pm_mathGamma::setGammaIncLow Interface Reference

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

Detailed Description

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 Cumulative Distribution Function (CDF) of the univariate Gamma distribution with the specified shape parameter and standardized x (with the scale parameter of unity).

Parameters

[out]	gammaIncLow	: The output scalar of the same type and kind as the input argument x representing the Lower Incomplete Gamma function for the specified kappa and lower limit. Note that gammaIncLow is, by definition, always positive in the range \([0, 1]\).
[in]	x	: The input scalar of the 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]	logGammaKappa	: The input scalar or array of the same shape as other input arguments, of the same type and kind as the input x, representing the precomputed \(\log(\Gamma(\kappa))\) which can be computed by calling the Fortran intrinsic function log_gamma(kappa).
[in]	kappa	: The input scalar of the same type and kind as x, representing the shape parameter ( \(\kappa\)) of the Lower Incomplete Gamma function \(P(\kappa,x)\).
[out]	info	: The input scalar or array of the same shape as other input arguments of type integer of default kind IK. On output, it is set to (positive) number of iterations taken for the algorithm to converge or its negative if the algorithm fails to converge. A convergence failure is likely to happen if the input value for kappa is too large. A negative value implies the lack of convergence.
[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, if it is missing, the default is set by setGammaIncUppContFrac or setGammaIncLowSeries).

Possible calling interfaces ⛓

use pm_mathGamma, only: setGammaIncLow
 
call setGammaIncLow(gammaIncLow, x, logGammaKappa, kappa, info, tol = tol)

Warning

The kappa and x input arguments must be positive real numbers with logGammaKappa = log_gamma(kappa) where log_gamma() is a Fortran intrinsic function.
Furthermore, tol << 1. must hold, if present as an input argument.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1.
By default, these procedures are pure in release build and impure in debug and testing builds.

Remarks

The procedures under discussion are elemental.

These procedures are particularly useful and needed in computing the PDF of the Gamma and related distributions.
The logic behind pre-computing and passing logGammaKappa = log_gamma(kappa) is to speed up the calculations since log_gamma() is computationally expensive and its recomputation can be avoided in repeated calls to setGammaIncLow with the same shape parameter but different x.

See also

getGammaIncUpp
setGammaIncUpp
getGammaIncLow
setGammaIncLowSeries
setGammaIncUppContFrac
See also The Numerical Recipes by Press et al. 1992 for further details about the Incomplete Gamma function.

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_mathGamma, only: setGammaIncLow
 
    implicit none
 
    integer(IK) , parameter :: NP = 1000_IK
    real(RKH)               :: gamIncLow_RKH, x_RKH, kappa_RKH
    real(RKD)               :: gamIncLow_RKD, x_RKD, kappa_RKD
    real(RKS)               :: gamIncLow_RKS, x_RKS, kappa_RKS
    integer(IK)             :: info
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    kappa_RKH = 1.5_RKH
    kappa_RKD = 1.5_RKD
    kappa_RKS = 1.5_RKS
 
    x_RKH = 2._RKH
    x_RKD = 2._RKD
    x_RKS = 2._RKS
 
    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("x_RKS")
    call disp%show( x_RKS )
    call disp%show("kappa_RKS")
    call disp%show( kappa_RKS )
    call disp%show("call setGammaIncLow(gamIncLow_RKS, x_RKS, logGammaKappa = log_gamma(kappa_RKS), kappa = kappa_RKS, info = info)")
                    call setGammaIncLow(gamIncLow_RKS, x_RKS, logGammaKappa = log_gamma(kappa_RKS), kappa = kappa_RKS, info = info)
    call disp%show("gamIncLow_RKS")
    call disp%show( gamIncLow_RKS )
    call disp%show("info")
    call disp%show( info )
    call disp%skip()
 
