ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation.
pm_mathGammaGil::setGammaIncGil Interface Reference

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

Detailed Description

Return the regularized Lower and Upper Incomplete Gamma function values 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.
The Regularized Upper Incomplete Gamma Function can be readily obtained as,

\begin{equation} \large Q(\kappa, x) = 1 - P(\kappa, x) ~, \end{equation}

However, the sake of numerical accuracy of either integral near 1, both lower and upper incomplete gamma values are returned by the procedures of this generic interface.

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 Incomplete Gamma function for the specified kappa and lower limit.
Note that gammaIncLow is, by definition, always positive in the range \([0, 1]\).
[out]gammaIncUpp: The output scalar of the same type and kind as the input argument x representing the Upper Incomplete Gamma function for the specified kappa and lower limit.
Note that gammaIncUpp 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 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 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 zero if the algorithm succeeds to converge or a negative value if the algorithm fails to converge.
A convergence failure could happen if the input value for kappa is too large.
A negative value implies the lack of convergence.


Possible calling interfaces ⛓

use pm_mathGammaGil, only: setGammaIncGil
call setGammaIncGil(gammaIncLow, gammaIncUpp, x, kappa, info)
pm_mathGammaGil::setGammaIncGil
Return the regularized Lower and Upper Incomplete Gamma function values for the specified shape param...
Definition: pm_mathGammaGil.F90:421
pm_mathGammaGil
This module contains procedures and generic interfaces for the Lower and Upper Incomplete Gamma funct...
Definition: pm_mathGammaGil.F90:57
Warning
The condition 0 < x must hold for the corresponding input arguments.
The condition 0 < kappa must hold for the corresponding input arguments.
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.
See also
getGammaIncLowGil
setGammaIncGil


