Return the regularized Inverse Incomplete Beta Function $I_x(\alpha, \beta)$ as defined in the details section of pm_mathBeta. More...

Detailed Description

Return the regularized Inverse Incomplete Beta Function $I_x(\alpha, \beta)$ as defined in the details section of pm_mathBeta.

Parameters

[in]	betaInv	: The output scalar (or array of the same shape as other array-like arguments) of the same type and kind as the input argument `betaInc` containing the regularized Inverse Incomplete Beta Function.
[in]	betaInc	: 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 value at which the function must be computed.
[in]	alpha	: The input scalar (or array of the same shape as other array-like arguments) of the same type and kind as `betaInc`.
[in]	beta	: The input scalar (or array of the same shape as other array-like arguments) of the same type and kind as `betaInc`.
[in]	logFuncBeta	: The input scalar (or array of the same shape as other array-like arguments) of the same type and kind as `betaInc`, representing the natural logarithm of the Beta Function as returned by getLogBeta(alpha, beta). Providing this argument can lead to significant runtime performance gains when the interface is called repeatedly for different `betaInc` values but with identical `alpha` and `beta`.
[in]	signed	: The input scalar (or array of the same shape as other array-like arguments) of type `logical` of default kind LK. If `signed = .false.`, the input `betaInc` must be in range `0 <= betaInc <= 1` and the output `betaInv` will be the expected incomplete Beta function in range `0 <= betaInv <= 1`. If `signed = .true.`, then the following rules hold: If the condition `betaInc < 0` holds, then the value `betaInc = 1 - betaInc < 0` will be used instead of `betaInc`. If the output `betaInv` is near `1`, the output will be returned as `betaInv = betaInv - 1 < 0` instead of `betaInv`. Therefore, the user is expected to be aware of the convention and apply the necessary correction (addition by `1`) before using the output value. Specifying `signed = .true.` can lead to considerably more accurate calculations (by orders of magnitudes) for values of `betaInc` and `betaInv` that are near `1`. The loss of precision near `1` occurs because of inadequate resolution of `real` number representations in digital computers near `1` which is orders of magnitude worse than the precision near `0`.
[out]	info	: The output scalar (or array of the same shape as other array-like arguments) of type `integer` of default kind IK. It is set to `0` if and only if the algorithm returns successfully. It is set to a non-zero if and only if the computation of the inverse Incomplete Beta Function at the specified value fails to converge. See the documentation of the corresponding argument `info` of setBetaInc for possible output values and their meanings.

Returns

Possible calling interfaces ⛓

: use pm_mathBeta, only: setBetaInv

call setBetaInv(betaInv, betaInc, alpha, beta, logFuncBeta, signed, info)

pm_mathBeta::setBetaInv
Return the regularized Inverse Incomplete Beta Function as defined in the details section of pm_math...
Definition: pm_mathBeta.F90:826

pm_mathBeta
This module contains classes and procedures for computing the mathematical Beta Function and its inve...
Definition: pm_mathBeta.F90:83

Warning: The conditions 0 <= betaInc .and. betaInc <= 1 must hold for the corresponding input arguments.
The conditions 0 < alpha .and. 0 < beta 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: getLogBeta
setBetaInv
getBetaLogPDF

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK, LK

4 use pm_kind, only: RKG => RK ! all processor kinds are supported.

5 use pm_io, only: display_type

6 use pm_mathBeta, only: getBetaInv

7 use pm_mathBeta, only: setBetaInv

8 use pm_distUnif, only: getUnifRand

9 use pm_mathBeta, only: getBetaInc

10 use pm_mathBeta, only: getLogBeta

11 use pm_arrayResize, only: setResized

12

13 implicit none

14

15 logical(LK) :: signed

16 integer(IK) :: info, infos(3)

17 real(RKG) :: alpha, beta, betaInc, betaInv

18 real(RKG), allocatable :: alphas(:), betas(:), betaIncs(:), betaInvs(:)

19

20 type(display_type) :: disp

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

22

23 signed = .false.

