Fast evaluation of the hypergeometric function p F p−1(a; b; z) at the singular point z = 1 by means of the Hurwitz zeta function ζ(α, s)

Bogolubsky, A. I.; Skorokhodov, S. L.

doi:10.1134/s0361768806030054

Cited by 4 publications

(17 citation statements)

References 6 publications

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“…In order to evaluate derivatives with respect to s of ζ(s, a), we substitute s → s + x ∈ C[[x]] and evaluate (5)(6)(7)(8)(9) with the corresponding arithmetic operations done on formal power series (which may be truncated at some arbitrary finite order in an implementation). For example, the summand in (6) becomes…”

Section: Evaluation Using the Euler-maclaurin Formulamentioning

confidence: 99%

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

Johansson

2014

Numer Algor

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We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ζ(s, a) for s, a ∈ C, along with an arbitrary number of derivatives with respect to s, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.

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Section: Evaluation Using the Euler-maclaurin Formulamentioning

confidence: 99%

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

Johansson

2014

Numer Algor

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“…The timing results of [7] were for an AMD Athlon64 3500+ processor; our processor is comparable if a little faster (it is a 4600+ model).…”

Section: Zeta Function Accelerationmentioning

confidence: 99%

Acceleration of generalized hypergeometric functions through precise remainder asymptotics

Willis

2011

Numer Algor

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We express the asymptotics of the remainders of the partial sums {s n } of the generalized hypergeometric function q+1 F q α 1 ,...,α q+1 β 1 ,...,β q z through an inverse power series z n n λ c k n k , where the exponent λ and the asymptotic coefficients {c k } may be recursively computed to any desired order from the hypergeometric parameters and argument. From this we derive a new series acceleration technique that can be applied to any such function, even with complex parameters and at the branch point z = 1. For moderate parameters (up to approximately ten) a C implementation at fixed precision is very effective at computing these functions; for larger parameters an implementation in higher than machine precision would be needed. Even for larger parameters, however, our C implementation is able to correctly determine whether or not it has converged; and when it converges, its estimate of its error is accurate.

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“…for convergence acceleration of series [27]. In fact, such methods apply to slowly converging hypergeometric series [7,56], though we do not pursue this approach here.…”

Section: Bernoulli Numbers and Zeta Constantsmentioning

confidence: 99%

“…Formula (6), which allows computing U when |z| is too small to use the asymptotic series, is [21, 13.2.42]. Formula (7), which in effect gives us the asymptotic expansion for 1 F 1 , can be derived from the connection formula between 1 F 1 and U given in [21, 13.2.41]. The advantage of using U * and the form (7) instead of U is that it behaves continuously in interval arithmetic when z straddles the real axis.…”

Section: Connection Formulasmentioning

confidence: 99%

“…The advantage of using U * and the form (7) instead of U is that it behaves continuously in interval arithmetic when z straddles the real axis. The discrete jumps that occur in (7) when crossing branch cuts on the right-hand side only contribute by an exponentially small amount: when |z| is large enough for the asymptotic expansion to be used, z a−b is continuous where e z dominates and (−z) −a is continuous where e z is negligible.…”

Section: Connection Formulasmentioning

confidence: 99%

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Computing Hypergeometric Functions Rigorously

Johansson

2019

ACM Trans. Math. Softw.

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We present an efficient implementation of hypergeometric functions in arbitraryprecision interval arithmetic. The functions 0 (or the Kummer U -function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function p F q and computation of high-order parameter derivatives.

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Fast evaluation of the hypergeometric function p F p−1(a; b; z) at the singular point z = 1 by means of the Hurwitz zeta function ζ(α, s)

Cited by 4 publications

References 6 publications

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

Acceleration of generalized hypergeometric functions through precise remainder asymptotics

Computing Hypergeometric Functions Rigorously

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