pm_quadPack::isFailedQuad Interface Reference

Compute the 1D integral of the input scalar (potentially singular) integrand getFunc on a finite or semi/fully-infinite interval (a, b) and estimate its absolute error via the requested adaptive global quadrature rule. More...

Detailed Description

Compute the 1D integral of the input scalar (potentially singular) integrand getFunc on a finite or semi/fully-infinite interval (a, b) and estimate its absolute error via the requested adaptive global quadrature rule.

This interface provides a simple nimble powerful wrapper for the higher-performance but lower-level getQuadErr interface, much like the \(\ms{integral}\) routine of MATLAB.
The algorithm uses an Adaptive Global Quadrature with Gauss-Kronrod 10-21 quadrature rules combined with the Epsilon extrapolation method of Wynn (1961) to evaluate the integral.
See the documentation of getQuadErr for further details on the integration methodologies and implementations.

Parameters

	getFunc	: See the description of the corresponding argument in the documentation of getQuadErr.
[in]	lb	: The input scalar of type real of the same kind as integral, representing the lower limit of integration. Set lb = -huge(lb) or to the IEEE-compliant negative infinity (lb =getInfNeg(lb)) to imply \(-\infty\) as the lower bound of integration.
[in]	ub	: The input scalar of type real of the same kind as integral, representing the upper limit of integration. Set ub = huge(ub) or to the IEEE-compliant positive infinity (ub =getInfPos(lb)) to imply \(+\infty\) as the upper bound of integration.
[out]	integral	: See the description of the corresponding argument in the documentation of getQuadErr.
[in]	abstol	: See the description of the corresponding argument in the documentation of getQuadErr. (optional, default = 0)
[in]	reltol	: See the description of the corresponding argument in the documentation of getQuadErr. (optional, default = epsilon(real(0, kind(integral)))**(2./3.))
[in]	help	: See the description of the corresponding argument in the documentation of getQuadErr. (optional. If missing, the procedure attempts to compute the integral without explicit assumptions about the singularities or discontinuities.)
[out]	abserr	: See the description of the corresponding argument in the documentation of getQuadErr. (optional. If missing, the estimated error of the integral will not be returned.)
[out]	neval	: See the description of the corresponding argument in the documentation of getQuadErr. (optional. If missing, the number of function evaluations will not be returned.)
[out]	nint	: See the description of the corresponding argument in the documentation of getQuadErr. (optional. If missing, the number of subintervals formed will not be returned.)
[out]	msg	: The output scalar argument of type character of default kind SK of arbitrary length type parameter that is set to a diagnostic message if the integration fails to converge. A length type parameter of 127 is sufficient to capture all error messages. If msg has shorter length parameter, the output message will be trimmed from the end, otherwise padded with blanks as necessary. (optional. If missing, no diagnostic message will be returned.)

Returns

failed : The output scalar of type logical of default kind LK, that is set to .true. if and only if the integration fails to converge within the requested tolerances.
Otherwise, it is set .false. if the integration succeeds with no errors.
See the description of the output argument err of getQuadErr for information on the kinds of integration failures that can happen.

Possible calling interfaces ⛓

use pm_quadPack, only: isFailedQuad
use pm_kind, only: LK
logical(LK) :: failed
 
failed = isFailedQuad(getFunc, lb, ub       , integral, abserr = abserr, abstol = abstol, reltol = reltol, neval = neval, nint = nint, msg = msg)
failed = isFailedQuad(getFunc, lb, ub, help , integral, abserr = abserr, abstol = abstol, reltol = reltol, neval = neval, nint = nint, msg = msg)

Warning

All conditions in the warning section of the documentation of getQuadErr also apply to this interface.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

If the target function contains points of difficulties, singularities, or discontinuities, user must ensure the abscissas of the specified Gauss-Kronrod rule do not match such points.
Particularly, computing an integrand at its singularities can lead to undefined values that can lead to unexpected segmentation fault or propagation of NaN values within the computational flow or other strange errors that can be extremely difficult to debug.
A simple check can be added within the target integrand implementations to ensure no such difficulty point matches an input value at which the function must be evaluated.
Alternatively, one should consider using the adaptive integration routines isFailedQuad or getQuadErr while setting their input help arguments to the points of difficulties of the integrand.

Remarks

The procedures under discussion are impure.

