This is the indicator type for generating instances of objects that indicate the integration interval is open and, the integrand has an Integrable square-root type of Singularity at the finite Lower Bound of integration (LBIS). More...

Public Attributes
real	exponent = 0.5

Detailed Description

This is the indicator type for generating instances of objects that indicate the integration interval is open and, the integrand has an Integrable square-root type of Singularity at the finite Lower Bound of integration (LBIS).

This is an empty derived type that exists solely for generating unique objects that are distinguishable as input arguments to procedures under the generic interface getQuadRomb.

Possible calling interfaces ⛓

: use pm_quadRomb, only: ubis_type

type(ubis_type) :: UBIS

pm_quadRomb
This module contains classes and procedures to perform numerical integrations.
Definition: pm_quadRomb.F90:31

pm_quadRomb::ubis_type
This is the indicator type for generating instances of objects that indicate the integration interval...
Definition: pm_quadRomb.F90:300

See also: lbis_type
nexp_type
open_type
pexp_type
pwrl_type
ubis_type
getQuadRomb

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK, SP => RKS, DP => RKD, QP => RKH ! all real kinds are supported.

4 use pm_mathBeta, only: getBetaInc

5 use pm_distBeta, only: getBetaPDF

6 use pm_quadRomb, only: getQuadRomb, lbis_type, ubis_type

7 use pm_io, only: display_type

8

9 implicit none

10

11 integer(IK) :: neval

12

13 real(SP) :: quad_SP, quadref_SP, relerr_SP, alpha_SP, beta_SP

14 real(DP) :: quad_DP, quadref_DP, relerr_DP, alpha_DP, beta_DP

15 real(QP) :: quad_QP, quadref_QP, relerr_QP, alpha_QP, beta_QP

16

17 type(display_type) :: disp

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

19

20 call disp%skip()

21 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

22 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

23 call disp%show("! Compute the Cumulative Distribution Function (CDF) over an open interval of the Beta distribution with singular support bounds.")

24 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

25 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

26 call disp%skip()

27

28 call disp%skip()

29 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

30 call disp%show("! Compute the numerical integration with single precision.")

31 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

32 call disp%skip()

33

34 call disp%skip()

35 call disp%show("alpha_SP = .5_SP")

36 alpha_SP = .5_SP

37 call disp%show("beta_SP = .5_SP")

38 beta_SP = .5_SP

39 call disp%show("quadref_SP = getBetaInc(1._SP, alpha = alpha_SP, beta = beta_SP) - getBetaInc(0._SP, alpha = alpha_SP, beta = beta_SP)")

40 quadref_SP = getBetaInc(1._SP, alpha = alpha_SP, beta = beta_SP) - getBetaInc(0._SP, alpha = alpha_SP, beta = beta_SP)

41 call disp%show("quadref_SP")

42 call disp%show( quadref_SP )

43 call disp%show("quad_SP = getQuadRomb(getBetaPDF_SP, 0._SP, .5_SP, epsilon(1.) * 100, 4_IK, lbis_type(real(alpha_SP) - 1.)) + getQuadRomb(getBetaPDF_SP, .5_SP, 1._SP, epsilon(1._SP) * 100, 4_IK, ubis_type(real(beta_SP) - 1.))")

44 quad_SP = getQuadRomb(getBetaPDF_SP, 0._SP, .5_SP, epsilon(1.) * 100, 4_IK, lbis_type(real(alpha_SP) - 1.)) + getQuadRomb(getBetaPDF_SP, .5_SP, 1._SP, epsilon(1._SP) * 100, 4_IK, ubis_type(real(beta_SP) - 1.))

45 call disp%show("quad_SP ! integral")

46 call disp%show( quad_SP )

47 call disp%skip()

48

49 call disp%skip()

50 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

51 call disp%show("! Compute the numerical integration with double precision.")

