This is the indicator type for generating instances of objects that indicate the integration interval is open \((a, b)\) and, the intervals should be spaced assuming an integrand that behaves like,. More...

Detailed Description

This is the indicator type for generating instances of objects that indicate the integration interval is open \((a, b)\) and, the intervals should be spaced assuming an integrand that behaves like,.

a decreasing Power-Law (PWL) on a positive support \((a > 0, b > 0)\), such that the upper limit of integration is allowed to be \(b = +\infty\), or
an increasing Power-Law (PWL) on a negative support \((a < 0, b < 0)\), such that the lower limit of integration is allowed to be \(a = -\infty\).

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: pwrl_type

type(pwrl_type) :: PWRL

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

pm_quadRomb::pwrl_type
This is the indicator type for generating instances of objects that indicate the integration interval...
Definition: pm_quadRomb.F90:125

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_distPareto, only: getParetoLogCDF

5 use pm_distPareto, only: getParetoLogPDF

6 use pm_distNorm, only: getNormLogPDF

7 use pm_quadRomb, only: getQuadRomb, pwrl_type

8 use pm_io, only: display_type

9

10 implicit none

11

12 integer(IK) :: neval

13

14 real(SP) :: quad_SP, quadref_SP, relerr_SP, avg_SP, std_SP, logMinX_SP, alpha_SP

15 real(DP) :: quad_DP, quadref_DP, relerr_DP, avg_DP, std_DP, logMinX_DP, alpha_DP

16 real(QP) :: quad_QP, quadref_QP, relerr_QP, avg_QP, std_QP, logMinX_QP, alpha_QP

17

18 type(display_type) :: disp

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

20

21 call disp%skip()

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

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

24 call disp%show("! Compute the Cumulative Distribution Function (CDF) over the semi-infinite interval of Pareto distribution.")

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

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

27 call disp%skip()

28

29 call disp%skip()

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

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

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

33 call disp%skip()

34

35 call disp%skip()

36 call disp%show("logMinX_SP = 3._SP; alpha_SP = -1._SP")

37 logMinX_SP = 3._SP; alpha_SP = -1._SP

38 call disp%show("quadref_SP = getParetoLogCDF(logx = log(huge(0._SP)), logMinX = logMinX_SP, alpha = alpha_SP)")

39 quadref_SP = getParetoLogCDF(logx = log(huge(0._SP)), logMinX = logMinX_SP, alpha = alpha_SP)

40 call disp%show("quadref_SP")

41 call disp%show( quadref_SP )

42 call disp%show("quad_SP = getQuadRomb(getFunc = getParetoPDF_SP, lb = logMinX_SP, ub = huge(0._SP), tol = epsilon(1._SP) * 100, nref = 4_IK, interval = pwrl_type(), relerr = relerr_SP, neval = neval)")

43 quad_SP = getQuadRomb(getFunc = getParetoPDF_SP, lb = logMinX_SP, ub = huge(0._SP), tol = epsilon(1._SP) * 100, nref = 4_IK, interval = pwrl_type(), relerr = relerr_SP, neval = neval)

44 call disp%show("if (relerr_SP < 0._SP) error stop 'Integration failed to converge.'")

45 if (relerr_SP < 0._SP) error stop 'Integration failed to converge.'

46 call disp%show("relerr_SP ! < 0. if integration fails.")

47 call disp%show( relerr_SP )

48 call disp%show("neval ! # calls to the integrand.")

