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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.
Notation
Introduction and Overview
Introduction: Key Themes in Rootfinding
The Chebyshev-Proxy Rootfinder and Its Generalizations
The Chebyshev-Proxy/Companion Matrix Rootfinder
Adaptive Chebyshev Interpolation
Adaptive Fourier Interpolation and Rootfinding
Complex Zeros: Interpolation on a Disk, the Delves-Lyness Algorithm, and Contour Integrals
Fundamentals: Iterations, Bifurcation, and Continuation
Newton Iteration and Its Kin
Bifurcation Theory
Continuation in a Parameter
Polynomials
Polynomial Equations and the Irony of Galois Theory
The Quadratic Equation
Roots of a Cubic Polynomial
Roots of a Quartic Polynomial
Analytical Methods
Methods for Explicit Solutions
Regular Perturbation Methods for Roots
Singular Perturbation Methods: Fractional Powers, Logarithms, and Exponential Asymptotics
Classics, Special Functions, Inverses, and Oracles
Classic Methods for Solving One Equation in One Unknown
Special Algorithms for Special Functions
Inverse Functions of One Unknown
Oracles: Theorems and Algorithms for Determining the Existence, Nonexistence, and Number of Zeros
Bivariate Systems
Two Equations in Two Unknowns
Challenges
Past and Future
Companion Matrices
Chebyshev Interpolation and Quadrature
Marching Triangles
Imbricate-Fourier Series and the Poisson Summation Theorem