This module contains procedures and generic interfaces for the Lower and Upper Incomplete Gamma functions.
More...
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interface | getGammaIncLowNR |
| Generate and return the regularized Lower Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and upper limit of the integral x . More...
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interface | getGammaIncUppNR |
| Generate and return the regularized Upper Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and lower limit of the integral x . More...
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interface | setGammaIncLowNR |
| Return the regularized Lower Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and upper limit of the integral x . More...
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interface | setGammaIncLowSeriesNR |
| Return the regularized Lower Incomplete Gamma function for the specified upper limit x and shape parameter, evaluated by the series representation of the Incomplete Gamma function. More...
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interface | setGammaIncUppContFracNR |
| Return the regularized Upper Incomplete Gamma function for the specified lower limit x and shape parameter, evaluated by the Legendre continued fraction representation of the Incomplete Gamma function. More...
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interface | setGammaIncUppNR |
| Return the regularized Upper Incomplete Gamma function for the specified shape parameter ( \(\kappa\)) and lower limit of the integral x . More...
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This module contains procedures and generic interfaces for the Lower and Upper Incomplete Gamma functions.
This module provides multiple function and subroutine procedures for computing the Lower and Upper Incomplete Gamma functions. These routines mostly differ only in terms of performance and usage convenience.
- If performance is important, use the subroutine interface setGammaIncLowNR and setGammaIncUppNR to compute the Lower and Upper Incomplete Gamma function.
- If ease of use matters more than performance, use the function interfaces getGammaIncLowNR and getGammaIncUppNR to compute the Lower and Upper Incomplete Gamma function.
- setGammaIncLowSeriesNR and setGammaIncUppContFracNR are not meant to be used directly to compute directly the incomplete Gamma functions since their performance depends on the values of the input parameters.
Rather they are low-level implementations to be called only by the above higher-level interfaces.
See below for the relevant benchmark.
- Warning
- Although all generic interfaces of this module are available for all processor
real
kinds, the accuracy and performance of the implemented algorithms are optimized for IEEE double precision.
In particular, the algorithms may not accurately compute the lower incomplete gamma function in extended precision (e.g., 128 bits) mode corresponding to RKH kind type parameter.
- Note
- The computations of this module are an implementation of the algorithm proposed approach by:
Numerical Recipes in Fortran 77: The Art of Scientific Computing 2nd Edition William H. Press (Author), Brian P. Flannery (Author), Saul A. Teukolsky (Author), William T. Vetterling (Author)
- Benchmarks:
Benchmark :: The runtime performance of setGammaIncLowSeries vs. setGammaIncUppContFracNR ⛓
4 use iso_fortran_env,
only:
error_unit
13 integer(IK) :: fileUnit
14 integer(IK) ,
parameter :: NPNT
= 20_IK
15 integer(IK) ,
parameter :: NBENCH
= 2_IK
16 real(RKG) ,
parameter :: kappa
= 1._RKG
17 real(RKG) ,
parameter :: logGammaKappa
= log_gamma(kappa)
19 real(RKG) :: Point(NPNT)
20 real(RKG) :: dummy
= 0._RKG
23 type(bench_type) :: bench(NBENCH)
25 bench(
1)
= bench_type(name
= SK_
"setGammaIncLowSeriesNR", exec
= setGammaIncLowSeriesNR, overhead
= setOverhead)
26 bench(
2)
= bench_type(name
= SK_
"setGammaIncUppContFracNR", exec
= setGammaIncUppContFracNR, overhead
= setOverhead)
28 call setLogSpace(Point, logx1
= log(
0.04_RKG), logx2
= log(
100._RKG))
31 write(
*,
"(*(g0,:,' '))")
32 write(
*,
"(*(g0,:,' vs. '))") (bench(i)
%name, i
= 1, NBENCH)
33 write(
*,
"(*(g0,:,' '))")
35 open(newunit
= fileUnit, file
= "main.out", status
= "replace")
37 write(fileUnit,
"(*(g0,:,','))")
"x / (kappa + 1)", (bench(i)
%name, i
= 1, NBENCH)
39 loopOverPoint:
do ipnt
= 1, NPNT
41 write(
*,
"(*(g0,:,' '))")
"Benchmarking with point", Point(ipnt)
44 bench(i)
%timing
= bench(i)
%getTiming(minsec
= 0.05_RK)
47 write(fileUnit,
"(*(g0,:,','))") Point(ipnt)
/ (kappa
+ 1), (bench(i)
%timing
%mean, i
= 1, NBENCH)
51 write(
*,
"(*(g0,:,' '))") dummy
52 write(
*,
"(*(g0,:,' '))")
62 subroutine setOverhead()
67 if (info
< 0_IK)
error stop
68 dummy
= dummy
+ gammaInc
71 subroutine setGammaIncLowSeriesNR()
76 , logGammaKappa
= logGammaKappa
&
85 subroutine setGammaIncUppContFracNR()
90 , logGammaKappa
= logGammaKappa
&
Return the logSpace output argument with size(logSpace) elements of logarithmically-evenly-spaced val...
