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Arbitrary-precision computation of the gamma function
[article]

2021
*
arXiv
*
pre-print

We discuss

arXiv:2109.08392v1
fatcat:jq7vdq6xijf5de5gaqdehvy5uu
*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*...##
###
Computing the Incomplete Gamma Function to Arbitrary Precision
[chapter]

2003
*
Lecture Notes in Computer Science
*

I consider an

doi:10.1007/3-540-44839-x_83
fatcat:wirmkrjhsbeynb33y4igpbbnoy
*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] . ...##
###
Efficient and Accurate Parallel Inversion of the Gamma Distribution

2015
*
SIAM Journal on Scientific Computing
*

We show

doi:10.1137/14095875x
fatcat:bl2ormnb2jffvcgau5n6lqlsii
*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. ...##
###
Towards real-time hardware gamma correction for dynamic contrast enhancement

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 ...

##
###
Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation

2016
*
Applied and Computational Mathematics
*

To overcome this challenge,

doi:10.11648/j.acm.20160502.16
fatcat:cyzuegeygndjnksz3jea3d5rki
*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] . ...##
###
A Matlab Implementation of an Algorithm for Computing Integrals of Products of Bessel Functions
[chapter]

2006
*
Lecture Notes in Computer Science
*

We present a Matlab program that

doi:10.1007/11832225_29
fatcat:h4r5tolztvc2xg3ztwmljtaa4a
*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*. ...##
###
Computing hypergeometric functions rigorously
[article]

2016
*
arXiv
*
pre-print

We present an efficient implementation

arXiv:1606.06977v2
fatcat:ulmwaxmmybdalhihlzbggwb7qm
*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*...##
###
Numerical approximations for speeding up MCMC inference in the infinite relational model

2015
*
2015 23rd European Signal Processing Conference (EUSIPCO)
*

We investigate how using numerical approximations

doi:10.1109/eusipco.2015.7362891
dblp:conf/eusipco/SchmidtA15
fatcat:f4gur6snd5cdba7vcoc3t7ktzm
*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*. ...##
###
Computation of Gauss-Jacobi Quadrature Nodes and Weights with Arbitrary Precision

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

doi:10.15439/2018f107
dblp:conf/fedcsis/Brzezinski18
fatcat:xqqlwl2i3zdg7pdrsghqix2zum
*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. ...##
###
ESTIMATION OF COMPUTING ALGORITHMS QUICKLY DEVELOPMENT BY USING HIGH-PRECISION ARITHMETIC SOFTWARE

2019
*
Математичне моделювання
*

are used;

doi:10.31319/2519-8106.2(41)2019.185037
fatcat:i4jg6cd5tjgr7ffby6kcerec6e
*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. ...##
###
A Flexible Architecture for Precise Gamma Correction

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. ...

##
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The Polylogarithm Function in Julia
[article]

2020
*
arXiv
*
pre-print

Only a handful

arXiv:2010.09860v1
fatcat:fsxiudbvfndebgosm4bv6xn5cq
*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. ...##
###
Hyperterminants I

1996
*
Journal of Computational and Applied Mathematics
*

Furthermore, we consider how these convergent expansions can be used to

doi:10.1016/s0377-0427(96)00108-2
fatcat:e74mskiiefbxzdqtddxaazw3he
*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. ...##
###
A practical, effective calculation of gamma difference distributions with open data science tools
[article]

2021
*
arXiv
*
pre-print

At

arXiv:2105.04427v2
fatcat:7gngix6a2jg4zib3rbxtgtxpjy
*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. ...##
###
Gaussian Quadrature Formulae for Arbitrary Positive Measures

2006
*
Evolutionary Bioinformatics
*

In addition, for certain well-known density measures (

doi:10.1177/117693430600200010
fatcat:ysnr7s4mj5a7ndgup7dyzdx5py
*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. ...
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