pm_mathGammaAM::getGammaIncUppAM Interface Reference

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

Detailed Description

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

The regularized Upper Incomplete Gamma function is defined as,

\begin{equation} \large Q(\kappa, x) = \frac{1}{\Gamma(\kappa)} \int_x^{+\infty}~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 lower limit in the integral of the Upper Incomplete Gamma function.
Note that this integral is bounded between zero and one ( \([0,1]\)).
The regularized Upper 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 lower 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 \(P(\kappa,x)\).

Returns

gammaIncUpp : The output scalar of the same type and kind as the output argument x representing the Upper Incomplete Gamma function for the specified kappa and lower limit.
Note that gammaIncUpp is, by definition, always positive.
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_mathGammaAM, only: getGammaIncUppAM
 
gammaIncUpp = getGammaIncUppAM(x, kappa)

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 setGammaIncUppAM.
As such they are slower than the corresponding subroutine version.

See also

setGammaIncUppAM
getGammaIncLowAM
setGammaIncLowAM

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_mathGammaAM, only: getGammaIncUppAM
 
    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 Upper Incomplete Gamma Function using its series representation.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = 1.5_RKS, kappa = 2._RKS)")
    call disp%show( getGammaIncUppAM(x = 1.5_RKS, kappa = 2._RKS) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = 1.5_RKD, kappa = 2._RKD)")
    call disp%show( getGammaIncUppAM(x = 1.5_RKD, kappa = 2._RKD) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = 1.5_RKH, kappa = 2._RKH)")
    call disp%show( getGammaIncUppAM(x = 1.5_RKH, kappa = 2._RKH) )
    call disp%skip()
 
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the regularized Upper Incomplete Gamma Function for a vector of points.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the regularized Upper Incomplete Gamma Function for a vector of kappa parameters.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = 1._RKS, kappa = [0.1_RKS, 1._RKS, 10._RKS])")
    call disp%show( getGammaIncUppAM(x = 1._RKS, kappa = [0.1_RKS, 1._RKS, 10._RKS]) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = 1._RKD, kappa = [0.1_RKD, 1._RKD, 10._RKD])")
    call disp%show( getGammaIncUppAM(x = 1._RKD, kappa = [0.1_RKD, 1._RKD, 10._RKD]) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = 1._RKH, kappa = [0.1_RKH, 1._RKH, 10._RKH])")
    call disp%show( getGammaIncUppAM(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 Upper Incomplete Gamma Function for a vector of points and kappa parameters.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = [1._RKS, 1._RKS, 1._RKS], kappa = [0.1_RKS, 1._RKS, 10._RKS])")
    call disp%show( getGammaIncUppAM(x = [1._RKS, 1._RKS, 1._RKS], kappa = [0.1_RKS, 1._RKS, 10._RKS]) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = [1._RKD, 1._RKD, 1._RKD], kappa = [0.1_RKD, 1._RKD, 10._RKD])")
    call disp%show( getGammaIncUppAM(x = [1._RKD, 1._RKD, 1._RKD], kappa = [0.1_RKD, 1._RKD, 10._RKD]) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = [1._RKH, 1._RKH, 1._RKH], kappa = [0.1_RKH, 1._RKH, 10._RKH])")
    call disp%show( getGammaIncUppAM(x = [1._RKH, 1._RKH, 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 Upper Incomplete Gamma Function for a fixed point and kappa parameter, but with different accuracies.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("getGammaIncUppAM(x = 1._RKH, kappa = 2._RKH)")
    call disp%show( getGammaIncUppAM(x = 1._RKH, kappa = 2._RKH) )
    call disp%skip()
 
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example array of the regularized Upper 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 = "getGammaIncUppAM.RK.txt")
        do i = 1, NP
            write(fileUnit, "(*(g0,:,' '))") x_RKS(i), getGammaIncUppAM(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 Upper Incomplete Gamma Function using its series representation.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
getGammaIncUppAM(x = 1.5_RKS, kappa = 2._RKS)
+0.557825446
 
 
getGammaIncUppAM(x = 1.5_RKD, kappa = 2._RKD)
+0.55782540037107442
 
 
getGammaIncUppAM(x = 1.5_RKH, kappa = 2._RKH)
+0.557825400371074572333201176910031586
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the regularized Upper Incomplete Gamma Function for a vector of points.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the regularized Upper Incomplete Gamma Function for a vector of kappa parameters.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
getGammaIncUppAM(x = 1._RKS, kappa = [0.1_RKS, 1._RKS, 10._RKS])
+0.241273176E-1, +0.367879391, +0.999999881
 
 
getGammaIncUppAM(x = 1._RKD, kappa = [0.1_RKD, 1._RKD, 10._RKD])
+0.24127343726327546E-1, +0.36787944117144245, +0.99999988857452171
 
 
getGammaIncUppAM(x = 1._RKH, kappa = [0.1_RKH, 1._RKH, 10._RKH])
+0.241273437263277773829694356333550562E-1, +0.367879441171442321595523770161461162, +0.999999888574521661279322646949312805
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the regularized Upper Incomplete Gamma Function for a vector of points and kappa parameters.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
getGammaIncUppAM(x = [1._RKS, 1._RKS, 1._RKS], kappa = [0.1_RKS, 1._RKS, 10._RKS])
+0.241273176E-1, +0.367879391, +0.999999881
 
 
getGammaIncUppAM(x = [1._RKD, 1._RKD, 1._RKD], kappa = [0.1_RKD, 1._RKD, 10._RKD])
+0.24127343726327546E-1, +0.36787944117144245, +0.99999988857452171
 
 
getGammaIncUppAM(x = [1._RKH, 1._RKH, 1._RKH], kappa = [0.1_RKH, 1._RKH, 10._RKH])
+0.241273437263277773829694356333550562E-1, +0.367879441171442321595523770161461162, +0.999999888574521661279322646949312805
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the regularized Upper Incomplete Gamma Function for a fixed point and kappa parameter, but with different accuracies.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
getGammaIncUppAM(x = 1._RKH, kappa = 2._RKH)
+0.735758882342884643191047540322921938
 
 

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 Upper\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_mathGammaAM

Remarks

See Numerical Recipes by Press et al. 1992 for further details of the Incomplete Gamma function.

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 279 of file pm_mathGammaAM.F90.

---

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

• src/fortran/main/pm_mathGammaAM.F90