See also

getQuadGK
getQuadErr
getQuadRomb
isFailedQuad

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK, RKH, RKG => RKH ! All other real kinds are also supported.
    use pm_quadTest, only: int1_type
    use pm_quadTest, only: int2_type
    use pm_quadTest, only: int3_type
    use pm_quadTest, only: int4_type
    use pm_quadTest, only: int5_type
    use pm_quadTest, only: int6_type
    use pm_quadTest, only: int7_type
    use pm_quadTest, only: int8_type
    use pm_quadTest, only: int9_type
    use pm_quadTest, only: intGamUpp_type
    use pm_quadTest, only: intSinCos_type
    use pm_quadTest, only: intNormPDF_type
    use pm_quadTest, only: intDoncker1_type
    use pm_quadTest, only: intDoncker2_type
    use pm_quadTest, only: intLogNormPDF_type
    use pm_quadTest, only: intPentaGammaInf_type
    use pm_quadTest, only: intGenExpGammaPDF_type
    use pm_quadTest, only: test_isFailedQuad
    use pm_quadTest, only: intCauchy1_type
    use pm_quadTest, only: intCauchy2_type
    use pm_io, only: display_type
 
    implicit none
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compare integration via the Adaptive Global Gauss-Kronrod Quadrature method with (QAG) and without (QAGS) extrapolation acceleration.")
    call disp%show("! Note the significant efficiency improvement in the integration via QAGS when the integrand has integrable singularity at some points.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call test_isFailedQuad(disp, int1_type(-4._RKH, +4._RKH))
    call test_isFailedQuad(disp, int2_type(+2._RKH, +3._RKH))
    call test_isFailedQuad(disp, int3_type(ub = +3._RKH))
    call test_isFailedQuad(disp, int4_type())
    call test_isFailedQuad(disp, intSinCos_type(-4_IK, +4_IK, a = 10._RKH, b = -10._RKH))
    call test_isFailedQuad(disp, intNormPDF_type(-3._RKH, +3._RKH))
    call test_isFailedQuad(disp, intLogNormPDF_type(lb = exp(-6._RKG), ub = exp(+6._RKG)))
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compare integration via:")
    call disp%show("!       the Adaptive Global Gauss-Kronrod Quadrature method with automatic break point handling and extrapolation acceleration (QAGS),")
    call disp%show("! with,")
    call disp%show("!       the Adaptive Global Gauss-Kronrod Quadrature method with specified break point handling and extrapolation acceleration (QAGP).")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call test_isFailedQuad(disp, int5_type(0._RKG, 3._RKG))
    call test_isFailedQuad(disp, int5_type(-2._RKG, +5._RKG))
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compare integration via:")
    call disp%show("!       the Adaptive Global Gauss-Kronrod Quadrature method without extrapolation on a semi-infinite range `(lb, +Inf)`,")
    call disp%show("! with,")
    call disp%show("!       the Adaptive Global Gauss-Kronrod Quadrature method with extrapolation on a semi-infinite range `(lb, +Inf)`.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call test_isFailedQuad(disp, intGamUpp_type())
    !call disp%show("call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = +1.0_RKG, beta = 3._RKG))")
    !                call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = +1.0_RKG, beta = 3._RKG))
    !call disp%show("call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = +0.5_RKG, beta = 3._RKG))")
    !                call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = +0.5_RKG, beta = 3._RKG))
    !call disp%show("call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = +0.0_RKG, beta = 3._RKG))")
    !                call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = +0.0_RKG, beta = 3._RKG))
    !call disp%show("call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = -0.5_RKG, beta = 3._RKG))")
    !                call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = -0.5_RKG, beta = 3._RKG))
    !call disp%show("call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = -1.0_RKG, beta = 3._RKG))")
    !                call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = -1.0_RKG, beta = 3._RKG))
    !call disp%show("call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = -1.5_RKG, beta = 3._RKG))")
    !                call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = -1.5_RKG, beta = 3._RKG))
    !call disp%show("call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = -2.0_RKG, beta = 3._RKG))")
    !                call test_isFailedQuad(disp, intGamUpp_type(lb = 1._RKG, alpha = -2.0_RKG, beta = 3._RKG))
    call test_isFailedQuad(disp, intLogNormPDF_type())
    call test_isFailedQuad(disp, intDoncker1_type())
    call test_isFailedQuad(disp, int6_type())
    call test_isFailedQuad(disp, int7_type())
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compare integration via:")
    call disp%show("!       the Adaptive Global Gauss-Kronrod Quadrature method without extrapolation on a semi-infinite range `(-Inf, ub)`,")
    call disp%show("! with,")
    call disp%show("!       the Adaptive Global Gauss-Kronrod Quadrature method with extrapolation on a semi-infinite range `(-Inf, ub)`.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call test_isFailedQuad(disp, int8_type())
    call test_isFailedQuad(disp, intDoncker2_type())
    call test_isFailedQuad(disp, intDoncker2_type(ub = -1._RKG))
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compare integration via:")
    call disp%show("!       the Adaptive Global Gauss-Kronrod Quadrature method without extrapolation on a full-infinite range `(-Inf, +Inf)`,")
    call disp%show("! with,")
    call disp%show("!       the Adaptive Global Gauss-Kronrod Quadrature method with extrapolation on a full-infinite range `(-Inf, +Inf)`.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call test_isFailedQuad(disp, int9_type())
    call test_isFailedQuad(disp, intNormPDF_type())
    call test_isFailedQuad(disp, intGenExpGammaPDF_type())
    call test_isFailedQuad(disp, intGenExpGammaPDF_type(kappa = 5._RKH, invOmega = 0.1_RKH, logSigma = 2._RKH))
    call test_isFailedQuad(disp, intPentaGammaInf_type())
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the Cauchy Principal Value over a finite interval.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call test_isFailedQuad(disp, intCauchy1_type())
    call test_isFailedQuad(disp, intCauchy2_type())
 