Generate and return an object of type timing_type containing the benchmark timing information and sta...
Return the regularized Lower Incomplete Gamma function for the specified upper limit x and shape para...
Return the regularized Upper Incomplete Gamma function for the specified lower limit x and shape para...
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
This module contains abstract interfaces and types that facilitate benchmarking of different procedur...
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
integer, parameter RK
The default real kind in the ParaMonte library: real64 in Fortran, c_double in C-Fortran Interoperati...
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 C-Fortran Interoper...
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
This module contains procedures and generic interfaces for the Lower and Upper Incomplete Gamma funct...
This is the class for creating benchmark and performance-profiling objects.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort -fpp -standard-semantics -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 /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Postprocessing of the benchmark output ⛓
3import matplotlib.pyplot
as plt
8dirname = os.path.basename(os.getcwd())
12df = pd.read_csv(
"main.out", delimiter =
",")
13colnames = list(df.columns.values)
19ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
22for colname
in colnames[1:]:
23 plt.plot( df[colnames[0]].values
28plt.xticks(fontsize = fontsize)
29plt.yticks(fontsize = fontsize)
30ax.set_xlabel(colnames[0], fontsize = fontsize)
31ax.set_ylabel(
"Runtime [ seconds ]", fontsize = fontsize)
32ax.set_title(
" vs. ".join(colnames[1:])+
"\nLower is better.", fontsize = fontsize)
36plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"-")
37ax.tick_params(axis =
"y", which =
"minor")
38ax.tick_params(axis =
"x", which =
"minor")
39ax.legend ( colnames[1:]
46plt.savefig(
"benchmark." + dirname +
".runtime.png")
52ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
55plt.plot( df[colnames[0]].values
56 , np.ones(len(df[colnames[0]].values))
61for colname
in colnames[2:]:
62 plt.plot( df[colnames[0]].values
63 , df[colname].values / df[colnames[1]].values
67plt.xticks(fontsize = fontsize)
68plt.yticks(fontsize = fontsize)
69ax.set_xlabel(colnames[0], fontsize = fontsize)
70ax.set_ylabel(
"Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
71ax.set_title(
"Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}().".format(colnames[1]), fontsize = fontsize)
75plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"-")
76ax.tick_params(axis =
"y", which =
"minor")
77ax.tick_params(axis =
"x", which =
"minor")
78ax.legend ( colnames[1:]
85plt.savefig(
"benchmark." + dirname +
".runtime.ratio.png")
Visualization of the benchmark output ⛓
Benchmark moral ⛓
- The procedures under the generic interface setGammaIncLowSeriesNR compute the Lower Incomplete Gamma function suing the series representation of the Incomplete Gamma function while the procedures under the generic interface setGammaIncUppContFracNR use the Continued Fraction representation of the Gamma function.
The Legendre continued fraction representation is known to converge faster at \(x > \kappa + 1\) whereas the series representation converges faster at \(x < \kappa + 1\).
This performance difference dependence on the specific values of \(x\) and \(\kappa\) is well illustrated by this benchmark.
- See also
- pm_mathGammaNR for detailed description of the (Regularized Incomplete) Gamma Function.
- Test:
- test_pm_mathGammaNR
- Todo:
- Critical Priority: The implementation of the algorithms of this module must be properly changed to allow reliable extended-precision computations of the incomplete Gamma function.
This would require significant investment in making the original algorithms of Gil et al. kind-agnostic.
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, July 22, 2024, 11:45 AM, NASA Goddard Space Flight Center, Washington, D.C.