pm_mathGammaNR::getGammaIncLowNR Interface Reference

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

Detailed Description

Generate and return the regularized Lower Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and upper limit of the integral x.

The regularized Lower Incomplete Gamma function is defined as,

\begin{equation} \large P(\kappa, x) = \frac{1}{\Gamma(\kappa)} \int_0^{x}~t^{\kappa-1}{\mathrm e}^{-t} ~ dt ~, \end{equation}

where \((\kappa > 0, x > 0)\) should hold, with \(\kappa\) representing the shape parameter of the Gamma function (or distribution) and \(x\) representing the upper limit in the integral of the Lower Incomplete Gamma function.
Note that this integral is bounded between zero and one ( \([0,1]\)).
The regularized Lower Incomplete Gamma function also represents the complement of the Cumulative Distribution Function (CDF) of the univariate Gamma distribution with the specified shape parameter and standardized x (with the scale parameter of unity).

Parameters

[in]	x	: The input scalar of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), representing the upper limit in the integral of the Lower Incomplete Gamma function \(P(\kappa,x)\).
[in]	kappa	: The input scalar of the same type and kind as x, representing the shape parameter ( \(\kappa\)) of the Lower Incomplete Gamma function \(P(\kappa,x)\).
[in]	tol	: The input scalar of the same type and kind as x, representing the relative accuracy in the convergence checking of the Gamma series or its Legendre continued fraction representation. (optional, default is set by setGammaIncLowNR())

Returns

gammaIncLow : The output scalar of the same type and kind as the output argument x representing the Lower Incomplete Gamma function for the specified kappa and upper limit.
Note that gammaIncLow is, by definition, always positive in the range \([0, 1]\).
Note that the procedure will abruptly end the program by calling error stop if the computation of the Incomplete Gamma function fails to converge**.

Possible calling interfaces ⛓

use pm_mathGammaNR, only: getGammaIncLowNR
 
gammaIncLow = getGammaIncLowNR(x, kappa, tol = tol)

Warning

The kappa and x input arguments must be positive real numbers.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

Remarks

The procedures under discussion are impure.

The procedures under discussion are elemental.

The procedures under this generic interface solely provide a more convenient method of calling the subroutine equivalents setGammaIncLowNR.
As such they are slower than the corresponding subroutine version.

See also

setGammaIncLowNR
getGammaIncLowNR
setGammaIncLowNR
setGammaIncLowSeriesNR
setGammaIncUppContFracNR
Numerical Recipes by Press et al. 1992

Example usage ⛓

program example
 
    use pm_kind, only: SK
    use pm_kind, only: IK
    use pm_kind, only: LK
    use pm_kind, only: RKS, RKD, RKH
    use pm_io, only: display_type
    use pm_mathGammaNR, only: getGammaIncLowNR
 
    implicit none
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the regularized Lower Incomplete Gamma Function using its series representation.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = 1.5_RKS, kappa = 2._RKS)")
    call disp%show( getGammaIncLowNR(x = 1.5_RKS, kappa = 2._RKS) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = 1.5_RKD, kappa = 2._RKD)")
    call disp%show( getGammaIncLowNR(x = 1.5_RKD, kappa = 2._RKD) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = 1.5_RKH, kappa = 2._RKH)")
    call disp%show( getGammaIncLowNR(x = 1.5_RKH, kappa = 2._RKH) )
    call disp%skip()
 
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the regularized Lower Incomplete Gamma Function for a vector of points.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = [0._RKS, 1._RKS, 10._RKS], kappa = 2._RKS)")
    call disp%show( getGammaIncLowNR(x = [0._RKS, 1._RKS, 10._RKS], kappa = 2._RKS) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = [0._RKD, 1._RKD, 10._RKD], kappa = 2._RKD)")
    call disp%show( getGammaIncLowNR(x = [0._RKD, 1._RKD, 10._RKD], kappa = 2._RKD) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = [0._RKH, 1._RKH, 10._RKH], kappa = 2._RKH)")
    call disp%show( getGammaIncLowNR(x = [0._RKH, 1._RKH, 10._RKH], kappa = 2._RKH) )
    call disp%skip()
 
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the regularized Lower Incomplete Gamma Function for a vector of shape parameters.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = 1._RKS, kappa = [0.1_RKS, 1._RKS, 10._RKS])")
    call disp%show( getGammaIncLowNR(x = 1._RKS, kappa = [0.1_RKS, 1._RKS, 10._RKS]) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = 1._RKD, kappa = [0.1_RKD, 1._RKD, 10._RKD])")
    call disp%show( getGammaIncLowNR(x = 1._RKD, kappa = [0.1_RKD, 1._RKD, 10._RKD]) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = 1._RKH, kappa = [0.1_RKH, 1._RKH, 10._RKH])")
    call disp%show( getGammaIncLowNR(x = 1._RKH, kappa = [0.1_RKH, 1._RKH, 10._RKH]) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the regularized Lower Incomplete Gamma Function for a fixed point and shape parameter, but with different accuracies.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncLowNR(x = 1._RKH, kappa = 2._RKH, tol = [1.e-8_RKH, 1.e-16_RKH, 1.e-32_RKH])")
    call disp%show( getGammaIncLowNR(x = 1._RKH, kappa = 2._RKH, tol = [1.e-8_RKH, 1.e-16_RKH, 1.e-32_RKH]) )
    call disp%skip()
 