end program example

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example output ⛓

 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compare integration via the Adaptive Global Gauss-Kronrod Quadrature method with (QAG) and without (QAGS) extrapolation acceleration.
! Note the significant efficiency improvement in the integration via QAGS when the integrand has integrable singularity at some points.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
integrand%desc
"int1_type: an algebraic integrand of the form f(x) = x**2 / (x**2 + 1) / (x**2 + 4) for x in (lb, ub)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int1_type: an algebraic integrand of the form f(x) = x**2 / (x**2 + 1) / (x**2 + 4) for x in (lb, ub)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.592319847946765693983261139952398973, 0.592319847946765693983261139951656719, 0.122995853428115837177070158359807283E-22, 0.742253400566840697767773899072230989E-30 (unbiased)? TRUE
numFuncEval
+441
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.592319847946765693983261139952398973, 0.592319847946765693983261139951656623, 0.122995853428115837177070158261927716E-22, 0.742349697064060059560426697969360235E-30 (unbiased)? TRUE
numFuncEval
+441
 
integrand%desc
"int2_type: an algebraic integrand of the form f(x) = 1 / sqrt(a - b * x) for x in (0, a / b) with a > 0 and b > 0 with a singularity at the upper bound of integration"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int2_type: an algebraic integrand of the form f(x) = 1 / sqrt(a - b * x) for x in (0, a / b) with a > 0 and b > 0 with a singularity at the upper bound of integration
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.942809041582063365867792482806465323, 0.942809041582063358489159110539769991, 0.274189625236721766830515122145815584E-15, 0.737863337226669533225571680774701636E-17 (unbiased)? TRUE
numFuncEval
+4263
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.942809041582063365867792482806465323, 0.942809041582063365867792482806451456, 0.747866619686254516023298199584791888E-27, 0.138666955995880981420030411866114767E-31 (unbiased)? TRUE
numFuncEval
+399
 
integrand%desc
"int3_type: an algebraic integrand of the form f(x) = log(x) / sqrt(x) for x in (0, ub) with a singularity at the lower limit of integration"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int3_type: an algebraic integrand of the form f(x) = log(x) / sqrt(x) for x in (0, ub) with a singularity at the lower limit of integration
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-3.12249862669012531099145873075559453, -3.12249862669012531099145800385027096, 0.871742302516757365494019872303469907E-22, 0.726905323565232669836577649328480277E-24 (unbiased)? TRUE
numFuncEval
+8127
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-3.12249862669012531099145873075559453, -3.12249862669012531099145873075553329, 0.638742935999207969456768107027576605E-25, 0.612445722315141001271800985742006887E-31 (unbiased)? TRUE
numFuncEval
+567
 
integrand%desc
"int4_type: an algebraic integrand of the form f(x) = log(x) / (1. + log(x)**2)**2 for x in (0, 1) with a singularity at the lower limit"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int4_type: an algebraic integrand of the form f(x) = log(x) / (1. + log(x)**2)**2 for x in (0, 1) with a singularity at the lower limit
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.189275187882093321180367135892330336, -0.189275187882093321180367139193492826, 0.439350295463286972817507956603694862E-23, 0.330116248974192359134514995223265581E-26 (unbiased)? TRUE
numFuncEval
+2457
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.189275187882093321180367135892330336, -0.189275187882093321180367392427969773, 0.530509501355781083785828386429659476E-23, 0.256535639436815431080742250597631077E-24 (unbiased)? TRUE
numFuncEval
+1029
 
