Wynn%20cross%20rule
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1: 20 Theta Functions
Chapter 20 Theta Functions
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Fast construction of the Fejér and Clenshaw-Curtis quadrature rules.
BIT 46 (1), pp. 195–202.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs.
North-Holland Publishing Co., Amsterdam.
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Upon systems of recursions which obtain among the quotients of the Padé table.
Numer. Math. 8 (3), pp. 264–269.
3: 8 Incomplete Gamma and Related
Functions
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4: 28 Mathieu Functions and Hill’s Equation
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5: 18.40 Methods of Computation
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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.
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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 … ►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). …6: 23 Weierstrass Elliptic and Modular
Functions
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7: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Maplesoft.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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8: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Arb: A C Library for Arbitrary Precision Ball Arithmetic.
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Center for Algorithms and Interactive Scientific Software.
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9: 3.11 Approximation Techniques
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►The Padé approximants can be computed by Wynn’s cross rule.
Any five approximants arranged in the Padé table as
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►For the recursive computation of by Wynn’s epsilon algorithm, see (3.9.11) and the subsequent text.
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10: Bibliography G
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Recurrence relations for cross-products of Bessel functions.
Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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GNU Scientific Library
The GNU Project.
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