Generate and return the integral of the input function getFunc() in the closed range [lb, ub] using the Adaptive Romberg extrapolation method. More...

Detailed Description

Generate and return the integral of the input function getFunc() in the closed range [lb, ub] using the Adaptive Romberg extrapolation method.

This Romberg integration method is quite powerful for sufficiently smooth (e.g., analytic) integrands getFunc(), integrated over bounded intervals which contain no singularities, and where the end points are also nonsingular.

Parameters

	getFunc	: The input function to be integrated (i.e., the integrand). On entry, it must take an input scalar of the same type and kind as `quadRomb`. On exit, it must generate an input scalar of the same type and kind as `quadRomb`, representing the corresponding function value. The following illustrates the general interface of `getFunc`: function getFunc(x) result(func) use pm_kind, only: RK => RKG real(RK) , intent(in) :: x real(RK) :: func end function 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 where `RKG` refers to any desired `real` kind supported by the processor.
[in]	lb	: The input scalar of the same type and kind as the output `quadRomb`, containing the lower bound of the integration.
[in]	ub	: The input scalar of the same type and kind as the output `quadRomb`, containing the upper bound of the integration.
[in]	tol	: The input scalar of the same type and kind as the output `quadRomb`, containing the relative error the integration. The algorithm converges if \(\ms{relerr} ~(~\equiv \|\Delta\ms{quadRomb}\|~) ~\leq~ \ms{tol} \times \| \ms{quadRomb} \|\). Note that `tol > epsilon(0._RKG)` must hold at all times for integration to converge. Here `RKG` is the desired `real` kind of the output. Ideally, set `tol` to a value such that `tol < epsilon(0._RKG) * 100` holds to ensure convergence.
[in]	nref	: The input scalar `integer` of default kind IK, representing the number of refinements to be used in the Romberg method. Think of `nref` as the maximum possible degree of the polynomial extrapolation used for approximating the integral at any stage. The smaller values of `nref` can delay an accurate estimation of the integral via the Romberg method. The larger values of `nref` can delay the first estimation of the integral by requiring more function evaluations. The computational precision of the `real` kind used in this procedure imposes an upper limit on the value of `nref`. The maximum value for `nref` is roughly equal to `int(log(epsilon(1._RKG)) / log(0.25))` with `RKG` representing the `real` kind used for the integration. The maximum `nref` for `real32`, `real64`, `real128` are respectively `12`, `26`, `56`. If the specified `nref` is larger than the maximum possible value, the integration will fail to converge. The number `nref = 2` corresponds to the famous Simpson integration rule. A number between 4-6 is frequently a reasonable choice.
[out]	relerr	: The output scalar of the same type and kind as the output `quadRomb` containing the final estimated relative error in the result. By definition, this is always a positive value if the integration converges. Specify the `relerr` optional output argument to monitor convergence. If `relerr < 0.`, then the integration has failed to converge. (optional. If missing and the integration fails to converge, the program will halt by calling `error stop`.)
[out]	neval	: The output scalar `integer` of default kind IK, representing the number of function evaluations made within the integrator. (optional. It can be present if and only if `relerr` argument is also present.)

Returns: quadRomb : The output scalar real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the integration result.

Possible calling interfaces ⛓

: use pm_quadRomb, only: getQuadRomb

quadRomb = getQuadRomb(getFunc, lb, ub, tol, nref)

quadRomb = getQuadRomb(getFunc, lb, ub, tol, nref, relerr)

quadRomb = getQuadRomb(getFunc, lb, ub, tol, nref, relerr, neval)

quadRomb = getQuadRomb(getFunc, lb, ub, tol, nref, interval)

quadRomb = getQuadRomb(getFunc, lb, ub, tol, nref, interval, relerr)

quadRomb = getQuadRomb(getFunc, lb, ub, tol, nref, interval, relerr, neval)

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

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

Warning

The procedures of this generic interface will behave differently if the integration fails to converge:

If the optional relerr output argument is missing, the program will halt by calling error stop upon integration convergence failure.
If the optional relerr output argument is present, the program will return relerr < 0. to indicate integration convergence failure.

