Philadelphia: SIAM, 2014. - 446p.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 NotationIntroduction and OverviewIntroduction: Key Themes in RootfindingThe Chebyshev-Proxy Rootfinder and Its GeneralizationsThe 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 IntegralsFundamentals: Iterations, Bifurcation, and ContinuationNewton Iteration and Its Kin Bifurcation Theory Continuation in a ParameterPolynomialsPolynomial Equations and the Irony of Galois Theory The Quadratic Equation Roots of a Cubic Polynomial Roots of a Quartic PolynomialAnalytical MethodsMethods for Explicit Solutions Regular Perturbation Methods for Roots Singular Perturbation Methods: Fractional Powers, Logarithms, and Exponential AsymptoticsClassics, Special Functions, Inverses, and OraclesClassic 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 ZerosBivariate Systems Two Equations in Two UnknownsChallenges Past and FutureCompanion Matrices Chebyshev Interpolation and Quadrature Marching Triangles Imbricate-Fourier Series and the Poisson Summation Theorem Glossary Bibliography Index
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