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