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

Detailed Description

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

[out]	gammaIncUpp	: The output scalar of 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]	x	: The input scalar of the 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]	logGammaKappa	: The input scalar 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 could 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 setGammaIncUppContFracNR or setGammaIncLowSeriesNR).

Possible calling interfaces ⛓

: use pm_mathGammaNR, only: setGammaIncUppNR

call setGammaIncUppNR(gammaIncUpp, x, logGammaKappa, kappa, info, tol = tol)

pm_mathGammaNR::setGammaIncUppNR
Return the regularized Upper Incomplete Gamma function for the specified shape parameter ( ) and lowe...
Definition: pm_mathGammaNR.F90:649

pm_mathGammaNR
This module contains procedures and generic interfaces for the Lower and Upper Incomplete Gamma funct...
Definition: pm_mathGammaNR.F90:86

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 setGammaIncUppNR with the same shape parameter but different x.

See also: getGammaIncUppNR
getGammaIncLowNR

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK

4 use pm_kind, only: IK

5 use pm_kind, only: LK

6 use pm_kind, only: RKS, RKD, RKH

7 use pm_io, only: display_type

8 use pm_mathGammaNR, only: setGammaIncUppNR

9

10 implicit none

11

12 integer(IK) , parameter :: NP = 1000_IK

13 real(RKH) :: gamIncUpp_RKH, x_RKH, kappa_RKH

14 real(RKD) :: gamIncUpp_RKD, x_RKD, kappa_RKD

15 real(RKS) :: gamIncUpp_RKS, x_RKS, kappa_RKS

16 integer(IK) :: info

17

18 type(display_type) :: disp

19 disp = display_type(file = "main.out.F90")

20

21 kappa_RKH = 1.5_RKH

22 kappa_RKD = 1.5_RKD

23 kappa_RKS = 1.5_RKS

24

25 x_RKH = 2._RKH

26 x_RKD = 2._RKD

27 x_RKS = 2._RKS

28

29 call disp%skip()

30 call disp%show("x_RKS")

31 call disp%show( x_RKS )

32 call disp%show("kappa_RKS")

33 call disp%show( kappa_RKS )

34 call disp%show("call setGammaIncUppNR(gamIncUpp_RKS, x_RKS, logGammaKappa = log_gamma(kappa_RKS), kappa = kappa_RKS, info = info)")

35 call setGammaIncUppNR(gamIncUpp_RKS, x_RKS, logGammaKappa = log_gamma(kappa_RKS), kappa = kappa_RKS, info = info)

36 call disp%show("gamIncUpp_RKS")

37 call disp%show( gamIncUpp_RKS )

38 call disp%show("info")

39 call disp%show( info )

40 call disp%skip()

41

42 call disp%skip()

43 call disp%show("x_RKD")

44 call disp%show( x_RKD)

45 call disp%show("kappa_RKD")

46 call disp%show( kappa_RKD)

47 call disp%show("call setGammaIncUppNR(gamIncUpp_RKD, x_RKD, logGammaKappa = log_gamma(kappa_RKD), kappa = kappa_RKD, info = info)")

48 call setGammaIncUppNR(gamIncUpp_RKD, x_RKD, logGammaKappa = log_gamma(kappa_RKD), kappa = kappa_RKD, info = info)

49 call disp%show("gamIncUpp_RKD")

50 call disp%show( gamIncUpp_RKD)

51 call disp%show("info")

52 call disp%show( info )

53 call disp%skip()

54

55 call disp%skip()

56 call disp%show("x_RKH")

57 call disp%show( x_RKH )

58 call disp%show("kappa_RKH")

59 call disp%show( kappa_RKH )

60 call disp%show("call setGammaIncUppNR(gamIncUpp_RKH, x_RKH, logGammaKappa = log_gamma(kappa_RKH), kappa = kappa_RKH, info = info)")

61 call setGammaIncUppNR(gamIncUpp_RKH, x_RKH, logGammaKappa = log_gamma(kappa_RKH), kappa = kappa_RKH, info = info)

62 call disp%show("gamIncUpp_RKH")

63 call disp%show( gamIncUpp_RKH )

64 call disp%show("info")

65 call disp%show( info )

66 call disp%skip()

67

68 block

69 use pm_kind, only: RKG => RKS

70 integer(IK) :: i, rprecision

71 integer(IK), allocatable :: exprange(:), info(:)

