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Arbitrary-precision computation of the gamma function [article]

Fredrik Johansson
2021 arXiv   pre-print
We discuss the best methods available for computing the gamma function Γ(z) in arbitrary-precision arithmetic with rigorous error bounds.  ...  The methods also cover the log-gamma function logΓ(z), the digamma function ψ(z), and derivatives Γ^(n)(z) and ψ^(n)(z).  ...  However, despite an extensive literature on gamma function computation, 1 there have been surprisingly few attempts to investigate the best available techniques from the point of view of arbitrary-precision  ... 
arXiv:2109.08392v1 fatcat:jq7vdq6xijf5de5gaqdehvy5uu

Computing the Incomplete Gamma Function to Arbitrary Precision [chapter]

Serge Winitzki
2003 Lecture Notes in Computer Science  
I consider an arbitrary-precision computation of the incomplete Gamma function from the Legendre continued fraction.  ...  Using the method of generating functions, I compute the convergence rate of the continued fraction and find a direct estimate of the necessary number of terms.  ...  Several methods have been described in the literature for the computation of the incomplete Gamma function [2] [3] [4] [5] .  ... 
doi:10.1007/3-540-44839-x_83 fatcat:wirmkrjhsbeynb33y4igpbbnoy

Efficient and Accurate Parallel Inversion of the Gamma Distribution

Thomas Luu
2015 SIAM Journal on Scientific Computing  
We show the performance of a CUDA GPU implementation of our algorithm (called Quantus) is within an order of magnitude of the time to compute the normal quantile function.  ...  Inversion of the gamma distribution is traditionally accomplished using some form of root finding. This is known to be computationally expensive.  ...  Asad Munir clarified various aspects of quantile function theory, for which the author is very grateful.  ... 
doi:10.1137/14095875x fatcat:bl2ormnb2jffvcgau5n6lqlsii

Towards real-time hardware gamma correction for dynamic contrast enhancement

Jesse Scott, Michael Pusateri
2009 2009 IEEE Applied Imagery Pattern Recognition Workshop (AIPR 2009)  
The gamma correction process is a non-linear mapping of intensity from input to output where the parameter gamma can be adjusted to improve the imagery's visual appeal.  ...  The focal plane array "sees" the world in a fundamentally different way than the human eye. Gamma correction has been historically used to help bridge the gap.  ...  Both the computation of a gamma update and the application of the gamma correction were simplified to basic arithmetic operations and two specialized functions, logarithm and exponentiation of a constant  ... 
doi:10.1109/aipr.2009.5466305 dblp:conf/aipr/ScottP09 fatcat:qpzfip3npzhexpygy35kzegiku

Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation

Richard Eneojo Amobeda
2016 Applied and Computational Mathematics  
To overcome this challenge, the Lanczos approximation of the Gamma function which presents an efficient and easy way to compute algorithms for approximating the Gamma function to an arbitrary precision  ...  To produce an accurate result using the integral definition of the Gamma function, a large number of terms would have to be added during the numerical integration procedure; this makes computation of Gamma  ...  However, Stirling series, Spouge series and Lanczos approximation present efficient and easy way to compute algorithms for approximating the Gamma function to an arbitrary precision [7] .  ... 
doi:10.11648/j.acm.20160502.16 fatcat:cyzuegeygndjnksz3jea3d5rki

A Matlab Implementation of an Algorithm for Computing Integrals of Products of Bessel Functions [chapter]

Joris Van Deun, Ronald Cools
2006 Lecture Notes in Computer Science  
We present a Matlab program that computes infinite range integrals of an arbitrary product of Bessel functions of the first kind.  ...  The algorithm uses an integral representation of the upper incomplete Gamma function to integrate the tail of the integrand.  ...  The (machine-dependent) constant t GJ is the relative efficiency of computing the incomplete Gamma function compared to computing the Bessel function.  ... 
doi:10.1007/11832225_29 fatcat:h4r5tolztvc2xg3ztwmljtaa4a

Computing hypergeometric functions rigorously [article]

Fredrik Johansson
2016 arXiv   pre-print
We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic.  ...  We also partially cover the generalized hypergeometric function _pF_q and computation of high-order parameter derivatives.  ...  Based on these very general functions, we are in turn able to compute incomplete gamma and beta functions, Bessel functions, Legendre functions, and others, covering a large portion of the special functions  ... 
arXiv:1606.06977v2 fatcat:ulmwaxmmybdalhihlzbggwb7qm

Numerical approximations for speeding up MCMC inference in the infinite relational model

Mikkel N. Schmidt, Kristoffer Jon Albers
2015 2015 23rd European Signal Processing Conference (EUSIPCO)  
We investigate how using numerical approximations of the log-Gamma function in evaluating the likelihood of the IRM can improve the computational speed of MCMC inference, and how it affects the performance  ...  We demonstrate that the computational time for MCMC inference can be reduced by a factor of two without affecting the performance, making it worthwhile in practical situations when on a computational budget  ...  By increasing the number of terms in the Lanczos' approximation to N (requiring two logarithms, one multiplication and N divisions) the log-Gamma function can be computed to arbitrary precision.  ... 
doi:10.1109/eusipco.2015.7362891 dblp:conf/eusipco/SchmidtA15 fatcat:f4gur6snd5cdba7vcoc3t7ktzm