24

25 call disp%skip()

26 call disp%show("signed")

27 call disp%show( signed )

28 call disp%show("betaInc = 0.5_RKG")

29 betaInc = 0.5_RKG

30 call disp%show("alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)")

31 alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)

32 call disp%show("[alpha, beta]")

33 call disp%show( [alpha, beta] )

34 call disp%show("call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)")

35 call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)

36 call disp%show("if (info /= 0) error stop 'Beta inversion failed.'")

37 if (info /= 0) error stop 'Beta inversion failed.'

38 call disp%show("betaInv")

39 call disp%show( betaInv )

40 call disp%show("getBetaInc(betaInv, alpha, beta, signed)")

41 call disp%show( getBetaInc(betaInv, alpha, beta, signed) )

42 call disp%skip()

43

44 call disp%skip()

45 call disp%show("signed")

46 call disp%show( signed )

47 call disp%show("betaInc = 0.5_RKG")

48 betaInc = 0.5_RKG

49 call disp%show("alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)")

50 alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)

51 call disp%show("[alpha, beta]")

52 call disp%show( [alpha, beta] )

53 call disp%show("call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)")

54 call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)

55 call disp%show("if (info /= 0) error stop 'Beta inversion failed.'")

56 if (info /= 0) error stop 'Beta inversion failed.'

57 call disp%show("betaInv")

58 call disp%show( betaInv )

59 call disp%show("getBetaInc(betaInv, alpha, beta, signed)")

60 call disp%show( getBetaInc(betaInv, alpha, beta, signed) )

61 call disp%skip()

62

63 call disp%skip()

64 call disp%show("signed")

65 call disp%show( signed )

66 call disp%show("betaInc = 1._RKG - epsilon(1._RKG)")

67 betaInc = 1._RKG - epsilon(1._RKG)

68 call disp%show("alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)")

69 alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)

70 call disp%show("[alpha, beta]")

71 call disp%show( [alpha, beta] )

72 call disp%show("call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)")

73 call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)

74 call disp%show("if (info /= 0) error stop 'Beta inversion failed.'")

75 if (info /= 0) error stop 'Beta inversion failed.'

76 call disp%show("betaInv")

77 call disp%show( betaInv )

78 call disp%show("getBetaInc(betaInv, alpha, beta, signed)")

79 call disp%show( getBetaInc(betaInv, alpha, beta, signed) )

80 call disp%skip()

81

82 call disp%skip()

83 call disp%show("signed")

84 call disp%show( signed )

85 call disp%show("betaIncs = [0._RKG, .5_RKG, 1._RKG]")

86 betaIncs = [0._RKG, .5_RKG, 1._RKG]

87 call disp%show("alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)")

88 alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)

89 call disp%show("[alpha, beta]")

90 call disp%show( [alpha, beta] )

91 call disp%show("call setResized(betaInvs, size(betaIncs, 1, IK))")

92 call setResized(betaInvs, size(betaIncs, 1, IK))

93 call disp%show("call setBetaInv(betaInvs, betaIncs, alpha, beta, getLogBeta(alpha, beta), signed, infos)")

94 call setBetaInv(betaInvs, betaIncs, alpha, beta, getLogBeta(alpha, beta), signed, infos)

95 call disp%show("if (any(infos /= 0)) error stop 'Beta inversion failed.'")

96 if (any(infos /= 0)) error stop 'Beta inversion failed.'