72 real(RKG), allocatable :: gamIncUpp(:)

73 call disp%skip()

74 call disp%show("rprecision = precision(0._RKG) / 2")

75 rprecision = precision(0._RKG) / 2

76 call disp%show("rprecision")

77 call disp%show( rprecision )

78 call disp%show("exprange = [(i, i = -rprecision, rprecision)]")

79 exprange = [(i, i = -rprecision, rprecision)]

80 call disp%show("exprange")

81 call disp%show( exprange )

82 call disp%show("allocate(gamIncUpp(size(exprange)), info(size(exprange)))")

83 allocate(gamIncUpp(size(exprange)), info(size(exprange)))

84 call disp%show("call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)")

85 call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)

86 call disp%show("gamIncUpp")

87 call disp%show( gamIncUpp )

88 call disp%show("info")

89 call disp%show( info )

90 call disp%skip()

91 end block

92

93 block

94 use pm_kind, only: RKG => RKD

95 integer(IK) :: i, rprecision

96 integer(IK), allocatable :: exprange(:), info(:)

97 real(RKG), allocatable :: gamIncUpp(:)

98 call disp%skip()

99 call disp%show("rprecision = precision(0._RKG) / 2")

100 rprecision = precision(0._RKG) / 2

101 call disp%show("rprecision")

102 call disp%show( rprecision )

103 call disp%show("exprange = [(i, i = -rprecision, rprecision)]")

104 exprange = [(i, i = -rprecision, rprecision)]

105 call disp%show("exprange")

106 call disp%show( exprange )

107 call disp%show("allocate(gamIncUpp(size(exprange)), info(size(exprange)))")

108 allocate(gamIncUpp(size(exprange)), info(size(exprange)))

109 call disp%show("call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)")

110 call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)

111 call disp%show("gamIncUpp")

112 call disp%show( gamIncUpp )

113 call disp%show("info")

114 call disp%show( info )

115 call disp%skip()

116 end block

117

118 block

119 use pm_kind, only: RKG => RKH

120 integer(IK) :: i, rprecision

121 integer(IK), allocatable :: exprange(:), info(:)

122 real(RKG), allocatable :: gamIncUpp(:)

123 call disp%skip()

124 call disp%show("rprecision = precision(0._RKG) / 2")

125 rprecision = precision(0._RKG) / 2

126 call disp%show("rprecision")

127 call disp%show( rprecision )

128 call disp%show("exprange = [(i, i = -rprecision, rprecision)]")

129 exprange = [(i, i = -rprecision, rprecision)]

130 call disp%show("exprange")

131 call disp%show( exprange )

132 call disp%show("allocate(gamIncUpp(size(exprange)), info(size(exprange)))")

133 allocate(gamIncUpp(size(exprange)), info(size(exprange)))

134 call disp%show("call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)")

135 call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)

136 call disp%show("gamIncUpp")

137 call disp%show( gamIncUpp )

138 call disp%show("info")

139 call disp%show( info )

140 call disp%skip()

141 end block

142

143 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

144 ! Output an example array of the regularized Upper Incomplete Gamma function for visualization.

145 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

146

147 block

148

149 use pm_arraySpace, only: setLinSpace

150 real(RKS) :: x_RKS(NP)

151 integer :: fileUnit, i

152

153 call setLinSpace(x_RKS, 0._RKS, 8._RKS)

154 open(newunit = fileUnit, file = "setGammaIncUppNR.RK.txt")

155 do i = 1, NP

156 call setGammaIncUppNR(gamIncUpp_RKS, x_RKS(i), logGammaKappa = log_gamma(kappa_RKS), kappa = kappa_RKS, info = info)

157 write(fileUnit,"(2(g0,:,' '))") x_RKS(i), gamIncUpp_RKS

158 end do

159 close(fileUnit)

160

161 end block

162

163end 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

2x_RKS

3+2.00000000

4kappa_RKS

5+1.50000000

6call setGammaIncUppNR(gamIncUpp_RKS, x_RKS, logGammaKappa = log_gamma(kappa_RKS), kappa = kappa_RKS, info = info)

7gamIncUpp_RKS

8+0.261464179

9info

10+11

11

12

13x_RKD

14+2.0000000000000000

15kappa_RKD

16+1.5000000000000000

17call setGammaIncUppNR(gamIncUpp_RKD, x_RKD, logGammaKappa = log_gamma(kappa_RKD), kappa = kappa_RKD, info = info)

