Description
Society for Industrial & Applied Mathematics,U.S. Solving Transcendental Equations The Chebyshev Polynomial Proxy and Other Numerical Rootfinders Perturbation Series and Oracles 2014 Edition by John P. Boyd
Transcendental equations arise in every branch of science and engineering. While some of these equations are easy to solve, many are not. When confronted with such an equation, this book serves as an indispensable resource. The author's goal is to teach the art of finding the root of a single algebraic equation or a pair of such equations. This book is the first to describe the Chebyshev-proxy rootfinder, the most reliable way to find all zeros of a smooth function on an interval, and the spectrally enhanced Weyl bisection/marching triangles method for bivariate rootfinding. Unlike other books on numerical rootfinding, it includes three chapters on analytical methods - explicit solutions, regular perturbation expansions, and singular perturbation series (including hyperasymptotics). While this book is written for specialists in numerical analysis, it can be used for introductory and advanced numerical analysis classes, and as a reference for anyone working with difficult equations. Table of contents :- Preface; Notation; Part I. Introduction and Overview: 1. Introduction: key themes in rootfinding; Part II. The Chebyshev-Proxy Rootfinder and Its Generalizations. 2. The Chebyshev-proxy/companion matrix rootfinder; 3. Adaptive Chebyshev interpolation; 4. Adaptive Fourier interpolation and rootfinding; 5. Complex zeros: interpolation on a disk, the Delves-Lyness algorithm, and contour integrals; Part III. Fundamentals: Iterations, Bifurcation, and Continua: 6. Newton iteration and its kin; 7. Bifurcation theory; 8. Continuation in a parameter; Part IV. Polynomials: 9. Polynomial equations and the irony of Galois theory; 10. The quadratic equation; 11. Roots of a cubic polynomial; 12. Roots of a quartic polynomial; Part V. Analytical methods: 13. Methods for explicit solutions; 14. Regular perturbation methods for roots; 15. Singular perturbation methods: fractional powers, logarithms, and exponential asymptotics; Part VI. Classics, Special Functions, Inverses, and Oracles: 16. Classic methods for solving one equation in one unknown; 17. Special algorithms for special functions; 18. Inverse functions of one unknown; 19. Oracles: theorems and algorithms for determining the existence, nonexistence, and number of zeros; Part VII. Bivariate Systems: 20. Two equations in two unknowns; Part VIII. Challenges: 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 formula.