97 call disp%show("betaInvs")

98 call disp%show( betaInvs )

99 call disp%show("getBetaInc(betaInvs, alpha, beta, signed)")

100 call disp%show( getBetaInc(betaInvs, alpha, beta, signed) )

101 call disp%skip()

102

103 call disp%skip()

104 call disp%show("signed")

105 call disp%show( signed )

106 call disp%show("betaIncs = [0._RKG, .5_RKG, 1._RKG]")

107 betaIncs = [0._RKG, .5_RKG, 1._RKG]

108 call disp%show("alphas = [.1_RKG, 1._RKG, 10._RKG]")

109 alphas = [.1_RKG, 1._RKG, 10._RKG]

110 call disp%show("beta = 3._RKG")

111 beta = 3._RKG

112 call disp%show("call setResized(betaInvs, size(betaIncs, 1, IK))")

113 call setResized(betaInvs, size(betaIncs, 1, IK))

114 call disp%show("call setBetaInv(betaInvs, betaIncs, alphas, beta, getLogBeta(alpha, beta), signed, infos)")

115 call setBetaInv(betaInvs, betaIncs, alphas, beta, getLogBeta(alpha, beta), signed, infos)

116 call disp%show("if (any(infos /= 0)) error stop 'Beta inversion failed.'")

117 if (any(infos /= 0)) error stop 'Beta inversion failed.'

118 call disp%show("betaInvs")

119 call disp%show( betaInvs )

120 call disp%show("getBetaInc(betaInvs, alpha = [.1_RKG, 1._RKG, 10._RKG], beta = 3._RKG, signed = signed)")

121 call disp%show( getBetaInc(betaInvs, alpha = [.1_RKG, 1._RKG, 10._RKG], beta = 3._RKG, signed = signed) )

122 call disp%skip()

123

124 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

125 ! Output an example array for visualization.

126 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

127

128 block

129

130 use pm_kind, only: RKG => RKH, RKH

131 use pm_arraySpace, only: setLinSpace

132 integer(IK) , parameter :: NP = 1000

133 real(RKG) , parameter :: alpha(*) = [0.1_RKG, 10._RKG, 1._RKG, 0.1_RKG, 10._RKG], beta(*) = [0.1_RKG, 0.1_RKG, 1._RKG, 10._RKG, 10._RKG]

134 real(RKG) :: betaInv(max(size(alpha), size(beta)))

135 real(RKG) :: betaInvRef(size(betaInv))

136 real(RKG) :: betaInc(NP)

137 integer(IK) :: infos(size(betaInv))

138 integer :: fileUnit, i, j

139

140 call setLinSpace(betaInc, 0._RKG, 1._RKG, fopen = .true._LK, lopen = .true._LK)

141 open(newunit = fileUnit, file = "setBetaInv.RK.txt")

142 do i = 1, NP

143 call setBetaInv(betaInv, betaInc(i), alpha, beta, getLogBeta(alpha, beta), signed, infos)

144 if (any(infos /= 0)) error stop "Beta inversion failed."

145 write(fileUnit, "(*(g0,:,','))") betaInc(i), betaInv

146 end do

147 close(fileUnit)

148

149 block

150 use pm_kind, only: RKG => RKS

151 character(*), parameter :: RKSTR = "RKS"

152 real(RKG) :: betaInv(max(size(alpha), size(beta)))

153 open(newunit = fileUnit, file = "setBetaInv."//RKSTR//".abserr.txt")

154 do i = 1, NP

155 call setBetaInv(betaInv, real(betaInc(i), RKG), real(alpha, RKG), real(beta, RKG), getLogBeta(real(alpha, RKG), real(beta, RKG)), signed, infos)

156 if (any(infos /= 0)) error stop "Beta inversion failed."

157 betaInvRef = getBetaInv(betaInc(i), alpha, beta, signed = .true._LK)

158 write(fileUnit, "(*(g0,:,','))") betaInc(i), abs(betaInv - merge(1 + betaInvRef, betaInvRef, betaInvRef < 0))

159 end do

160 close(fileUnit)

161 end block

162

163 block

164 use pm_kind, only: RKG => RKD

165 character(*), parameter :: RKSTR = "RKD"

166 real(RKG) :: betaInv(max(size(alpha), size(beta)))

167 open(newunit = fileUnit, file = "setBetaInv."//RKSTR//".abserr.txt")

168 do i = 1, NP

169 call setBetaInv(betaInv, real(betaInc(i), RKG), real(alpha, RKG), real(beta, RKG), getLogBeta(real(alpha, RKG), real(beta, RKG)), signed, infos)

170 if (any(infos /= 0)) error stop "Beta inversion failed."

