Return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x
within the support of the distribution \(x \in (0,+\infty)\).
More...
Return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x
within the support of the distribution \(x \in (0,+\infty)\).
See the documentation of pm_distGamma for more information on the Gamma CDF.
 Parameters

[out]  cdf  : The output scalar or array of the same shape as any input arraylike argument, of the same type and kind the input argument x , containing the CDF of the distribution at the specified x .

[in]  x  : The input scalar or array of the same shape as other array like arguments, of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the values at which the CDF must be computed.

[in]  kappa  : The input scalar or array of the same shape as other arraylike arguments, of the same type and kind as x , containing the shape parameter of the distribution.
(optional, default = 1. . It must be present if invSigma is also present.) 
[in]  invSigma  : The input scalar or array of the same shape as other arraylike arguments, of the same type and kind as x , containing the rate (inverse scale) parameter of the distribution.
(optional, default = 1. . It can be present only if all previous arguments are also present.) 
[out]  info  : The output scalar of type integer of default kind IK.
On output, it is set to positive the number of iterations taken for the series representation of the Gamma function to converge.
If the algorithm fails to converge, then info is set to the negative of the number of iterations taken by the algorithm.
An negative output value signifies the lack of convergence and failure to compute the CDF.
This is likely to happen if the input value for kappa is too large.