See also: getQuadErr

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK, QP => RKH

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, parameter :: SP = kind(0.e0), DP = kind(0.d0)

12

13 integer(IK) :: neval

14

15 real(SP) :: quad_sp, quadref_sp, relerr_sp, alpha_sp, beta_sp

16 real(DP) :: quad_dp, quadref_dp, relerr_dp, alpha_dp, beta_dp

17 real(QP) :: quad_qp, quadref_qp, relerr_qp, alpha_qp, beta_qp

18

19 type(display_type) :: disp

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

21

22 call disp%skip()

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

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

25 call disp%show("! Compute the Cumulative Distribution Function (CDF) over a closed interval of the Beta distribution by numerical integration.")

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

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

28 call disp%skip()

29

30 call disp%skip()

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

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

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

34 call disp%skip()

35

36 call disp%skip()

37 call disp%show("alpha_sp = 2.")

38 alpha_sp = 2.

39 call disp%show("beta_sp = 5.")

40 beta_sp = 5.

41 call disp%show("quadref_sp = getBetaInc(1., alpha = alpha_sp, beta = beta_sp) - getBetaInc(0., alpha = alpha_sp, beta = beta_sp)")

42 quadref_sp = getBetaInc(1., alpha = alpha_sp, beta = beta_sp) - getBetaInc(0., alpha = alpha_sp, beta = beta_sp)

43 call disp%show("quadref_sp")

44 call disp%show( quadref_sp )

45 call disp%show("quad_sp = getQuadRomb(getFunc = getBetaPDF_SP, lb = 0., ub = 1., tol = epsilon(1.) * 100, nref = 4_IK, relerr = relerr_sp, neval = neval)")

46 quad_sp = getQuadRomb(getFunc = getBetaPDF_SP, lb = 0., ub = 1., tol = epsilon(1.) * 100, nref = 4_IK, relerr = relerr_sp, neval = neval)

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

48 if (relerr_sp < 0.) error stop 'Integration failed to converge.'

49 call disp%show("relerr_sp ! < 0. if integration fails.")

50 call disp%show( relerr_sp )

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

52 call disp%show( neval )

53 call disp%show("quad_sp ! integral")

54 call disp%show( quad_sp )

55 call disp%skip()

56

57 call disp%skip()

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

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

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

61 call disp%skip()

62

63 call disp%skip()

64 call disp%show("alpha_dp = 2.d0")

65 alpha_dp = 2.d0

66 call disp%show("beta_dp = 5.d0")

67 beta_dp = 5.d0

68 call disp%show("quadref_dp = getBetaInc(1.d0, alpha = alpha_dp, beta = beta_dp) - getBetaInc(0.d0, alpha = alpha_dp, beta = beta_dp)")

69 quadref_dp = getBetaInc(1.d0, alpha = alpha_dp, beta = beta_dp) - getBetaInc(0.d0, alpha = alpha_dp, beta = beta_dp)

70 call disp%show("quadref_dp")

71 call disp%show( quadref_dp )

72 call disp%show("quad_dp = getQuadRomb(getFunc = getBetaPDF_DP, lb = 0.d0, ub = 1.d0, tol = epsilon(1.d0) * 100, nref = 4_IK, relerr = relerr_dp, neval = neval)")

73 quad_dp = getQuadRomb(getFunc = getBetaPDF_DP, lb = 0.d0, ub = 1.d0, tol = epsilon(1.d0) * 100, nref = 4_IK, relerr = relerr_dp, neval = neval)

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

75 if (relerr_dp < 0.) error stop 'Integration failed to converge.'

76 call disp%show("relerr_dp ! < 0. if integration fails.")

77 call disp%show( relerr_dp )

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

79 call disp%show( neval )

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

81 call disp%show( quad_dp )

82 call disp%skip()

83

84 call disp%skip()

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

86 call disp%show("! Compute the numerical integration with quadro precision.")

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

88 call disp%skip()

89

90 call disp%skip()

91 call disp%show("alpha_qp = 2._qp")

92 alpha_qp = 2._qp

93 call disp%show("beta_qp = 5._qp")

94 beta_qp = 5._qp

95 call disp%show("quadref_qp = getBetaInc(1._QP, alpha = alpha_qp, beta = beta_qp) - getBetaInc(0._QP, alpha = alpha_qp, beta = beta_qp)")

96 quadref_qp = getBetaInc(1._QP, alpha = alpha_qp, beta = beta_qp) - getBetaInc(0._QP, alpha = alpha_qp, beta = beta_qp)

97 call disp%show("quadref_qp")

98 call disp%show( quadref_qp )

99 call disp%show("quad_qp = getQuadRomb(getFunc = getBetaPDF_QP, lb = 0._QP, ub = 1._QP, tol = epsilon(1._QP) * 100, nref = 4_IK, relerr = relerr_qp, neval = neval)")

100 quad_qp = getQuadRomb(getFunc = getBetaPDF_QP, lb = 0._QP, ub = 1._QP, tol = epsilon(1._QP) * 100, nref = 4_IK, relerr = relerr_qp, neval = neval)

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

102 if (relerr_qp < 0.) error stop 'Integration failed to converge.'