49 call disp%show( neval )

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

51 call disp%show( quad_SP )

52 call disp%skip()

53

54 call disp%skip()

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

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

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

58 call disp%skip()

59

60 call disp%skip()

61 call disp%show("logMinX_DP = 3._DP; alpha_DP = -1._DP")

62 logMinX_DP = 3._DP; alpha_DP = -1._DP

63 call disp%show("quadref_DP = getParetoLogCDF(logx = log(huge(0._DP)), logMinX = logMinX_DP, alpha = alpha_DP)")

64 quadref_DP = getParetoLogCDF(logx = log(huge(0._DP)), logMinX = logMinX_DP, alpha = alpha_DP)

65 call disp%show("quadref_DP")

66 call disp%show( quadref_DP )

67 call disp%show("quad_DP = getQuadRomb(getFunc = getParetoPDF_DP, lb = logMinX_DP, ub = huge(0._DP), tol = epsilon(1._DP) * 100, nref = 7_IK, interval = pwrl_type(), relerr = relerr_DP, neval = neval)")

68 quad_DP = getQuadRomb(getFunc = getParetoPDF_DP, lb = logMinX_DP, ub = huge(0._DP), tol = epsilon(1._DP) * 100, nref = 7_IK, interval = pwrl_type(), relerr = relerr_DP, neval = neval)

69 call disp%show("if (relerr_DP < 0.) error stop 'Integration failed to converge.'")

70 if (relerr_DP < 0.) error stop 'Integration failed to converge.'

71 call disp%show("relerr_DP ! < 0. if integration fails.")

72 call disp%show( relerr_DP )

73 call disp%show("neval ! # calls to the integrand.")

74 call disp%show( neval )

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

76 call disp%show( quad_DP )

77 call disp%skip()

78

79 call disp%skip()

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

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

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

83 call disp%skip()

84

85 call disp%skip()

86 call disp%show("logMinX_QP = 1._QP; alpha_QP = -3._QP")

87 logMinX_QP = 1._QP; alpha_QP = -3._QP

88 call disp%show("quadref_QP = getParetoLogCDF(logx = log(huge(0._QP)), logMinX = logMinX_QP, alpha = alpha_QP)")

89 quadref_QP = getParetoLogCDF(logx = log(huge(0._QP)), logMinX = logMinX_QP, alpha = alpha_QP)

90 call disp%show("quadref_QP")

91 call disp%show( quadref_QP )

92 call disp%show("quad_QP = getQuadRomb(getFunc = getParetoPDF_QP, lb = logMinX_QP, ub = huge(0._QP), tol = epsilon(1._QP)*100, nref = 10_IK, interval = pwrl_type(), relerr = relerr_QP, neval = neval)")

93 quad_QP = getQuadRomb(getFunc = getParetoPDF_QP, lb = logMinX_QP, ub = huge(0._QP), tol = epsilon(1._QP)*100, nref = 10_IK, interval = pwrl_type(), relerr = relerr_QP, neval = neval)

94 call disp%show("if (relerr_QP < 0.) error stop 'Integration failed to converge.'")

95 if (relerr_QP < 0.) error stop 'Integration failed to converge.'

96 call disp%show("relerr_QP ! < 0. if integration fails.")

97 call disp%show( relerr_QP )

98 call disp%show("neval ! # calls to the integrand.")

99 call disp%show( neval )

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

101 call disp%show( quad_QP )

102 call disp%skip()

103

104contains

105

106 function getParetoPDF_SP(x) result(expPDF)

107 real(SP), intent(in) :: x

108 real(SP) :: expPDF

109 expPDF = exp(getParetoLogPDF(x, alpha = alpha_SP, logMinX = logMinX_SP))

110 end function

111

112 function getParetoPDF_DP(x) result(expPDF)

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

114 real(DP) :: expPDF

115 expPDF = exp(getParetoLogPDF(x, alpha = alpha_DP, logMinX = logMinX_DP))

116 end function

117

118 function getParetoPDF_QP(x) result(expPDF)

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

120 real(QP) :: expPDF

121 expPDF = exp(getParetoLogPDF(x, alpha = alpha_QP, logMinX = logMinX_QP))

122 end function

123

124 function getNormPDF_SP(x) result(expPDF)

125 real(SP), intent(in) :: x

126 real(SP) :: expPDF

127 expPDF = exp(getNormLogPDF(x, mu = logMinX_SP))

128 end function

129

130 function getNormPDF_DP(x) result(expPDF)