Example usage ⛓

1program example
2
3 use pm_kind, only: SK, IK, LK
4 use pm_kind, only: RKS, RKD, RKH
5 use pm_mathGammaGil, only: setGammaIncGil
6 use pm_io, only: display_type
7
8 implicit none
9
10 integer(IK) , parameter :: NP = 1000_IK
11 real(RKH) :: gamIncLow_RKH, gamIncUpp_RKH, x_RKH, kappa_RKH
12 real(RKD) :: gamIncLow_RKD, gamIncUpp_RKD, x_RKD, kappa_RKD
13 real(RKS) :: gamIncLow_RKS, gamIncUpp_RKS, x_RKS, kappa_RKS
14 integer(IK) :: info
15
16 type(display_type) :: disp
17 disp = display_type(file = "main.out.F90")
18
19 kappa_RKH = 1.5_RKH
20 kappa_RKD = 1.5_RKD
21 kappa_RKS = 1.5_RKS
22
23 x_RKH = 2._RKH
24 x_RKD = 2._RKD
25 x_RKS = 2._RKS
26
27 call disp%skip()
28 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
29 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
30 call disp%show("! Compute the regularized Lower Incomplete Gamma Function using its series representation.")
31 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
32 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
33 call disp%skip()
34
35 call disp%skip()
36 call disp%show("x_RKS")
37 call disp%show( x_RKS )
38 call disp%show("kappa_RKS")
39 call disp%show( kappa_RKS )
40 call disp%show("call setGammaIncGil(gamIncLow_RKS, gamIncUpp_RKS, x_RKS, kappa = kappa_RKS, info = info)")
41 call setGammaIncGil(gamIncLow_RKS, gamIncUpp_RKS, x_RKS, kappa = kappa_RKS, info = info)
42 call disp%show("[gamIncLow_RKS, gamIncUpp_RKS]")
43 call disp%show( [gamIncLow_RKS, gamIncUpp_RKS] )
44 call disp%show("info")
45 call disp%show( info )
46 call disp%skip()
47
48 call disp%skip()
49 call disp%show("x_RKD")
50 call disp%show( x_RKD )
51 call disp%show("kappa_RKD")
52 call disp%show( kappa_RKD )
53 call disp%show("call setGammaIncGil(gamIncLow_RKD, gamIncUpp_RKD, x_RKD, kappa = kappa_RKD, info = info)")
54 call setGammaIncGil(gamIncLow_RKD, gamIncUpp_RKD, x_RKD, kappa = kappa_RKD, info = info)
55 call disp%show("[gamIncLow_RKD, gamIncUpp_RKD]")
56 call disp%show( [gamIncLow_RKD, gamIncUpp_RKD] )
57 call disp%show("info")
58 call disp%show( info )
59 call disp%skip()
60
61 call disp%skip()
62 call disp%show("x_RKH")
63 call disp%show( x_RKH )
64 call disp%show("kappa_RKH")
65 call disp%show( kappa_RKH )
66 call disp%show("call setGammaIncGil(gamIncLow_RKH, gamIncUpp_RKH, x_RKH, kappa = kappa_RKH, info = info)")
67 call setGammaIncGil(gamIncLow_RKH, gamIncUpp_RKH, x_RKH, kappa = kappa_RKH, info = info)
68 call disp%show("[gamIncLow_RKH, gamIncUpp_RKH]")
69 call disp%show( [gamIncLow_RKH, gamIncUpp_RKH] )
70 call disp%show("info")
71 call disp%show( info )
72 call disp%skip()
73
74 call disp%skip()
75 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
76 call disp%show("! Compute the regularized Lower Incomplete Gamma Function for a vector of points and shape parameters.")
77 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
78 call disp%skip()
79
80 block
81 use pm_kind, only: RKG => RKD
82 integer(IK) :: i, rprecision
83 integer(IK), allocatable :: exprange(:), info(:)
84 real(RKG), allocatable :: gamIncLow(:), gamIncUpp(:)
85 call disp%skip()
86 call disp%show("rprecision = precision(0._RKG) * 2")
87 rprecision = precision(0._RKG) * 2
88 call disp%show("rprecision")
89 call disp%show( rprecision )
90 call disp%show("exprange = [(i, i = -rprecision, rprecision)]")
91 exprange = [(i, i = -rprecision, rprecision)]
92 call disp%show("exprange")
93 call disp%show( exprange )
94 call disp%show("allocate(gamIncLow(size(exprange)), gamIncUpp(size(exprange)), info(size(exprange)))")
95 allocate(gamIncLow(size(exprange)), gamIncUpp(size(exprange)), info(size(exprange)))
96 call disp%show("call setGammaIncGil(gamIncLow, gamIncUpp, x = 10._RKG**exprange, kappa = 10._RKG**exprange, info = info)")
97 call setGammaIncGil(gamIncLow, gamIncUpp, x = 10._RKG**exprange, kappa = 10._RKG**exprange, info = info)
98 call disp%show("reshape([10._RKG**exprange, gamIncLow, real(info, RKG)], shape = [size(info), 3])")
99 call disp%show( reshape([10._RKG**exprange, gamIncLow, real(info, RKG)], shape = [size(info), 3]) )
100 call disp%skip()
101 end block
102
103 block
104 use pm_kind, only: RKG => RKH
105 integer(IK) :: i, rprecision
106 integer(IK), allocatable :: exprange(:), info(:)
107 real(RKG), allocatable :: gamIncLow(:), gamIncUpp(:)
108 call disp%skip()
109 call disp%show("rprecision = precision(0._RKG) * 2")
110 rprecision = precision(0._RKG) * 2
111 call disp%show("rprecision")
112 call disp%show( rprecision )
113 call disp%show("exprange = [(i, i = -rprecision, rprecision)]")
114 exprange = [(i, i = -rprecision, rprecision)]
115 call disp%show("exprange")
116 call disp%show( exprange )
117 call disp%show("allocate(gamIncLow(size(exprange)), gamIncUpp(size(exprange)), info(size(exprange)))")
118 allocate(gamIncLow(size(exprange)), gamIncUpp(size(exprange)), info(size(exprange)))
119 call disp%show("call setGammaIncGil(gamIncLow, gamIncUpp, x = 10._RKG**exprange, kappa = 10._RKG**exprange, info = info)")
120 call setGammaIncGil(gamIncLow, gamIncUpp, x = 10._RKG**exprange, kappa = 10._RKG**exprange, info = info)
121 call disp%show("reshape([10._RKG**exprange, gamIncLow, real(info, RKG)], shape = [size(info), 3])")
122 call disp%show( reshape([10._RKG**exprange, gamIncLow, real(info, RKG)], shape = [size(info), 3]) )
123 call disp%skip()
124 end block
125
126 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
127 ! Output an example array of the regularized Lower Incomplete Gamma function for visualization.
128 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
129
130 block
131
132 use pm_arraySpace, only: setLinSpace
133 real(RKS) :: x_RKS(NP)
134 integer :: fileUnit, i
135
136 call setLinSpace(x_RKS, 0._RKS, 8._RKS)
137 open(newunit = fileUnit, file = "setGammaIncGil.RK.txt")
138 do i = 1, NP
139 call setGammaIncGil(gamIncLow_RKS, gamIncUpp_RKS, x_RKS(i), kappa = kappa_RKS, info = info)
140 write(fileUnit,"(2(g0,:,' '))") x_RKS(i), gamIncLow_RKS
141 end do
142 close(fileUnit)
143
144 end block
145
146end program example
pm_arraySpace::setLinSpace
Return the linSpace output argument with size(linSpace) elements of evenly-spaced values over the int...
Definition: pm_arraySpace.F90:324
pm_io::show
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726
pm_io::skip
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508
pm_arraySpace
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
Definition: pm_arraySpace.F90:33
pm_io
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252
pm_io::disp
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393
pm_kind
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
pm_kind::LK
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
Definition: pm_kind.F90:541
pm_kind::IK
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540
pm_kind::RKD
integer, parameter RKD
The double precision real kind in Fortran mode. On most platforms, this is an 64-bit real kind.
Definition: pm_kind.F90:568
pm_kind::SK
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539
pm_kind::RKH
integer, parameter RKH
The scalar integer constant of intrinsic default kind, representing the highest-precision real kind t...
Definition: pm_kind.F90:858
pm_kind::RKS
integer, parameter RKS
The single-precision real kind in Fortran mode. On most platforms, this is an 32-bit real kind.
Definition: pm_kind.F90:567
pm_io::display_type
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