171 betaInvRef = getBetaInv(betaInc(i), alpha, beta, signed = .true._LK)

172 write(fileUnit, "(*(g0,:,','))") betaInc(i), abs(betaInv - merge(1 + betaInvRef, betaInvRef, betaInvRef < 0))

173 end do

174 close(fileUnit)

175 end block

176

177 block

178 use pm_kind, only: RKG => RKH

179 character(*), parameter :: RKSTR = "RKH"

180 real(RKG) :: betaInv(max(size(alpha), size(beta)))

181 open(newunit = fileUnit, file = "setBetaInv."//RKSTR//".abserr.txt")

182 do i = 1, NP

183 call setBetaInv(betaInv, real(betaInc(i), RKG), real(alpha, RKG), real(beta, RKG), getLogBeta(real(alpha, RKG), real(beta, RKG)), signed, infos)

184 if (any(infos /= 0)) error stop "Beta inversion failed."

185 betaInvRef = getBetaInv(betaInc(i), alpha, beta, signed = .true._LK)

186 write(fileUnit, "(*(g0,:,','))") betaInc(i), abs(betaInv - merge(1 + betaInvRef, betaInvRef, betaInvRef < 0))

187 end do

188 close(fileUnit)

189 end block

190

191 end block

192

193end program example

pm_arrayResize::setResized
Allocate or resize (shrink or expand) an input allocatable scalar string or array of rank 1....
Definition: pm_arrayResize.F90:249

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_distUnif::getUnifRand
Generate and return a scalar or a contiguous array of rank 1 of length s1 of randomly uniformly distr...
Definition: pm_distUnif.F90:4150

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_mathBeta::getBetaInc
Generate and return the regularized Incomplete Beta Function as defined in the details section of pm...
Definition: pm_mathBeta.F90:279

pm_mathBeta::getBetaInv
Generate and return the regularized Inverse Incomplete Beta Function as defined in the details secti...
Definition: pm_mathBeta.F90:677

pm_mathBeta::getLogBeta
Generate and return the natural logarithm of the Beta Function as defined in the details section of ...
Definition: pm_mathBeta.F90:143

pm_arrayResize
This module contains procedures and generic interfaces for resizing allocatable arrays of various typ...
Definition: pm_arrayResize.F90:81

pm_arraySpace
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
Definition: pm_arraySpace.F90:33

pm_distUnif
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distUnif.F90:274

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::RK
integer, parameter RK
The default real kind in the ParaMonte library: real64 in Fortran, c_double in C-Fortran Interoperati...
Definition: pm_kind.F90:543

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

2signed

3F

4betaInc = 0.5_RKG

5alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)

6[alpha, beta]

7+3.3586226449696928, +1.6777439673037353

8call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)

9if (info /= 0) error stop 'Beta inversion failed.'

10betaInv

11+0.69034855656417282

12getBetaInc(betaInv, alpha, beta, signed)

13+0.50000000000000022

14

15

16signed

17F

18betaInc = 0.5_RKG

19alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)

20[alpha, beta]

21+9.5141715968095379, +5.0634513784404760

22call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)

23if (info /= 0) error stop 'Beta inversion failed.'

24betaInv

25+0.65980966876983149

26getBetaInc(betaInv, alpha, beta, signed)

27+0.50000000000000544

28

29

30signed

31F

32betaInc = 1._RKG - epsilon(1._RKG)

33alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)

34[alpha, beta]

35+0.67793192402639557, +0.60257550338669730

36call setBetaInv(betaInv, betaInc, alpha, beta, getLogBeta(alpha, beta), signed, info)

37if (info /= 0) error stop 'Beta inversion failed.'

