Numerical Analysis PDF

PREFACE

This book is designed for a first course in numerical analysis. It differs considerably from other such texts in its choice of topics. Our concern has been with the needs of science, engineering and applied mathematics students, who, in increasing numbers, form a majority of the students in numerical analysis courses. Thus, we have presented a broad spectrum of topics of applied interest, which, in fact, bring the reader to various frontiers of the subject. We have been able to do this by purging the traditional curriculum of many topics which are not likely to be encountered in either science or technology. These topics include, for example, number systems, computer operations, the secant method, Weddle’s rule, Richardson extrapolation, and the methods of Graeffe and Milne. Related decisions were guided by a desire to include nonlinear equations, in addition to linear ones.

TTie need to solve a wide spectrum of nonlinear problems has been increasing since the end of the nineteenth century. For example, accurate simulation of dynamical processes related to planetary motion, rocket propulsion, turbulent flow, chemical oscillators, diffusion processes, and elastic stress demand the ability to solve nonlinear equations. Accurate fitting of data derived by use of laser technology, atomic clocks, electron microscopes and/or radio telescopes requires greater sophistication than that available from a linear least square fit. Quantitative methodology applicable to large classes of nonlinear problems became available only with the development of modem digital computers, and the result has been, and probably will continue to be, an explosion of knowledge.

In the final chapter, we have included a study of the Navier-Stokes equations, a fully nonlinear system of fluid dynamical equations which, interestingly enough, can be derived from both the macro, or hydrodynamic, approach and a micro, or molecular, approach. These equations are among the most challenging in contemporary numerical analysis and are fundamental to diverse studies relating to such areas as weather prediction, aerodynamics, petroleum recovery, cardiovascular circulation, heat convection, ocean currents, and the not-so-ordinary flow of ordinary water in pipes. Interest in the Navier- Stokes equations is so broad that we felt their inclusion to be appropriate.

One of the major goals in writing this book was to develop methodology for which the numerical solution of a given problem has the same qualitative behavior as the analytical solution, for, thereby, the numerical solution preserves the physics of a given mathematical model. Another major goal was to develop numerical analysis in such a fashion that the reader would be able to apply the methods thoughtfully and within a reasonably short time to problems which he or she finds both interesting and significant. For this reason, methodology, theory and intuition have been interwoven throughout.

The book is suitable for a junior, senior or first year graduate course. Only a familiarity with computer programming and ordinary differential equations is assumed throughout. We have set stars before various sections to denote material of relative difficulty. Chapters 1-5, exclusive of the starred sections, provide ample material for a one semester undergraduate course, assuming computer implementation by the student. In their entirety, these chapters can be used for a one semester graduate course. Chapters 6-9 axe designed for the second half of a full year course. These chapters deal with partial differential equations, and, in each, the first few sections develop the necessary mathematical background. Note that a star has been affixed to the title of Chapter 9 to indicate the advanced nature of the entire chapter. Observe also that the exercises have been divided into two sets, basic ones and supplementary ones. Among the supplementary ones are several unexpected surprises.

In our own teaching, our philosophy has been that computer implementation by the student is essential. As a consequence, the time required for the study of various topics has been greater than that required in a purely theoretical lecture course. Nevertheless, from an applied point of view, a numerical algorithm that does not run on a computer is useless, and the student should verify an algorithm’s viability by direct computer implementation. Theory is important in numerical analysis, but theory, alone, will fail to meet the needs of the majority of our students.

Finally, we wish to thank those undergraduate students at the University of Trento and those undergraduate and graduate students at the University of Texas at Arlington who contributed in so many ways to the final structure of the book. Unfortunately, the list of names is too long for inclusion.

Author: Donald Greenspan