Solving transcendental equations : the Chebyshev polynomial proxy and other numerical rootfinders, perturbation series, and oracles /
Boyd, John P.(John Philip),1951-author.
Includes bibliographical references and index.
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.
Transcendental functions.
Chebyshev polynomials.
Transcendental numbers.