103 call disp%show("relerr_qp ! < 0. if integration fails.")

104 call disp%show( relerr_qp )

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

106 call disp%show( neval )

107 call disp%show("quad_qp ! integral")

108 call disp%show( quad_qp )

109 call disp%skip()

110

111contains

112

113 function getBetaPDF_SP(x) result(betaPDF)

114 real , intent(in) :: x

115 real :: betaPDF

116 betaPDF = getBetaPDF(x, alpha = alpha_sp, beta = beta_sp)

117 end function

118

119 function getBetaPDF_DP(x) result(betaPDF)

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

121 real(DP) :: betaPDF

122 betaPDF = getBetaPDF(x, alpha = alpha_dp, beta = beta_dp)

123 end function

124

125 function getBetaPDF_QP(x) result(betaPDF)

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

127 real(QP) :: betaPDF

128 betaPDF = getBetaPDF(x, alpha = alpha_qp, beta = beta_qp)

129 end function

130

131end 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_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::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::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_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

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

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 a closed interval of the Beta distribution by numerical integration.

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

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

7

8

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

10! Compute the numerical integration with single precision.

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

12

13

14alpha_sp = 2.

15beta_sp = 5.

16quadref_sp = getBetaInc(1., alpha = alpha_sp, beta = beta_sp) - getBetaInc(0., alpha = alpha_sp, beta = beta_sp)

17quadref_sp

18+1.00000000

19quad_sp = getQuadRomb(getFunc = getBetaPDF_SP, lb = 0., ub = 1., tol = epsilon(1.) * 100, nref = 4_IK, relerr = relerr_sp, neval = neval)

20if (relerr_sp < 0.) error stop 'Integration failed to converge.'

21relerr_sp ! < 0. if integration fails.

22+0.946105527E-9

23neval ! # calls to the integrand.

24+9

25quad_sp ! integral

26+1.00000012

27

28

29!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

30! Compute the numerical integration with double precision.

31!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

32

33

34alpha_dp = 2.d0

35beta_dp = 5.d0

36quadref_dp = getBetaInc(1.d0, alpha = alpha_dp, beta = beta_dp) - getBetaInc(0.d0, alpha = alpha_dp, beta = beta_dp)

37quadref_dp

38+1.0000000000000000

39quad_dp = getQuadRomb(getFunc = getBetaPDF_DP, lb = 0.d0, ub = 1.d0, tol = epsilon(1.d0) * 100, nref = 4_IK, relerr = relerr_dp, neval = neval)

40if (relerr_dp < 0.) error stop 'Integration failed to converge.'

41relerr_dp ! < 0. if integration fails.

42+0.11564823173178713E-17

43neval ! # calls to the integrand.

44+9

45quad_dp ! integral

46+1.0000000000000002

47

48

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

50! Compute the numerical integration with quadro precision.

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

52

53

54alpha_qp = 2._qp

55beta_qp = 5._qp

56quadref_qp = getBetaInc(1._QP, alpha = alpha_qp, beta = beta_qp) - getBetaInc(0._QP, alpha = alpha_qp, beta = beta_qp)

57quadref_qp

58+1.00000000000000000000000000000000000

59quad_qp = getQuadRomb(getFunc = getBetaPDF_QP, lb = 0._QP, ub = 1._QP, tol = epsilon(1._QP) * 100, nref = 4_IK, relerr = relerr_qp, neval = neval)

60if (relerr_qp < 0.) error stop 'Integration failed to converge.'

61relerr_qp ! < 0. if integration fails.

62+0.152851582887875861353649043062295812E-35

63neval ! # calls to the integrand.

64+9

65quad_qp ! integral

66+1.00000000000000000000000000000000019

67

68

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 Joshua Alexander Osborne, May 28, 2020, 8:58 PM, Arlington, TX

Definition at line 396 of file pm_quadRomb.F90.

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

src/fortran/main/pm_quadRomb.F90