integrand%desc
"intSinCos_typer(): a highly oscillatory integrand of the form f(x) = cos(a * sin(b * x)) for x in (lb, ub)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intSinCos_typer(): a highly oscillatory integrand of the form f(x) = cos(a * sin(b * x)) for x in (lb, ub)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-6.18103992684276605416490727881801341, -6.18103992684276605416490727881796180, 0.204459144829722446056783263048240039E-21, 0.516149225095779208619002088612760521E-31 (unbiased)? TRUE
numFuncEval
+40509
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-6.18103992684276605416490727881801341, -6.18103992684276605416490727881796180, 0.204459144829722446056783263048240039E-21, 0.516149225095779208619002088612760521E-31 (unbiased)? TRUE
numFuncEval
+40509
 
integrand%desc
"intNormPDF_type: f(x) = exp(-0.5 * (log(x) - mu)**2 / sigma**2) / (sigma * sqrt(2 * acos(-1.))) for x in (lb, ub)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intNormPDF_type: f(x) = exp(-0.5 * (log(x) - mu)**2 / sigma**2) / (sigma * sqrt(2 * acos(-1.))) for x in (lb, ub)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.997300203936739810946696370464810073, 0.997300203936739810946696370464810073, 0.863216418769238025171827058295059452E-23, 0.00000000000000000000000000000000000 (unbiased)? TRUE
numFuncEval
+147
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.997300203936739810946696370464810073, 0.997300203936739810946696370464810073, 0.863216418769238025171827058295059452E-23, 0.00000000000000000000000000000000000 (unbiased)? TRUE
numFuncEval
+147
 
integrand%desc
"intLogNormPDF_type: f(x) = exp(-0.5 * (log(x) - mu)**2 / sigma**2) / (x * sigma * sqrt(2 * acos(-1.))) for x in (0 <= lb, ub)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intLogNormPDF_type: f(x) = exp(-0.5 * (log(x) - mu)**2 / sigma**2) / (x * sigma * sqrt(2 * acos(-1.))) for x in (0 <= lb, ub)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.999999998026824709924603718598271714, 0.999999998026824709924603718597611697, 0.287685113478269648540938767224899488E-23, 0.660016191941505726842283640923854592E-30 (unbiased)? TRUE
numFuncEval
+1113
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.999999998026824709924603718598271714, 0.999999998026824709924603718597611601, 0.287685113534716101717891864391506212E-23, 0.660112488438725088634936439820983838E-30 (unbiased)? TRUE
numFuncEval
+1113
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compare integration via:
!       the Adaptive Global Gauss-Kronrod Quadrature method with automatic break point handling and extrapolation acceleration (QAGS),
! with,
!       the Adaptive Global Gauss-Kronrod Quadrature method with specified break point handling and extrapolation acceleration (QAGP).
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
integrand%desc
"int5_type: an algebraic integrand of the form f(x) = x**3 log(abs((x**2 - 1) * (x**2 - 2))) for x in (0., 3.) with 4 possible singularities: [-sqrt(2.), -1., 1., sqrt(2.)]"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int5_type: an algebraic integrand of the form f(x) = x**3 log(abs((x**2 - 1) * (x**2 - 2))) for x in (0., 3.) with 4 possible singularities: [-sqrt(2.), -1., 1., sqrt(2.)]
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977291765299254724, 0.172884396198263085498243239907807437E-20, 0.231903045085095141212385933523689371E-22 (unbiased)? TRUE
numFuncEval
+10101
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977292388198386134, 0.157545601084532647841636140405395188E-20, 0.390996086324092930047679517794757520E-22 (unbiased)? TRUE
numFuncEval
+9975
 
 
! Assisted adaptive global quadrature by specifying points of difficulties of the integrand.
 
break = integrand%break
break
+1.00000000000000000000000000000000000, +1.41421356237309504880168872420969798
if (isFailedQuad(getFunc, lb, ub, break, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977291997201071898, 0.145935898758650954111037868386060001E-21, 0.122791130278308118836119374489798694E-27 (unbiased)? TRUE
numFuncEval
+2163
 
integrand%desc
"int5_type: an algebraic integrand of the form f(x) = x**3 log(abs((x**2 - 1) * (x**2 - 2))) for x in (-2., 5.) with 4 possible singularities: [-sqrt(2.), -1., 1., sqrt(2.)]"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int5_type: an algebraic integrand of the form f(x) = x**3 log(abs((x**2 - 1) * (x**2 - 2))) for x in (-2., 5.) with 4 possible singularities: [-sqrt(2.), -1., 1., sqrt(2.)]
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
808.362636933094758891687400938039628, 808.362636933094758891687268473760062, 0.255379463719878359511890040859317480E-19, 0.132464279565495221439626229092202295E-21 (unbiased)? TRUE
numFuncEval
+19047
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
808.362636933094758891687400938039628, 808.362636933094758891681251389789229, 0.334323235794388144484139093037256979E-20, 0.614954825039840811287942669490845986E-20 (unbiased)? FALSE
numFuncEval
+17955
 