18gamIncUpp_RKD

19+0.26146412994911072

20info

21+21

22

23

24x_RKH

25+2.00000000000000000000000000000000000

26kappa_RKH

27+1.50000000000000000000000000000000000

28call setGammaIncUppNR(gamIncUpp_RKH, x_RKH, logGammaKappa = log_gamma(kappa_RKH), kappa = kappa_RKH, info = info)

29gamIncUpp_RKH

30+0.261464129949110622202822075975921249

31info

32+36

33

34

35rprecision = precision(0._RKG) / 2

36rprecision

37+3

38exprange = [(i, i = -rprecision, rprecision)]

39exprange

40-3, -2, -1, +0, +1, +2, +3

41allocate(gamIncUpp(size(exprange)), info(size(exprange)))

42call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)

43gamIncUpp

44+0.631248951E-2, +0.396528244E-1, +0.172448397, +0.367879391, +0.457929194, +0.486702442, +0.495647073

45info

46+2, +3, +5, +9, +19, +51, +144

47

48

49rprecision = precision(0._RKG) / 2

50rprecision

51+7

52exprange = [(i, i = -rprecision, rprecision)]

53exprange

54-7, -6, -5, -4, -3, -2, -1, +0, +1, +2, +3, +4, +5, +6, +7

55allocate(gamIncUpp(size(exprange)), info(size(exprange)))

56call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)

57gamIncUpp

58+0.15540868092411841E-5, +0.13238209089494468E-4, +0.10935130095757195E-3, +0.86295813100534247E-3, +0.63123532911392166E-2, +0.39652576478489632E-1, +0.17244824041414908, +0.36787944117144233, +0.45792971447185082, +0.48670120172086928, +0.49579475581981880, +0.49867019166480753, +0.49957947781728729, +0.49986701958132673, +0.49995795435734636

59info

60+3, +3, +3, +4, +5, +6, +9, +16, +35, +89, +255, +770, +2365, +7304, +22572

61

62

63rprecision = precision(0._RKG) / 2

64rprecision

65+16

66exprange = [(i, i = -rprecision, rprecision)]

67exprange

68-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, +0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10, +11, +12, +13, +14, +15, +16

69allocate(gamIncUpp(size(exprange)), info(size(exprange)))

70call setGammaIncUppNR(gamIncUpp, 10._RKG**exprange, logGammaKappa = log_gamma(10._RKG**exprange), kappa = 10._RKG**exprange, info = info)

71gamIncUpp

72+0.362641458230031325076828129542474766E-14, +0.339615607300085775277864670886475238E-13, +0.316589756370101134867553294329700904E-12, +0.293563905439781533950555715921222342E-11, +0.270538054506628836199548154653849778E-10, +0.247512203549880797881142114345206485E-9, +0.224486352400241094382073832681308249E-8, +0.201460499709356778860957793291373752E-7, +0.178434635053293640783839339393679058E-6, +0.155408680923619091775019872787391912E-5, +0.132382090896648326218821852228554586E-4, +0.109351300956329854758268594627563903E-3, +0.862958131006599938564454065107016909E-3, +0.631235329113970979344136715350855536E-2, +0.396525764784908016731185218404744951E-1, +0.172448240414149458549686503831842920, +0.367879441171442321595523770161460873, +0.457929714471852208314164857059333906, +0.486701201720851335142685743435971364, +0.495794755819784491496222156397972042, +0.498670191660044799617257748700958660, +0.499579477889634823306687420993386096, +0.499867019239127408755677182042854039, +0.499957947791276301666243835240964160, +0.499986701923985880128751122750594547, +0.499995794779129943037581224570753625, +1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000

73info

74+3, +3, +3, +3, +3, +3, +4, +4, +4, +5, +6, +6, +8, +10, +13, +18, +30, +59, +143, +402, +1211, +3747, +11701, +36641, +114821, +359862, -1000001, -1000001, -1000001, -1000001, -1000001, -1000001, -1000001

75

76

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 Upper\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_mathGammaNR

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.

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:: Fatemeh Bagheri, Monday 12:36 pm, August 16, 2021, Dallas TX

Definition at line 649 of file pm_mathGammaNR.F90.

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

src/fortran/main/pm_mathGammaNR.F90