Possible calling interfaces ⛓
Return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x within the...
This module contains classes and procedures for computing various statistical quantities related to t...
 Warning
 The condition
0 < x
must hold for the corresponding input arguments.
The condition 0 < kappa
must hold for the corresponding input arguments.
The condition 0 < invSigma
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.
 See also
 getGammaCDF
Example usage ⛓
11 integer(IK),
parameter :: NP
= 999_IK
12 real :: point(NP), CDF(NP)
13 integer(IK) :: info(NP)
15 type(display_type) :: disp
21 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
22 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
23 call disp%show(
"! Compute the Cumulative Distribution Function (CDF) of the Gamma distribution.")
24 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
25 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
31 call disp%show(
"call setGammaCDF(CDF(1), point(1), info(1))")
33 call disp%show(
"if (info(1) < 0) error stop 'setGammaCDF() failed.'")
34 if (info(
1)
< 0)
error stop 'setGammaCDF() failed.'
40 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
41 call disp%show(
"! Accelerate the runtime performance for repeated calls when `kappa` and `invSigma` are fixed.")
42 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
48 call disp%show(
"call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = 2., invSigma = 2., info = info(1:NP:NP/4))")
49 call setGammaCDF(CDF(
1:NP:NP
/4), point(
1:NP:NP
/4), kappa
= 2., invSigma
= 2., info
= info(
1:NP:NP
/4))
50 call disp%show(
"if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'")
51 if (
any(info(
1:NP:NP
/4)
< 0))
error stop 'setGammaCDF() failed.'
57 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
58 call disp%show(
"! A vector of CDF at different points with the same CDF parameters.")
59 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
65 call disp%show(
"call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = 0.5, invSigma = 5., info = info(1:NP:NP/4))")
66 call setGammaCDF(CDF(
1:NP:NP
/4), point(
1:NP:NP
/4), kappa
= 0.5, invSigma
= 5., info
= info(
1:NP:NP
/4))
67 call disp%show(
"if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'")
68 if (
any(info(
1:NP:NP
/4)
< 0))
error stop 'setGammaCDF() failed.'
74 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
75 call disp%show(
"! A vector of CDF at the same point but with different CDF parameters.")
76 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
82 call disp%show(
"call setGammaCDF(CDF(1:NP:NP/4), point(NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5), info = info(1:NP:NP/4))")
84 call disp%show(
"if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'")
85 if (
any(info(
1:NP:NP
/4)
< 0))
error stop 'setGammaCDF() failed.'
91 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
92 call disp%show(
"! A vector of CDF at different points with different CDF parameters.")
93 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
99 call disp%show(
"call setGammaCDF(CDF(1:NP:NP/4), point(1:NP:NP/4), kappa = getLinSpace(0.5, 5., 5), invSigma = getLinSpace(5., .5, 5), info = info(1:NP:NP/4))")
101 call disp%show(
"if (any(info(1:NP:NP/4) < 0)) error stop 'setGammaCDF() failed.'")
102 if (
any(info(
1:NP:NP
/4)
< 0))
error stop 'setGammaCDF() failed.'
112 integer(IK) :: fileUnit, i
113 real,
parameter :: kappa(
*)
= [
0.5,
1.0,
2.0,
7.5]
114 real,
parameter :: invSigma(
*)
= [
1.0,
0.5,
0.5,
1.0]
115 open(newunit
= fileUnit, file
= "setGammaCDF.RK.txt")
117 call setGammaCDF(CDF(
1:
4), point(i), kappa, invSigma, info(
1:
4))
118 if (
any(info(
1:
4)
< 0))
error stop 'setGammaCDF() failed.'
119 write(fileUnit,
"(5(g0,:,' '))") point(i), CDF(
1:
4)
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Return the logSpace output argument with size(logSpace) elements of logarithmicallyevenlyspaced val...
This is a generic method of the derived type display_type with pass attribute.
This is a generic method of the derived type display_type with pass attribute.
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
This module defines the relevant Fortran kind typeparameters frequently used in the ParaMonte librar...
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in CFortran Interoper...
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in CFortran Intero...
Generate and return an object of type display_type.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort fpp standardsemantics O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example Windows Batch compile command via Intel ifort
compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standardsemantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran cpp ffreelinelengthnone O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example output ⛓
12if (info(
1)
< 0)
error stop 'setGammaCDF() failed.'
23+0.673794653E2,
+0.816747844E1,
+0.990029752,
+12.0007629,
+145.468491
24call setGammaCDF(CDF(
1:NP:NP
/4), point(
1:NP:NP
/4), kappa
= 2., invSigma
= 2., info
= info(
1:NP:NP
/4))
25if (
any(info(
1:NP:NP
/4)
< 0))
error stop 'setGammaCDF() failed.'
27+0.899882580E4,
+0.119738970E1,
+0.588569999,
+1.00000000,
+1.00000000
36+0.673794653E2,
+0.816747844E1,
+0.990029752,
+12.0007629,
+145.468491
37call setGammaCDF(CDF(
1:NP:NP
/4), point(
1:NP:NP
/4), kappa
= 0.5, invSigma
= 5., info
= info(
1:NP:NP
/4))
38if (
any(info(
1:NP:NP
/4)
< 0))
error stop 'setGammaCDF() failed.'
40+0.204808801,
+0.633867264,
+0.998347461,
+1.00000000,
+1.00000000
51if (
any(info(
1:NP:NP
/4)
< 0))
error stop 'setGammaCDF() failed.'
53+0.631466985,
+0.860468596E1,
+0.307858689E2,
+0.173809440E4,
+0.870413130E9
62+0.673794653E2,
+0.816747844E1,
+0.990029752,
+12.0007629,
+145.468491
64if (
any(info(
1:NP:NP
/4)
< 0))
error stop 'setGammaCDF() failed.'
66+0.204808801,
+0.872944593E1,
+0.573785663,
+0.999996066,
+1.00000000
Postprocessing of the example output ⛓
3import matplotlib.pyplot
as plt
16xlab = {
"CK" :
"X ( real/imaginary components )"
17 ,
"IK" :
"X ( integervalued )"
18 ,
"RK" :
"X ( realvalued )"
20legends = [
r"$\kappa = 0.5,~\sigma = 1.0$"
21 ,
r"$\kappa = 1.0,~\sigma = 2.0$"
22 ,
r"$\kappa = 2.0,~\sigma = 2.0$"
23 ,
r"$\kappa = 7.5,~\sigma = 1.0$"
26for kind
in [
"IK",
"CK",
"RK"]:
28 pattern =
"*." + kind +
".txt"
29 fileList = glob.glob(pattern)
30 if len(fileList) == 1:
32 df = pd.read_csv(fileList[0], delimiter =
" ")
34 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
38 plt.plot( df.values[:, 0]
41 , linewidth = linewidth
44 plt.plot( df.values[:, 1]
47 , linewidth = linewidth
51 plt.plot( df.values[:, 0]
54 , linewidth = linewidth
61 plt.xticks(fontsize = fontsize  2)
62 plt.yticks(fontsize = fontsize  2)
63 ax.set_xlabel(xlab[kind], fontsize = 17)
64 ax.set_ylabel(
"Cumulative Distribution Function (CDF)", fontsize = 17)
67 plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"")
68 ax.tick_params(axis =
"y", which =
"minor")
69 ax.tick_params(axis =
"x", which =
"minor")
71 plt.savefig(fileList[0].replace(
".txt",
".png"))
73 elif len(fileList) > 1:
75 sys.exit(
"Ambiguous file list exists.")
Visualization of the example output ⛓
 Test:
 test_pm_distGamma
 Todo:
 Low Priority: This generic interface can be extended to
complex
arguments.
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, Oct 16, 2009, 11:14 AM, Michigan
Definition at line 917 of file pm_distGamma.F90.