52 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

53 call disp%skip()

54

55 call disp%skip()

56 call disp%show("alpha_DP = .5_DP")

57 alpha_DP = .5_DP

58 call disp%show("beta_DP = .5_DP")

59 beta_DP = .5_DP

60 call disp%show("quadref_DP = getBetaInc(1._DP, alpha = alpha_DP, beta = beta_DP) - getBetaInc(0._DP, alpha = alpha_DP, beta = beta_DP)")

61 quadref_DP = getBetaInc(1._DP, alpha = alpha_DP, beta = beta_DP) - getBetaInc(0._DP, alpha = alpha_DP, beta = beta_DP)

62 call disp%show("quadref_DP")

63 call disp%show( quadref_DP )

64 call disp%show("quad_DP = getQuadRomb(getBetaPDF_DP, 0._DP, .5_DP, epsilon(1._DP) * 100, 10_IK, lbis_type(real(alpha_DP) - 1.)) + getQuadRomb(getBetaPDF_DP, .5_DP, 1._DP, epsilon(1._DP) * 100, 10_IK, ubis_type(real(beta_DP) - 1.))")

65 quad_DP = getQuadRomb(getBetaPDF_DP, 0._DP, .5_DP, epsilon(1._DP) * 100, 10_IK, lbis_type(real(alpha_DP) - 1.)) + getQuadRomb(getBetaPDF_DP, .5_DP, 1._DP, epsilon(1._DP) * 100, 10_IK, ubis_type(real(beta_DP) - 1.))

66 call disp%show("quad_DP ! integral")

67 call disp%show( quad_DP )

68 call disp%skip()

69

70 call disp%skip()

71 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

72 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

73 call disp%show("! Compute the integral of 1/sqrt(abs(x)) = 4 over the interval [-1, 1] with alpha singularity at x = 0.")

74 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

75 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

76 call disp%skip()

77

78 call disp%skip()

79 call disp%show("quad_DP = getQuadRomb(getInvSqrtAbs_DP, -1._DP, 0._DP, epsilon(1._DP) * 100, 7_IK, ubis_type(-0.5)) + getQuadRomb(getInvSqrtAbs_DP, 0._DP, 1._DP, epsilon(1._DP) * 100, 10_IK, lbis_type(-0.5))")

80 quad_DP = getQuadRomb(getInvSqrtAbs_DP, -1._DP, 0._DP, epsilon(1._DP) * 100, 7_IK, ubis_type(-0.5)) + getQuadRomb(getInvSqrtAbs_DP, 0._DP, 1._DP, epsilon(1._DP) * 100, 10_IK, lbis_type(-0.5))

81 call disp%show("quad_DP ! integral")

82 call disp%show( quad_DP )

83 call disp%skip()

84

85 call disp%skip()

86 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

87 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

88 call disp%show("! Compute the integral of 1/sqrt(abs(x)) = 4 over the interval [-1, 1] with alpha singularity at x = 0.")

89 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

90 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

91 call disp%skip()

92

93 call disp%skip()

94 call disp%show("quad_QP = getQuadRomb(getInvSqrtAbs_DP, -1._QP, 0._QP, epsilon(1._QP) * 100, 7_IK, ubis_type(-0.5)) + getQuadRomb(getInvSqrtAbs_QP, 0._QP, 1._QP, epsilon(1._QP) * 100, 10_IK, lbis_type(-0.5))")

95 quad_QP = getQuadRomb(getInvSqrtAbs_QP, -1._QP, 0._QP, epsilon(1._QP) * 100, 7_IK, ubis_type(-0.5)) + getQuadRomb(getInvSqrtAbs_QP, 0._QP, 1._QP, epsilon(1._QP) * 100, 10_IK, lbis_type(-0.5))

96 call disp%show("quad_QP ! integral")

97 call disp%show( quad_QP )

98 call disp%skip()

99

100contains

101

102 function getBetaPDF_SP(x) result(betaPDF)

103 real , intent(in) :: x

104 real :: betaPDF

105 betaPDF = getBetaPDF(x, alpha = alpha_SP, beta = beta_SP)

106 end function

107

108 function getBetaPDF_DP(x) result(betaPDF)

109 real(DP), intent(in) :: x

110 real(DP) :: betaPDF

111 betaPDF = getBetaPDF(x, alpha = alpha_DP, beta = beta_DP)

112 end function

113

114 function getBetaPDF_QP(x) result(betaPDF)

115 real(QP), intent(in) :: x

116 real(QP) :: betaPDF

117 betaPDF = getBetaPDF(x, alpha = alpha_QP, beta = beta_QP)

118 end function

119

120 pure function getInvSqrtAbs_DP(x) result(invSqrtAbs)

121 real(DP), intent(in) :: x

122 real(DP) :: invSqrtAbs

123 invSqrtAbs = 1._DP / sqrt(abs(x))