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

132 real(DP) :: expPDF

133 expPDF = exp(getNormLogPDF(x, mu = logMinX_DP))

134 end function

135

136 function getNormPDF_QP(x) result(expPDF)

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

138 real(QP) :: expPDF

139 expPDF = exp(getNormLogPDF(x, mu = logMinX_QP))

140 end function

141

142end program example

pm_distNorm::getNormLogPDF
Generate the natural logarithm of probability density function (PDF) of the univariate Normal distrib...
Definition: pm_distNorm.F90:294

pm_distPareto::getParetoLogCDF
Generate and return the natural logarithm of the Cumulative Distribution Function (CDF) of the (Trunc...
Definition: pm_distPareto.F90:979

pm_distPareto::getParetoLogPDF
Generate and return the natural logarithm of the Probability Density Function (PDF) of the (Truncated...
Definition: pm_distPareto.F90:458

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

pm_distNorm
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distNorm.F90:179

pm_distPareto
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distPareto.F90:141

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_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

2!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

3!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

4! Compute the Cumulative Distribution Function (CDF) over the semi-infinite interval of Pareto distribution.

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

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

7

8

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

10! Compute the numerical integration with single precision.

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

12

13

14logMinX_SP = 3._SP; alpha_SP = -1._SP

15quadref_SP = getParetoLogCDF(logx = log(huge(0._SP)), logMinX = logMinX_SP, alpha = alpha_SP)

16quadref_SP

17+0.00000000

18quad_SP = getQuadRomb(getFunc = getParetoPDF_SP, lb = logMinX_SP, ub = huge(0._SP), tol = epsilon(1._SP) * 100, nref = 4_IK, interval = pwrl_type(), relerr = relerr_SP, neval = neval)

19if (relerr_SP < 0._SP) error stop 'Integration failed to converge.'

20relerr_SP ! < 0. if integration fails.

21+0.990457671E-8

22neval ! # calls to the integrand.

23+27

24quad_SP ! integral

25+0.248935577E-1

26

27

28!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

29! Compute the numerical integration with double precision.

30!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

31

32

33logMinX_DP = 3._DP; alpha_DP = -1._DP

34quadref_DP = getParetoLogCDF(logx = log(huge(0._DP)), logMinX = logMinX_DP, alpha = alpha_DP)

35quadref_DP

36+0.0000000000000000

37quad_DP = getQuadRomb(getFunc = getParetoPDF_DP, lb = logMinX_DP, ub = huge(0._DP), tol = epsilon(1._DP) * 100, nref = 7_IK, interval = pwrl_type(), relerr = relerr_DP, neval = neval)

38if (relerr_DP < 0.) error stop 'Integration failed to converge.'

39relerr_DP ! < 0. if integration fails.

40+0.15800940639871217E-19

41neval ! # calls to the integrand.

42+729

43quad_DP ! integral

44+0.24893534183931615E-1

45

46

47!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

48! Compute the numerical integration with double precision.

49!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

50

51

52logMinX_QP = 1._QP; alpha_QP = -3._QP

53quadref_QP = getParetoLogCDF(logx = log(huge(0._QP)), logMinX = logMinX_QP, alpha = alpha_QP)

54quadref_QP

55+0.00000000000000000000000000000000000

56quad_QP = getQuadRomb(getFunc = getParetoPDF_QP, lb = logMinX_QP, ub = huge(0._QP), tol = epsilon(1._QP)*100, nref = 10_IK, interval = pwrl_type(), relerr = relerr_QP, neval = neval)

57if (relerr_QP < 0.) error stop 'Integration failed to converge.'

58relerr_QP ! < 0. if integration fails.

59+0.180961348322211011195384953574198665E-35

60neval ! # calls to the integrand.

61+19683

62quad_QP ! integral

63+0.275909580878581741196642827620996662

64

65

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 125 of file pm_quadRomb.F90.

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

src/fortran/main/pm_quadRomb.F90