Example Unix compile command via Intel ifort compiler ⛓
1#!/usr/bin/env sh
2rm main.exe
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓
1del main.exe
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓
1#!/usr/bin/env sh
2rm main.exe
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example output ⛓
1
2!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
4! Compute the regularized Lower Incomplete Gamma Function using its series representation.
5!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
6!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
8
9x_RKS
10+2.00000000
11kappa_RKS
12+1.50000000
13call setGammaIncGil(gamIncLow_RKS, gamIncUpp_RKS, x_RKS, kappa = kappa_RKS, info = info)
14[gamIncLow_RKS, gamIncUpp_RKS]
15+0.738535821, +0.261464179
16info
17+0
18
19
20x_RKD
21+2.0000000000000000
22kappa_RKD
23+1.5000000000000000
24call setGammaIncGil(gamIncLow_RKD, gamIncUpp_RKD, x_RKD, kappa = kappa_RKD, info = info)
25[gamIncLow_RKD, gamIncUpp_RKD]
26+0.73853587005088861, +0.26146412994911145
27info
28+0
29
30
31x_RKH
32+2.00000000000000000000000000000000000
33kappa_RKH
34+1.50000000000000000000000000000000000
35call setGammaIncGil(gamIncLow_RKH, gamIncUpp_RKH, x_RKH, kappa = kappa_RKH, info = info)
36[gamIncLow_RKH, gamIncUpp_RKH]
37+0.738535870050889377797177924024077018, +0.261464129949110622202822075975923030
38info
39+0
40
41
42!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
43! Compute the regularized Lower Incomplete Gamma Function for a vector of points and shape parameters.
44!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
45
46
47rprecision = precision(0._RKG) * 2
48rprecision
49+30
50exprange = [(i, i = -rprecision, rprecision)]
51exprange
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53allocate(gamIncLow(size(exprange)), gamIncUpp(size(exprange)), info(size(exprange)))
54call setGammaIncGil(gamIncLow, gamIncUpp, x = 10._RKG**exprange, kappa = 10._RKG**exprange, info = info)
55reshape([10._RKG**exprange, gamIncLow, real(info, RKG)], shape = [size(info), 3])
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118
119rprecision = precision(0._RKG) * 2
120rprecision
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122exprange = [(i, i = -rprecision, rprecision)]
123exprange
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125allocate(gamIncLow(size(exprange)), gamIncUpp(size(exprange)), info(size(exprange)))
126call setGammaIncGil(gamIncLow, gamIncUpp, x = 10._RKG**exprange, kappa = 10._RKG**exprange, info = info)
127reshape([10._RKG**exprange, gamIncLow, real(info, RKG)], shape = [size(info), 3])
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Postprocessing of the example output ⛓
1#!/usr/bin/env python
2
3import matplotlib.pyplot as plt
4import pandas as pd
5import numpy as np
6import glob
7import sys
8
9fontsize = 17
10
11marker ={ "CK" : "-"
12 , "IK" : "."
13 , "RK" : "-"
14 }
15xlab = { "CK" : r"x ( real/imaginary )"
16 , "IK" : r"x ( integer-valued )"
17 , "RK" : r"x ( real-valued )"
18 }
19labels = [r"shape parameter: $\kappa = 2$"]
20
21for kind in ["IK", "CK", "RK"]:
22
23 pattern = "*." + kind + ".txt"
24 fileList = glob.glob(pattern)
25 if len(fileList) == 1:
26
27 df = pd.read_csv(fileList[0], delimiter = " ")
28
29 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
30 ax = plt.subplot()
31
32 if kind == "CK":
33 plt.plot( df.values[:, 0]
34 , df.values[:,2]
35 , marker[kind]
36 , color = "r"
37 )
38 plt.plot( df.values[:, 1]
39 , df.values[:,3]
40 , marker[kind]
41 , color = "blue"
42 )
43 else:
44 plt.plot( df.values[:, 0]
45 , df.values[:, 1]
46 , marker[kind]
47 , color = "r"
48 )
49
50 plt.xticks(fontsize = fontsize - 2)
51 plt.yticks(fontsize = fontsize - 2)
52 ax.set_xlabel(xlab[kind], fontsize = fontsize)
53 ax.set_ylabel("Regularized Lower\nIncomplete Gamma Function", fontsize = fontsize)
54
55 plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
56 ax.tick_params(axis = "y", which = "minor")
57 ax.tick_params(axis = "x", which = "minor")
58
59 ax.legend ( labels
60 , fontsize = fontsize
61 #, loc = "center left"
62 #, bbox_to_anchor = (1, 0.5)
63 )
64
65 plt.savefig(fileList[0].replace(".txt",".png"))
66
67 elif len(fileList) > 1:
68
69 sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓
Test:
test_pm_mathGammaGil


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 421 of file pm_mathGammaGil.F90.

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