124 end function

125

126 pure function getInvSqrtAbs_QP(x) result(invSqrtAbs)

127 real(QP), intent(in) :: x

128 real(QP) :: invSqrtAbs

129 invSqrtAbs = 1._QP / sqrt(abs(x))

130 end function

131

132end program example

pm_distBeta::getBetaPDF
Generate and return the Probability Density Function (PDF) of the Beta distribution for an input x wi...
Definition: pm_distBeta.F90:227

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.F90:280

pm_quadRomb::getQuadRomb
Generate and return the integral of the input function getFunc() in the closed range [lb,...
Definition: pm_quadRomb.F90:396

pm_distBeta
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distBeta.F90:99

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::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_mathBeta
This module contains classes and procedures for computing the mathematical Beta Function and its inve...
Definition: pm_mathBeta.F90:84

pm_io::display_type
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

pm_quadRomb::lbis_type
This is the indicator type for generating instances of objects that indicate the integration interval...
Definition: pm_quadRomb.F90:256

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 Cumulative Distribution Function (CDF) over an open interval of the Beta distribution with singular support bounds.

5!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

6!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

7

8

9!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

10! Compute the numerical integration with single precision.

11!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

12

13

14alpha_SP = .5_SP

15beta_SP = .5_SP

16quadref_SP = getBetaInc(1._SP, alpha = alpha_SP, beta = beta_SP) - getBetaInc(0._SP, alpha = alpha_SP, beta = beta_SP)

17quadref_SP

18+1.00000000

19quad_SP = getQuadRomb(getBetaPDF_SP, 0._SP, .5_SP, epsilon(1.) * 100, 4_IK, lbis_type(real(alpha_SP) - 1.)) + getQuadRomb(getBetaPDF_SP, .5_SP, 1._SP, epsilon(1._SP) * 100, 4_IK, ubis_type(real(beta_SP) - 1.))

20quad_SP ! integral

21+0.999999046

22

23

24!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

25! Compute the numerical integration with double precision.

26!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

27

28

29alpha_DP = .5_DP

30beta_DP = .5_DP

31quadref_DP = getBetaInc(1._DP, alpha = alpha_DP, beta = beta_DP) - getBetaInc(0._DP, alpha = alpha_DP, beta = beta_DP)

32quadref_DP

33+1.0000000000000000

34quad_DP = getQuadRomb(getBetaPDF_DP, 0._DP, .5_DP, epsilon(1._DP) * 100, 10_IK, lbis_type(real(alpha_DP) - 1.)) + getQuadRomb(getBetaPDF_DP, .5_DP, 1._DP, epsilon(1._DP) * 100, 10_IK, ubis_type(real(beta_DP) - 1.))

35quad_DP ! integral

36+1.0000000000022546

37

38

39!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

40!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

41! Compute the integral of 1/sqrt(abs(x)) = 4 over the interval [-1, 1] with alpha singularity at x = 0.

42!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

43!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

44

45

46quad_DP = getQuadRomb(getInvSqrtAbs_DP, -1._DP, 0._DP, epsilon(1._DP) * 100, 7_IK, ubis_type(-0.5)) + getQuadRomb(getInvSqrtAbs_DP, 0._DP, 1._DP, epsilon(1._DP) * 100, 10_IK, lbis_type(-0.5))

47quad_DP ! integral

48+4.0000000000000000

49

50

51!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

52!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

53! Compute the integral of 1/sqrt(abs(x)) = 4 over the interval [-1, 1] with alpha singularity at x = 0.

54!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

55!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

56

57

58quad_QP = getQuadRomb(getInvSqrtAbs_DP, -1._QP, 0._QP, epsilon(1._QP) * 100, 7_IK, ubis_type(-0.5)) + getQuadRomb(getInvSqrtAbs_QP, 0._QP, 1._QP, epsilon(1._QP) * 100, 10_IK, lbis_type(-0.5))

59quad_QP ! integral

60+4.00000000000000000000000000000000000

61

62

Test:: test_pm_quadRomb

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, September 1, 2017, 12:00 AM, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin

Definition at line 300 of file pm_quadRomb.F90.

Member Data Documentation

◆ exponent

real pm_quadRomb::ubis_type::exponent = 0.5

Definition at line 301 of file pm_quadRomb.F90.

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

src/fortran/main/pm_quadRomb.F90

Public Attributes

Detailed Description

Member Data Documentation

◆ exponent