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example array of the regularized Lower Incomplete Gamma function for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
 
        use pm_arraySpace, only: setLinSpace
        integer(IK) , parameter :: NP = 1000_IK
        real(RKS) :: x_RKS(NP)
        integer :: fileUnit, i
 
        call setLinSpace(x_RKS, 0._RKS, 10._RKS)
        open(newunit = fileUnit, file = "getGammaIncLowNR.RK.txt")
        do i = 1, NP
            write(fileUnit, "(*(g0,:,' '))") x_RKS(i), getGammaIncLowNR(x_RKS(i), kappa = [1.0_RKS, 2.5_RKS, 5.0_RKS])
        end do
        close(fileUnit)
 
    end block
 
end program example

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

Example output ⛓

 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the regularized Lower Incomplete Gamma Function using its series representation.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
getGammaIncLowNR(x = 1.5_RKS, kappa = 2._RKS)
+0.442174494
 
 
getGammaIncLowNR(x = 1.5_RKD, kappa = 2._RKD)
+0.44217459962892541
 
 
getGammaIncLowNR(x = 1.5_RKH, kappa = 2._RKH)
+0.442174599628925427666798823089968607
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the regularized Lower Incomplete Gamma Function for a vector of points.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
getGammaIncLowNR(x = [0._RKS, 1._RKS, 10._RKS], kappa = 2._RKS)
+0.00000000, +0.264241129, +0.999500573
 
 
getGammaIncLowNR(x = [0._RKD, 1._RKD, 10._RKD], kappa = 2._RKD)
+0.0000000000000000, +0.26424111765711533, +0.99950060077261271
 
 
getGammaIncLowNR(x = [0._RKH, 1._RKH, 10._RKH], kappa = 2._RKH)
+0.00000000000000000000000000000000000, +0.264241117657115356808952459677078255, +0.999500600772612666633108493328833987
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the regularized Lower Incomplete Gamma Function for a vector of shape parameters.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
getGammaIncLowNR(x = 1._RKS, kappa = [0.1_RKS, 1._RKS, 10._RKS])
+0.975872457, +0.632120609, +0.111425479E-6
 
 
getGammaIncLowNR(x = 1._RKD, kappa = [0.1_RKD, 1._RKD, 10._RKD])
+0.97587265627367248, +0.63212055882855767, +0.11142547833872053E-6
 
 
getGammaIncLowNR(x = 1._RKH, kappa = [0.1_RKH, 1._RKH, 10._RKH])
+0.975872656273672222617030564366644980, +0.632120558828557678404476229838539127, +0.111425478338720677353050687240252353E-6
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the regularized Lower Incomplete Gamma Function for a fixed point and shape parameter, but with different accuracies.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
getGammaIncLowNR(x = 1._RKH, kappa = 2._RKH, tol = [1.e-8_RKH, 1.e-16_RKH, 1.e-32_RKH])
+0.264241117593517583537611040683702485, +0.264241117657115356650199699569751946, +0.264241117657115356808952459677078207
 
 

Postprocessing of the example output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import glob
import sys
 
fontsize = 17
 
kind = "RK"
label = [ r"shape: $\kappa = 1.0$"
        , r"shape: $\kappa = 2.5$"
        , r"shape: $\kappa = 5.0$"
        ]
 
pattern = "*." + kind + ".txt"
fileList = glob.glob(pattern)
if len(fileList) == 1:
 
    df = pd.read_csv(fileList[0], delimiter = " ")
 
    fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
    ax = plt.subplot()
 
    for i in range(1,len(df.values[0,:]+1)):
 
            plt.plot( df.values[:, 0]
                    , df.values[:,i]
                    , linewidth = 2
                    )
 
    plt.xticks(fontsize = fontsize - 2)
    plt.yticks(fontsize = fontsize - 2)
    ax.set_xlabel("x", fontsize = fontsize)
    ax.set_ylabel("Regularized Lower\nIncomplete Gamma Function", fontsize = fontsize)
 
    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   ( label
                , fontsize = fontsize
               #, loc = "center left"
               #, bbox_to_anchor = (1, 0.5)
                )
 
    plt.savefig(fileList[0].replace(".txt",".png"))
 
else:
 
    sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓

Test:

test_pm_mathGammaNR

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.

1. 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.
   
2. 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 12:36 pm, August 16, 2021, Dallas TX

Definition at line 178 of file pm_mathGammaNR.F90.

---

The documentation for this interface was generated from the following file:

• src/fortran/main/pm_mathGammaNR.F90