Computing Hypergeometric Functions Rigorously

Johansson, Fredrik

doi:10.1145/3328732

Cited by 40 publications

(32 citation statements)

References 57 publications

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“…The point z = 1 is a branch point, and the interval [1, ∞] is the branch cut, with the evaluation on this cut taking the limit from below the cut. We need to look at [63,91,11]. The hypergeometric function 2 F 1 (a, b; c; H) can be rewritten 26 This function is also not analytic on C\R >0 , as it has a cut wherever…”

Section: Discussionmentioning

confidence: 99%

Stieltjes Moment Sequences for Pattern-Avoiding Permutations

Bostan

Price²,

Guttmann³

et al. 2020

Electron. J. Combin.

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A small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as log-convexity, which are useful to rigorously bound their growth constant from below. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences, $Av(123\ldots k)$, are Stieltjes moment sequences, and we explicitly find the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool. We first illustrate our approach on two basic examples, $Av(123)$ and $Av(1342)$, whose generating functions are algebraic. We next investigate the general (transcendental) case of $Av(123\ldots k)$, which counts permutations whose longest increasing subsequences have length at most $k-1$. We show that the generating functions of the sequences $\, Av(1234)$ and $\, Av(12345)$ correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian $\, _2F_1$ hypergeometric function, respectively to an order-two linear differential operator acting on the square of a classical modular form given as a pullback of a $\, _2F_1$ hypergeometric function. We demonstrate that the density function for the Stieltjes moment sequence $Av(123\ldots k)$ is closely, but non-trivially, related to the density attached to the distance traveled by a walk in the plane with $k-1$ unit steps in random directions. Finally, we study the challenging case of the $Av(1324)$ sequence and give compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, we show how rigorous lower bounds on the growth constant of this sequence can be constructed, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give an estimate of the (unknown) growth constant.

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Section: Discussionmentioning

confidence: 99%

Stieltjes Moment Sequences for Pattern-Avoiding Permutations

Bostan

Price²,

Guttmann³

et al. 2020

Electron. J. Combin.

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“…Bounds that cover wider classes of functions become more complicated as the number of parameters increases, yet adequate tail bounds are available in the literature and used in practice for common special functions depending on parameters. For example, Du and Yap [DY05,Section 3] or Johansson [Joh16,Section 4.1] present bounds that cover general hypergeometric functions.…”

Section: Differentially Finite Functions As Special Functionsmentioning

confidence: 99%

Truncation Bounds for Differentially Finite Series

Mezzarobba¹

2019

Annales Henri Lebesgue

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We describe a flexible symbolic-numeric algorithm for computing bounds on the tails of series solutions of linear differential equations with polynomial coefficients. Such bounds are useful in rigorous numerics, in particular in rigorous versions of the Taylor method of numerical integration of ODEs and related algorithms. The focus of this work is on obtaining tight bounds in practice at an acceptable computational cost, even for equations of high order with coefficients of large degree. Our algorithm fully covers the case of generalized series expansions at regular singular points. We provide a complete implementation in SageMath and use it to validate the method in practice. Résumé.-Nous décrivons un algorithme symbolique-numérique souple pour le calcul de bornes sur les restes de solutions séries d'équations différentielles linéaires à coefficients polynomiaux. Ces bornes sont destinées au calcul numérique rigoureux, et utiles notamment dans des versions rigoureuses de la méthode de Taylor d'intégration des EDO ou d'autres algorithmes apparentés. L'objectif principal de ce travail est d'obtenir des bornes fines en pratique pour un coût de calcul acceptable, y compris dans le cas d'équations d'ordre élevé à coefficients de grand degré. Notre algorithme couvre entièrement le cas des développement en séries généralisées au voisinage de points singuliers réguliers. Nous présentons une implémentation complète de la méthode en SageMath, et nous l'utilisons pour valider son bon comportement en pratique.

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“…The numerical calculation may again break down for large x or certain choices of parameters. Therefore we repeat the calculations with higher precision using the Arb C library for arbitrary-precision ball arithmetic [14] which implements the hypergeometric function [15]. For the computation we start with 128 bit precision and increase the precision up to 4096 bits trying to achieve at least 64 bits of relative precision for the computed result.…”

Section: Network Latencymentioning

confidence: 99%

Network-Wide Measurement of GPRS Bandwidth and Latency

Pfitzinger¹,

Baumann²,

Emde³

et al. 2019

Proceedings of the Annual Hawaii International Conference on System Sciences

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Complementing a recently collected large data set on the TCP connection termination latency in GPRS networks we analyze server-side log data generated in a large scale automatic toll system to observe the network bandwidth. After a recent architectural change the on-board units (OBUs) record GPS tracks and transmit track data to the central system for processing rather than transmitting the toll data after local processing. The bandwidth in upload direction is estimated from the server-side log entries and corrected for the network latency. The data collected allows comparing the performance of seven types of OBUs in three GPRS networks over time. While the three networks differ in the average bandwidth offered, the biggest performance impact is the OBU type where modems with the same specification yield different upload rates. In addition we update the GPRS network latency data by fitting two statistical distributions, improving markedly on the prior results.

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Computing Hypergeometric Functions Rigorously

Cited by 40 publications

References 57 publications

Stieltjes Moment Sequences for Pattern-Avoiding Permutations

Stieltjes Moment Sequences for Pattern-Avoiding Permutations

Truncation Bounds for Differentially Finite Series

Network-Wide Measurement of GPRS Bandwidth and Latency

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