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

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

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

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

interface	setGammaIncLowSeriesNR
	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	setGammaIncUppContFracNR
	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...

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

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

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 interface setGammaIncLowNR and setGammaIncUppNR to compute the Lower and Upper Incomplete Gamma function.
If ease of use matters more than performance, use the function interfaces getGammaIncLowNR and getGammaIncUppNR to compute the Lower and Upper Incomplete Gamma function.
setGammaIncLowSeriesNR and setGammaIncUppContFracNR 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.

Warning: Although all generic interfaces of this module are available for all processor real kinds, the accuracy and performance of the implemented algorithms are optimized for IEEE double precision.
In particular, the algorithms may not accurately compute the lower incomplete gamma function in extended precision (e.g., 128 bits) mode corresponding to RKH kind type parameter.

Note: The computations of this module are an implementation of the algorithm proposed approach by:
Numerical Recipes in Fortran 77: The Art of Scientific Computing 2nd Edition William H. Press (Author), Brian P. Flannery (Author), Saul A. Teukolsky (Author), William T. Vetterling (Author)

Benchmarks:

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

! Test the performance of `setGammaIncLowSeriesNR()` vs. `setGammaIncUppContFracNR()`.
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_"setGammaIncLowSeriesNR", exec = setGammaIncLowSeriesNR, overhead = setOverhead)
    bench(2) = bench_type(name = SK_"setGammaIncUppContFracNR", exec = setGammaIncUppContFracNR, 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 setGammaIncLowSeriesNR()
        block
            use pm_mathGammaNR, only: setGammaIncLowSeriesNR
            call setGammaIncLowSeriesNR( gammaIncLow = gammaInc &
                                        , x = Point(ipnt) &
                                        , logGammaKappa = logGammaKappa &
                                        , kappa = kappa &
                                        , info = info &
                                       !, tol = TOL &
                                        )
            call finalize()
        end block
    end subroutine
 
    subroutine setGammaIncUppContFracNR()
        block
            use pm_mathGammaNR, only: setGammaIncUppContFracNR
            call setGammaIncUppContFracNR( 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 setGammaIncLowSeriesNR compute the Lower Incomplete Gamma function suing the series representation of the Incomplete Gamma function while the procedures under the generic interface setGammaIncUppContFracNR 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.

See also: pm_mathGammaNR for detailed description of the (Regularized Incomplete) Gamma Function.

Test:: test_pm_mathGammaNR

Todo:: Critical Priority: The implementation of the algorithms of this module must be properly changed to allow reliable extended-precision computations of the incomplete Gamma function.
This would require significant investment in making the original algorithms of Gil et al. kind-agnostic.

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:: Amir Shahmoradi, July 22, 2024, 11:45 AM, NASA Goddard Space Flight Center, Washington, D.C.

Variable Documentation

◆ MODULE_NAME

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

Definition at line 92 of file pm_mathGammaNR.F90.

Data Types

Variables

Detailed Description

Variable Documentation

◆ MODULE_NAME