This book contains the proceedings of the Institute of Mathematics and its Applications (IMA) conference on Algorithms for the Approximation of Functions and Data (held in July 1985 at the Royal Military College of Science in Shrivenham, England). The primary objective of this conference was to establish a strong theoretical approach to the most important topics in approximation theory, especially those topics that concern major areas of current CAD research. Most of the papers were presented by well-known authorities in approximation theory, so the standard is high. The style is strictly mathematical and the reader will need a solid background. The book will greatly interest only advanced researchers and specialists in applied mathematics for CAD research.

Each of the book’s three parts is divided into several sections. Part 1, “Development of Algorithms,” defines the theoretical framework for the algorithms discussed throughout the book, and its sections cover spline approximation and smoothing, spline interpolation and shape preservation, multivariate interpolation, least squares methods, rational approximation, complex and nonlinear approximation, CAD, and blending. Most of the papers study the use of polynomial splines and related algorithms in the data-fitting and design process; scattered data fitting seems to be the central theme for most of the algorithms described in this part of the book. Accompanying the highly theoretical papers are review papers on subdivision algorithms, curve interpolation with shape control, radial basis functions for multivariate interpolation, and spaces of piecewise polynomials on triangulations. Even if CAD researchers, accustomed to thinking of free-form curves and surfaces in terms of vector-valued parametric functions, find it difficult to deal with the scalar-valued approximation used throughout this book, they will still discover useful approaches to new and important methods in CAD research.

Part 2, “Applications,” contains sections on applications in numerical analysis, partial differential equations, and other disciplines. This part explores special well-defined topic areas as particular instances of the general methodology developed throughout Part 1. These topics include multigrid methods, integral equations, gamma functions, fluid flow fields, vibrating structures, and meteorological data analysis.

Finally, Part 3, “Software,” contains descriptions of the NAG and NPL libraries of data approximation subroutines. All of the papers in this book benefit from the IMA’s excellent job of typesetting, which seems to underline the value and consistency of the book.