    call disp%skip()
    call disp%show("x_RKD")
    call disp%show( x_RKD)
    call disp%show("kappa_RKD")
    call disp%show( kappa_RKD)
    call disp%show("call setGammaIncLow(gamIncLow_RKD, x_RKD, logGammaKappa = log_gamma(kappa_RKD), kappa = kappa_RKD, info = info)")
                    call setGammaIncLow(gamIncLow_RKD, x_RKD, logGammaKappa = log_gamma(kappa_RKD), kappa = kappa_RKD, info = info)
    call disp%show("gamIncLow_RKD")
    call disp%show( gamIncLow_RKD)
    call disp%show("info")
    call disp%show( info )
    call disp%skip()
 
    call disp%skip()
    call disp%show("x_RKH")
    call disp%show( x_RKH )
    call disp%show("kappa_RKH")
    call disp%show( kappa_RKH )
    call disp%show("call setGammaIncLow(gamIncLow_RKH, x_RKH, logGammaKappa = log_gamma(kappa_RKH), kappa = kappa_RKH, info = info)")
                    call setGammaIncLow(gamIncLow_RKH, x_RKH, logGammaKappa = log_gamma(kappa_RKH), kappa = kappa_RKH, info = info)
    call disp%show("gamIncLow_RKH")
    call disp%show( gamIncLow_RKH )
    call disp%show("info")
    call disp%show( info )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example array of the regularized Lower Incomplete Gamma function for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
 
        use pm_arraySpace, only: setLinSpace
        real(RKS) :: x_RKS(NP)
        integer :: fileUnit, i
 
        call setLinSpace(x_RKS, 0._RKS, 8._RKS)
        open(newunit = fileUnit, file = "setGammaIncLow.RK.txt")
        do i = 1, NP
            call setGammaIncLow(gamIncLow_RKS, x_RKS(i), logGammaKappa = log_gamma(kappa_RKS), kappa = kappa_RKS, info = info)
            write(fileUnit,"(2(g0,:,' '))") x_RKS(i), gamIncLow_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.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
x_RKS
+2.00000000
kappa_RKS
+1.50000000
call setGammaIncLow(gamIncLow_RKS, x_RKS, logGammaKappa = log_gamma(kappa_RKS), kappa = kappa_RKS, info = info)
gamIncLow_RKS
+0.738535762
info
+9
 
 
x_RKD
+2.0000000000000000
kappa_RKD
+1.5000000000000000
call setGammaIncLow(gamIncLow_RKD, x_RKD, logGammaKappa = log_gamma(kappa_RKD), kappa = kappa_RKD, info = info)
gamIncLow_RKD
+0.73853587005088817
info
+40
 
 
x_RKH
+2.00000000000000000000000000000000000
kappa_RKH
+1.50000000000000000000000000000000000
call setGammaIncLow(gamIncLow_RKH, x_RKH, logGammaKappa = log_gamma(kappa_RKH), kappa = kappa_RKH, info = info)
gamIncLow_RKH
+0.738535870050889377797177924024075285
info
+181
 
 

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
 
marker ={ "CK" : "-"
        , "IK" : "."
        , "RK" : "-"
        }
xlab =  { "CK" : r"x ( real/imaginary )"
        , "IK" : r"x ( integer-valued )"
        , "RK" : r"x ( real-valued )"
        }
labels = [r"shape parameter: $\kappa = 2$"]
 
for kind in ["IK", "CK", "RK"]:
 
    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()
 
        if kind == "CK":
            plt.plot( df.values[:, 0]
                    , df.values[:,2]
                    , marker[kind]
                    , color = "r"
                    )
            plt.plot( df.values[:,1]
                    , df.values[:,3]
                    , marker[kind]
                    , color = "blue"
                    )
        else:
            plt.plot( df.values[:, 0]
                    , df.values[:,1]
                    , marker[kind]
                    , color = "r"
                    )
 
        plt.xticks(fontsize = fontsize - 2)
        plt.yticks(fontsize = fontsize - 2)
        ax.set_xlabel(xlab[kind], 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   ( labels
                    , fontsize = fontsize
                   #, loc = "center left"
                   #, bbox_to_anchor = (1, 0.5)
                    )
 
        plt.savefig(fileList[0].replace(".txt",".png"))
 
    elif len(fileList) > 1:
 
        sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓

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.

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 470 of file pm_mathGamma.F90.

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

• src/fortran/main/pm_mathGamma.F90