This module contains procedures and generic interfaces for the Lower and Upper Incomplete Gamma functions. More...

Data Types
interface	getGammaIncLow
	Generate and return the regularized Lower Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and upper limit of the integral `x`. More...

interface	getGammaIncUpp
	Generate and return the regularized Upper Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and lower limit of the integral `x`. More...

interface	setGammaIncLow
	Return the regularized Lower Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and upper limit of the integral `x`. More...

interface	setGammaIncLowSeries
	Return the regularized Lower Incomplete Gamma function for the specified upper limit `x` and shape parameter, evaluated by the series representation of the Incomplete Gamma function. More...

interface	setGammaIncUpp
	Return the regularized Upper Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and lower limit of the integral `x`. More...

interface	setGammaIncUppContFrac
	Return the regularized Upper Incomplete Gamma function for the specified lower limit `x` and shape parameter, evaluated by the Legendre continued fraction representation of the Incomplete Gamma function. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_mathGamma"

Detailed Description

This module contains procedures and generic interfaces for the Lower and Upper Incomplete Gamma functions.

This module provides multiple function and subroutine procedures for computing the Lower and Upper Incomplete Gamma functions. These routines mostly differ only in terms of performance and usage convenience.

If performance is important, use the subroutine interfaces setGammaIncLow and setGammaIncUpp to compute the Lower and Upper Incomplete Gamma functions, respectively.
If ease of use matters more than performance, use the function interfaces getGammaIncLow and getGammaIncUpp to compute the Lower and Upper Incomplete Gamma functions, respectively.
setGammaIncLowSeries and setGammaIncUppContFrac are not meant to be used directly to compute directly the incomplete Gamma functions since their performance depends on the values of the input parameters.
Rather they are low-level implementations to be called only by the above higher-level interfaces.
See below for the relevant benchmark.

Benchmarks:

Benchmark :: The runtime performance of setGammaIncLowSeries vs. setGammaIncUppContFrac ⛓

! Test the performance of `setGammaIncLowSeries()` vs. `setGammaIncUppContFrac()`.
program benchmark
 
    use iso_fortran_env, only: error_unit
    use pm_kind, only: IK, LK, RKG => RK, RK, SK
    use pm_arraySpace, only: setLogSpace
    use pm_bench, only: bench_type
 
    implicit none
 
    integer(IK)                         :: i
    integer(IK)                         :: ipnt
    integer(IK)                         :: fileUnit
    integer(IK)     , parameter         :: NPNT = 20_IK
    integer(IK)     , parameter         :: NBENCH = 2_IK
    real(RKG)       , parameter         :: kappa = 1._RKG
    real(RKG)       , parameter         :: logGammaKappa = log_gamma(kappa)
   !real(RKG)       , parameter         :: TOL = 1000 * epsilon(0._RKG)     !<  The tolerance.
    real(RKG)                           :: Point(NPNT)
    real(RKG)                           :: dummy = 0._RKG
    real(RKG)                           :: gammaInc
    integer(IK)                         :: info
    type(bench_type)                    :: bench(NBENCH)
 
    bench(1) = bench_type(name = SK_"setGammaIncLowSeries", exec = setGammaIncLowSeries , overhead = setOverhead)
    bench(2) = bench_type(name = SK_"setGammaIncUppContFrac", exec = setGammaIncUppContFrac , overhead = setOverhead)
 
    call setLogSpace(Point, logx1 = log(0.04_RKG), logx2 = log(100._RKG))
 
 
    write(*,"(*(g0,:,' '))")
    write(*,"(*(g0,:,' vs. '))") (bench(i)%name, i = 1, NBENCH)
    write(*,"(*(g0,:,' '))")
 
    open(newunit = fileUnit, file = "main.out", status = "replace")
 
        write(fileUnit, "(*(g0,:,','))") "x / (kappa + 1)", (bench(i)%name, i = 1, NBENCH)
 
        loopOverPoint: do ipnt = 1, NPNT
 
            write(*,"(*(g0,:,' '))") "Benchmarking with point", Point(ipnt)
 
            do i = 1, NBENCH
                bench(i)%timing = bench(i)%getTiming(minsec = 0.05_RK)
            end do
 
            write(fileUnit,"(*(g0,:,','))") Point(ipnt) / (kappa + 1), (bench(i)%timing%mean, i = 1, NBENCH)
 
        end do loopOverPoint
 
        write(*,"(*(g0,:,' '))") dummy
        write(*,"(*(g0,:,' '))")
 
    close(fileUnit)
 
contains
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! procedure wrappers.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    subroutine setOverhead()
        call finalize()
    end subroutine
 
    subroutine finalize()
        if (info < 0_IK) error stop
        dummy = dummy + gammaInc
    end subroutine
 
    subroutine setGammaIncLowSeries()
        block
            use pm_mathGamma, only: setGammaIncLowSeries
            call setGammaIncLowSeries( gammaIncLow = gammaInc &
                                        , x = Point(ipnt) &
                                        , logGammaKappa = logGammaKappa &
                                        , kappa = kappa &
                                        , info = info &
                                       !, tol = TOL &
                                        )
            call finalize()
        end block
    end subroutine
 
    subroutine setGammaIncUppContFrac()
        block
            use pm_mathGamma, only: setGammaIncUppContFrac
            call setGammaIncUppContFrac( gammaIncUpp = gammaInc &
                                        , x = Point(ipnt) &
                                        , logGammaKappa = logGammaKappa &
                                        , kappa = kappa &
                                        , info = info &
                                       !, tol = TOL &
                                        )
            call finalize()
        end block
    end subroutine
 
end program benchmark

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Postprocessing of the benchmark output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
 
import os
dirname = os.path.basename(os.getcwd()) 
 
fontsize = 14
 
df = pd.read_csv("main.out", delimiter = ",")
colnames = list(df.columns.values)
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for colname in colnames[1:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
ax.set_title(" vs. ".join(colnames[1:])+"\nLower is better.", fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( colnames[1:]
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.png")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
plt.plot( df[colnames[0]].values
        , np.ones(len(df[colnames[0]].values))
        , linestyle = "--"
       #, color = "black"
        , linewidth = 2
        )
for colname in colnames[2:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values / df[colnames[1]].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
ax.set_title("Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}().".format(colnames[1]), fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( colnames[1:]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

The procedures under the generic interface setGammaIncLowSeries compute the Lower Incomplete Gamma function suing the series representation of the Incomplete Gamma function while the procedures under the generic interface setGammaIncUppContFrac use the Continued Fraction representation of the Gamma function.
The Legendre continued fraction representation is known to converge faster at \(x > \kappa + 1\) whereas the series representation converges faster at \(x < \kappa + 1\).
This performance difference dependence on the specific values of \(x\) and \(\kappa\) is well illustrated by this benchmark.

Test:: test_pm_mathGamma

Final Remarks ⛓

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Copyright: Computational Data Science Lab

Author:: Fatemeh Bagheri, Monday 12:56 pm, August 16, 2021, Dallas TX

Variable Documentation

◆ MODULE_NAME

character(*, SK), parameter pm_mathGamma::MODULE_NAME = "@pm_mathGamma"

Definition at line 74 of file pm_mathGamma.F90.

Data Types

Variables

Detailed Description

Variable Documentation

◆ MODULE_NAME