Numerical methods for special functions

By Amparo Gil, Javier Segura, N. Temme. 2007

490 citations recorded by the source corpus.

Special functions arise in many problems of pure and applied mathematics, statistics, physics, and engineering. This book provides an up-to-date overview of methods for computing special functions and discusses when to use them in standard parameter domains, as well as in large and complex domains. The first part of the book covers convergent and divergent series, Chebyshev expansions, numerical quadrature, and recurrence relations. Its focus is on the computation of special functions. Pseudoalgorithms are given to help students write their own algorithms. In addition to these basic tools, the authors discuss methods for computing zeros of special functions, uniform asymptotic expansions, Pad approximations, and sequence transformations. The book also provides specific algorithms for computing several special functions (Airy functions and parabolic cylinder functions, among others). Audience: This book is intended for researchers in applied mathematics, scientific computing, physics, engineering, statistics, and other scientific disciplines in which special functions are used as computational tools. Some chapters can be used in general numerical analysis courses. Contents: List of Algorithms; Preface; Chapter 1: Introduction; Part I: Basic Methods. Chapter 2: Convergent and Divergent Series; Chapter 3: Chebyshev Expansions; Chapter 4: Linear Recurrence Relations and Associated Continued Fractions; Chapter 5: Quadrature Methods; Part II: Further Tools and Methods. Chapter 6: Numerical Aspects of Continued Fractions; Chapter 7: Computation of the Zeros of Special Functions; Chapter 8: Uniform Asymptotic Expansions; Chapter 9: Other Methods; Part III: Related Topics and Examples. Chapter 10: Inversion of Cumulative Distribution Functions; Chapter 11: Further Examples; Part IV: Software. Chapter 12: Associated Algorithms; Bibliography; Index.

Subfields: Mathematical Physics, Numerical Methods & Scientific Computing

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Citations in the corpus

Bounds for ratios of modified Bessel functions and associated Turán-type inequalities (2011, unknown): “7 of [11]).”
Incremental modeling of a new high-order polynomial surrogate model (2016, background): “the zeros of the first kind Chebyshev polynomials [30-34].”
Why Deep Neural Networks for Function Approximation? (2016, background): “The proof can be found in the reference (Gil et al., 2007).”
Tempered stable distributions and processes in finance: numerical analysis (2010, background): “This can be done by means of the analytic continuation, without having recourse neither to numerical integration nor to numerical solution of a differential equation [17] (for a complete table of the analytic continuation formulas for arbitrary values of z ∈ C and of the parameters a, b, c, see [3] ”
Scalable Log Determinants for Gaussian Process Kernel Learning (2017, background): “Chebyshev polynomials are deﬁned by the recursion For where δ j 0 is the Kronecker delta and x k = cos( π ( k +1 / 2) / ( m +1)) for k = 0 , 1 , 2 , . . . , m ; see [31].”
Fast, numerically stable computation of oscillatory integrals with stationary points (2010, methodology): “Such methods are typically based on a combination of asymptotic expansions, recurrence relationships and contour integration in the complex plane [10].”