38betaInv

39+1.0000000000000000

40getBetaInc(betaInv, alpha, beta, signed)

41+1.0000000000000000

42

43

44signed

45F

46betaIncs = [0._RKG, .5_RKG, 1._RKG]

47alpha = getUnifRand(0.1_RKG, 10._RKG); beta = getUnifRand(0.1_RKG, 10._RKG)

48[alpha, beta]

49+6.6260890908919174, +8.2940262014142299

50call setResized(betaInvs, size(betaIncs, 1, IK))

51call setBetaInv(betaInvs, betaIncs, alpha, beta, getLogBeta(alpha, beta), signed, infos)

52if (any(infos /= 0)) error stop 'Beta inversion failed.'

53betaInvs

54+0.0000000000000000, +0.44154198194786182, +1.0000000000000000

55getBetaInc(betaInvs, alpha, beta, signed)

56+0.0000000000000000, +0.49999999999999978, +1.0000000000000000

57

58

59signed

60F

61betaIncs = [0._RKG, .5_RKG, 1._RKG]

62alphas = [.1_RKG, 1._RKG, 10._RKG]

63beta = 3._RKG

64call setResized(betaInvs, size(betaIncs, 1, IK))

65call setBetaInv(betaInvs, betaIncs, alphas, beta, getLogBeta(alpha, beta), signed, infos)

66if (any(infos /= 0)) error stop 'Beta inversion failed.'

67betaInvs

68+0.0000000000000000, +0.22994604328204467E-2, +1.0000000000000000

69getBetaInc(betaInvs, alpha = [.1_RKG, 1._RKG, 10._RKG], beta = 3._RKG, signed = signed)

70+0.0000000000000000, +0.68825309020540924E-2, +1.0000000000000000

71

72

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

8import os

9

10fontsize = 17

11

12kind = "RK"

13label = [ r"$\alpha, \beta = .1, .1$"

14 , r"$\alpha, \beta = 10, .1$"

15 , r"$\alpha, \beta = 1., 1.$"

16 , r"$\alpha, \beta = .1, 10$"

17 , r"$\alpha, \beta = 10, 10$"

18 ]

19

20parent = os.path.basename(os.path.dirname(__file__))

21pattern = parent + "*.txt"

22fileList = glob.glob(pattern)

23for file in fileList:

24

25 df = pd.read_csv(file, delimiter = ",")

26

27 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)

28 ax = plt.subplot()

29

30 for i in range(1,len(df.values[0,:]+1)):

31

32 plt.plot( df.values[:, 0]

33 , df.values[:,i]

34 , linewidth = 2

35 )

36

37 plt.xticks(fontsize = fontsize - 2)

38 plt.yticks(fontsize = fontsize - 2)

39 if "abserr" in file:

40 nbit = file.split(".")[1][2:]

41 ax.set_xlabel("X : Regularized Incomplete Beta Function", fontsize = fontsize)

42 ax.set_ylabel("{}-bits Absolute Error: X - BetaInc(BetaInv(X))".format(nbit), fontsize = fontsize)

43 else:

44 ax.set_xlabel("x", fontsize = fontsize)

45 ax.set_ylabel("Regularized Inverse Beta Function", fontsize = fontsize)

46

47 ax.legend ( label

48 , fontsize = fontsize

49 #, loc = "center left"

50 #, bbox_to_anchor = (1, 0.5)

51 )

52

53 plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")

54 ax.tick_params(axis = "y", which = "minor")

55 ax.tick_params(axis = "x", which = "minor")

56 plt.tight_layout()

57

58 plt.savefig(file.replace(".txt",".png"))

Visualization of the example output ⛓

Test:: test_pm_mathBeta

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 826 of file pm_mathBeta.F90.

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

src/fortran/main/pm_mathBeta.F90