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Solving transcendental equations : the Chebyshev polynomial proxy and other numerical rootfinders, perturbation series, and oracles

Author: John P Boyd
Publisher: Philadelphia : SIAM, Society for Industrial and Applied Mathematics, [2014]
Series: Other titles in applied mathematics
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
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  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: John P Boyd
ISBN: 9781611973518 1611973511 161197352X 9781611973525
OCLC Number: 879329778
Description: xviii, 460 pages : illustrations ; 26 cm.
Contents: I: Introduction and overview --
Introduction: Key themes in rootfinding --
II: 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 --
III: Fundamentals: Interations, bifurcation, and continuation --
Newton iteration and its kin --
Bifurcation theory --
Continuation in a parameter --
IV: Polynomials --
Polynomial equations and the irony of Galois Theory --
The Quadratic Equation --
Roots of a cubic polynomial --
Roots of a quartic polynomial --
V: Analytical methods --
Methods for explicit solutions --
Regular perturbation methods for roots --
Singular perturbation methods: fractional powers, logarithms, and exponential asympototics --
VI: Classics, special functions, inverses, and oracles --
Classical 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 --
VII: Bivariate systems --
Two equations in two unknowns --
VIII: Challenges --
Past and future --
A: Companion matrices --
B: Chebyshev interpolation and quadrature --
Marching triangles --
D: Imbricate-Fourier series and the Poisson Summation Theorem.
Series Title: Other titles in applied mathematics
Responsibility: John P. Boyd, University of Michigan, Ann Arbor, Michigan.
More information:

Abstract:

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 -- Provided by publisher.

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