathematical Functions

Prentice-Hall. Englewood Cliffs, NJ, USA. 1968. 238 pages. Includes index. ISBN: 0131657208This book developed from lecture notes for a class that were taught for several years at the IBM Systems Research Institute. This is an eight-week class on mathematical methods used in function evaluation routines for digital computers. This was the first textbook to contain all the main topics relevant to this subject. Indeed, much of the material was not available in books at all and could be found only in original papers and in the documentation of existing computer programs. As the reader can see, the topics included in the book make a varied and interesting assortment and would probably not be grouped together under any other heading.A short course or seminar based on this book should fit nicely into almost any computer science curriculum. Besides being of special interest in computing, the material provides opportunities for students to employ techniques learned in other courses. Thus, in some exercises a numerical analysis student can exploit his knowledge of numerical methods for solving nonlinear equations. Similarly there are exercises in which a student can make use of his ability to program a language for algebraic manipulation (such as Formac). To be able to do some of the exercises properly, a student should be able to program in a language like Algol, Fortran, or PL/I, and should have access to a computer.The mathematical level of the book is for the most part that of a junior or senior undergraduate mathematics student. All nonelementary proofs are omitted and elementary ones are included selectively. References to suitable sources are provided for proofs that had to be omitted. A few proofs for which there was no space in the text were also made into exercises.Error in Function Evaluation Computations
Square-root and Cube-root Evaluation
Reducing the Argument Range
Polynomial Evaluation Methods
Minimax Polynomial Approximations
Chebyshev Polynomials and Chebyshev Series
Various Polynomial Approximation Methods
Rational-function Evaluation Methods
Minimax Rational Approximations
Various Rational Approximation Methods
Asymptotic Expansions