E-Book Overview
Transcendental equations arise in every branch of science and engineering. While most of these equations are easy to solve, some are not, and that is where this book serves as the mathematical equivalent of a skydiver's reserve parachute - not always needed, but indispensible when it is. The author s goal is to teach the art of finding the root of a single algebraic equation or a pair of such equations.
Solving Transcendental Equations is unique in that it is the first book to describe the Chebyshev-proxy rootfinder, which is the most reliable way to find all zeros of a smooth function on the interval, and the very reliable spectrally enhanced Weyl bisection/marching triangles method for bivariate rootfinding. It also includes three chapters on analytical methods - explicit solutions, regular pertubation expansions, and singular perturbation series (including hyperasymptotics) - unlike other books that give only numerical algorithms for solving algebraic and transcendental equations.
Audience: This book is written for specialists in numerical analysis and will also appeal to mathematicians in general. It can be used for introductory and advanced numerical analysis classes, and as a reference for engineers and others working with difficult equations.
Contents: Preface; Notation; Part I: Introduction and Overview; Chapter 1: Introduction: Key Themes in Rootfinding; Part II: The Chebyshev-Proxy Rootfinder and Its Generalizations; Chapter 2: The Chebyshev-Proxy/Companion Matrix Rootfinder; Chapter 3: Adaptive Chebyshev Interpolation; Chapter 4: Adaptive Fourier Interpolation and Rootfinding; Chapter 5: Complex Zeros: Interpolation on a Disk, the Delves-Lyness Algorithm, and Contour Integrals; Part III: Fundamentals: Iterations, Bifurcation, and Continuation; Chapter 6: Newton Iteration and Its Kin; Chapter 7: Bifurcation Theory; Chapter 8: Continuation in a Parameter; Part IV: Polynomials; Chapter 9: Polynomial Equations and the Irony of Galois Theory; Chapter 10: The Quadratic Equation; Chapter 11: Roots of a Cubic Polynomial; Chapter 12: Roots of a Quartic Polynomial; Part V: Analytical Methods; Chapter 13: Methods for Explicit Solutions; Chapter 14: Regular Perturbation Methods for Roots; Chapter 15: Singular Perturbation Methods: Fractional Powers, Logarithms, and Exponential Asymptotics; Part VI: Classics, Special Functions, Inverses, and Oracles; Chapter 16: Classic Methods for Solving One Equation in One Unknown; Chapter 17: Special Algorithms for Special Functions; Chapter 18: Inverse Functions of One Unknown; Chapter 19: Oracles: Theorems and Algorithms for Determining the Existence, Nonexistence, and Number of Zeros; Part VII: Bivariate Systems; Chapter 20: Two Equations in Two Unknowns; Part VIII: Challenges; Chapter 21: Past and Future; Appendix A: Companion Matrices; Appendix B: Chebyshev Interpolation and Quadrature; Appendix C: Marching Triangles; Appendix D: Imbricate-Fourier Series and the Poisson Summation Theorem; Glossary; Bibliography; Index
E-Book Content
SOLVING TRANSCENDENTAL EQUATIONS
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SOLVING TRANSCENDENTAL EQUATIONS The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series, and Oracles John P. Boyd
University of Michigan Ann Arbor, Michigan
Society for Industrial and Applied Mathematics Philadelphia
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Copyright © 2014 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Phil