Computation of Gauss-Jacobi Quadrature Nodes and Weights with Arbitrary Precision

Dariusz Brzeziński
2018 Proceedings of the 2018 Federated Conference on Computer Science and Information Systems  
They include an enhancement for standard iteration method for Jacobi polynomials zeros finding, weight function formula transformation for increased accuracy of fractional derivatives computation and arbitrary  ...  The results of numerical experiments presented in the paper prove high accuracy and efficiency of developed methods for computation of quadratures' nodes and weights, decreased amount of required iterations  ...  ACKNOWLEDGEMENT The work was created as a result of the research project no. DEC-2016/23/D/ST6/01709 financed from the funds of the National Science Center, Poland.  ... 
doi:10.15439/2018f107 dblp:conf/fedcsis/Brzezinski18 fatcat:xqqlwl2i3zdg7pdrsghqix2zum

ESTIMATION OF COMPUTING ALGORITHMS QUICKLY DEVELOPMENT BY USING HIGH-PRECISION ARITHMETIC SOFTWARE

Інна Іванівна Жульковська, Олег Олександрович Жульковський, Євген Олександрович Сугаль
2019 Математичне моделювання  
are used; the package includes routines for calculating algebraic, transcendental and some special functions, such as gamma function, incomplete gamma function, zeta function.  ...  The aim of this work is to study the degree of influence of a given accuracy on the speed of computational algorithms using modern means of high-precision arithmetic of the Java language.  ... 
doi:10.31319/2519-8106.2(41)2019.185037 fatcat:i4jg6cd5tjgr7ffby6kcerec6e

A Flexible Architecture for Precise Gamma Correction

Dong-U Lee, Ray C. C. Cheung, John D. Villasenor
2007 IEEE Transactions on Very Large Scale Integration (vlsi) Systems  
Arbitrary gamma values, input bit widths, and output bit widths are supported.  ...  The gamma correction curve is segmented via a combination of uniform segments and segments whose sizes vary by powers of two.  ...  The flexibility of the architecture allows the support of arbitrary gamma values, input bit widths, and output bit widths.  ... 
doi:10.1109/tvlsi.2007.893671 fatcat:ynupng5syfa5lfjisgq3owz5ie

The Polylogarithm Function in Julia [article]

Matthew Roughan
2020 arXiv   pre-print
Only a handful of formal publications consider the evaluation of the function, and most focus on a specific domain and/or presume arbitrary precision arithmetic will be used.  ...  The polylogarithm function is one of the constellation of important mathematical functions.  ...  We would like to thank the Australian Research Council for funding through the Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS), and grant DP110103505.  ... 
arXiv:2010.09860v1 fatcat:fsxiudbvfndebgosm4bv6xn5cq

Hyperterminants I

A.B. Olde Daalhuis
1996 Journal of Computational and Applied Mathematics  
Furthermore, we consider how these convergent expansions can be used to compute the hyperterminants to arbitrary precision.  ...  Hyperasymptotic expansions are in terms of certain multiple integrals, the so-called hyperterminants. In this paper we give new integral representations for the hyperterminants.  ...  The support of the Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged.  ... 
doi:10.1016/s0377-0427(96)00108-2 fatcat:e74mskiiefbxzdqtddxaazw3he

A practical, effective calculation of gamma difference distributions with open data science tools [article]

Martina Hančová, Andrej Gajdoš, Jozef Hanč
2021 arXiv   pre-print
At the double 53-bit precision, our tool outperformed the speed of the analytical computation based on Tricomi's U(a, b, z) function in CAS software (commercial Mathematica, open SageMath) by 1.5-2 orders  ...  At the precision of scientific numerical computational tools, it exceeded open SciPy, NumPy, and commercial MATLAB 5-10 times.  ...  inversion of characteristic functions and his MATLAB code CharFunTool.  ... 
arXiv:2105.04427v2 fatcat:7gngix6a2jg4zib3rbxtgtxpjy

Gaussian Quadrature Formulae for Arbitrary Positive Measures

Andrew D. Fernandes, William R. Atchley
2006 Evolutionary Bioinformatics  
In addition, for certain well-known density measures (the normal, gamma, log-normal, Student's t, inversegamma, beta, and Fisher's F) we present exact formulae for computing the respective quadrature scheme  ...  We present computational methods and subroutines to compute Gaussian quadrature integration formulas for arbitrary positive measures.  ...  Financial support was provided by the National Institutes of Health (GM45344) and the North Carolina State University.  ... 
doi:10.1177/117693430600200010 fatcat:ysnr7s4mj5a7ndgup7dyzdx5py
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