T. Watanabe, M. Natori, and T. Oguni (Eds.) (1994)Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs.
North-Holland Publishing Co., Amsterdam.
ⓘ
Notes:
Translated from the 1989 Japanese original.
With 1 IBM-PC floppy disk (5.25 inch)
containing a large general purpose mathematical software collection
written in Fortran, including some special function codes. Double precision.
…
►The Padé approximants can be computed by Wynn’s crossrule.
Any five approximants arranged in the Padé table as
…
►For the recursive computation of by Wynn’s epsilon algorithm, see (3.9.11) and the subsequent text.
…
…
►Gautschi (2004, p. 119–120) has explored the limit via the Wynn
-algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in , depending smoothly on , for , for an example involving first numerator Legendre OP’s.
…
►
Derivative Rule Approach
►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that
…
►►►Figure 18.40.2: Derivative Rule inversions for carried out via Lagrange and PWCF interpolations.
…
Magnify►Further, exponential convergence in , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate for these OP systems on and respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a).
…
G. Allasia and R. Besenghi (1991)Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule.
Rend. Sem. Mat. Univ. Politec. Torino49 (3), pp. 315–327.
An extended-precision C code for special functions. The library uses arbitrary-precision
ball arithmetic. Ball arithmetic is an efficient technique for performing numerical computations
with automatic and rigorous tracking of error bounds. Arb is designed with computer algebra
and computational number theory in mind, extending the principles behind FLINT to the
domain of real and complex numbers.
N. M. Temme (1978)The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions.
Report TW 183/78
Mathematisch Centrum, Amsterdam, Afdeling Toegepaste
Wiskunde.
W. J. Thompson (1997)Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners.
John Wiley & Sons Inc., New York.
ⓘ
Notes:
With CD-ROM containing a large collection of mathematical function software
written in Fortran 90 and Mathematica (an edition with the same software in C and Mathematica exists also).
The functions are computed for real variables only. Maximum accuracy 12D.
M. J. Gander and A. H. Karp (2001)Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer.
J. Quant. Spectrosc. Radiat. Transfer68 (2), pp. 213–223.
GSL (free C library)
GNU Scientific Library
The GNU Project.
ⓘ
Notes:
The GNU Scientific Library is a large general purpose numerical software library
with broad coverage of elementary and special functions.
Implementation is in double precision.
…
►A minimal cubature rule is a numerical integration rule that uses the smallest number of nodes among cubature rules of the same degree.
The nodes of these cubature rules are closely related to common zeros of OPs and they are often good points for polynomial interpolation.
Although Gaussian cubature rules rarely exist and they do not exist for centrally symmetric domains, minimal or near minimal cubature rules on the unit square are known and provide efficient numerical integration rules.
…
W. P. Reinhardt (2021a)Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering23 (4), pp. 91.
W. P. Reinhardt (2021b)Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering23 (3), pp. 56–64.
General purpose mathematical computing environment.
Includes more than 100 special functions.
Includes symbolic and arbitrary precision numerical computation.
General purpose mathematical computing environment with broad coverage of
special functions, including complex arguments and parameters.
Includes symbolic and arbitrary precision numerical computation.
General purpose mathematical computing environment.
Includes a collection of special functions.
Computations in double precision
with symbolic and extended-precision capabilities available.
System for the manipulation of symbolic and numerical expressions.
Includes a collection of special functions.
Utilizes exact fractions, arbitrary precision integers, and arbitrary
precision floating point numbers.
Mpmath is a pure-Python library for multiprecision floating-point arithmetic.
It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers,
interval arithmetic, numerical integration and differentiation, root-finding, linear algebra,
and much more.
►where is the unit vector normal to and whose direction is determined by the right-hand rule; see Figure 1.6.1.
►►►Figure 1.6.1: Vector notation.
Right-hand rule for cross products.
Magnify
…
►