 
! Assisted adaptive global quadrature by specifying points of difficulties of the integrand.
 
break = integrand%break
break
-1.41421356237309504880168872420969798, -1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000, +1.41421356237309504880168872420969798
if (isFailedQuad(getFunc, lb, ub, break, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
808.362636933094758891687400938039628, 808.362636933094758891687400937280645, 0.990216586020630180919255548708256071E-21, 0.758982798435765983281759345872700701E-27 (unbiased)? TRUE
numFuncEval
+3045
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compare integration via:
!       the Adaptive Global Gauss-Kronrod Quadrature method without extrapolation on a semi-infinite range `(lb, +Inf)`,
! with,
!       the Adaptive Global Gauss-Kronrod Quadrature method with extrapolation on a semi-infinite range `(lb, +Inf)`.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
integrand%desc
"intGamUpp_type: an algebraic integrand of the form f(x; lb, alpha, beta) = (x / lb)**alpha * exp(-beta * (x - lb)) for x in (lb, +Inf), lb > 0."
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intGamUpp_type: an algebraic integrand of the form f(x; lb, alpha, beta) = (x / lb)**alpha * exp(-beta * (x - lb)) for x in (lb, +Inf), lb > 0.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
2.00000000000000000000000000000000039, 2.00000000000000000000000000000000039, 0.503988260578339162149833493686791185E-24, 0.00000000000000000000000000000000000 (unbiased)? TRUE
numFuncEval
+483
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
2.00000000000000000000000000000000039, 2.00000000000000000000000000000000039, 0.503988260578338140171031197806738174E-24, 0.00000000000000000000000000000000000 (unbiased)? TRUE
numFuncEval
+483
 
integrand%desc
"intLogNormPDF_type: f(x) = exp(-0.5 * (log(x) - mu)**2 / sigma**2) / (x * sigma * sqrt(2 * acos(-1.))) for x in (0 <= lb, ub)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intLogNormPDF_type: f(x) = exp(-0.5 * (log(x) - mu)**2 / sigma**2) / (x * sigma * sqrt(2 * acos(-1.))) for x in (0 <= lb, ub)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.00000000000000000000000000000000000, 1.00000000000000000000000005509718469, 0.185875326636098607607964145008442373E-22, 0.550971846938518212456720189377984095E-25 (unbiased)? TRUE
numFuncEval
+735
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.00000000000000000000000000000000000, 1.00000000000000000000000005509718469, 0.185875326636098606689609183428530258E-22, 0.550971846938518212456720189377984095E-25 (unbiased)? TRUE
numFuncEval
+735
 
integrand%desc
"intDoncker1_type: f(x) = 1 / (1 + x) / sqrt(x) for x in (0 <= lb, ub) with a square-root singularity at 0"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intDoncker1_type: f(x) = 1 / (1 + x) / sqrt(x) for x in (0 <= lb, ub) with a square-root singularity at 0
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
3.14159265358979323846264338327950280, 3.14159265358979319764120376296608615, 0.114874526811124567747394666325693956E-14, 0.408214396203134166514689154707924291E-16 (unbiased)? TRUE
numFuncEval
+8463
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
3.14159265358979323846264338327950280, 3.14159265358979323846264338346019817, 0.105499447763580851205123272015818952E-22, 0.180695369614276997099759184920380085E-27 (unbiased)? TRUE
numFuncEval
+2121
 
integrand%desc
"int6_type: an algebraic integrand of the form f(x) = log(x) / (1 + 100 * x**2) for x in (0, +Inf) with a singularity at the lower limit"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int6_type: an algebraic integrand of the form f(x) = log(x) / (1 + 100 * x**2) for x in (0, +Inf) with a singularity at the lower limit
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.361689220620773240624502327513089541, -0.361689220620773240624502327908133835, 0.118682246318398965349923011739062857E-22, 0.395044294632030724453129549211143496E-27 (unbiased)? TRUE
numFuncEval
+6657
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.361689220620773240624502327513089541, -0.361689220620773240624502327510642406, 0.120941504830908329737177699048668687E-23, 0.244713473558703155578925197329697327E-29 (unbiased)? TRUE
numFuncEval
+1659
 
integrand%desc
"int7_type: an algebraic integrand of the form f(x) = -log(abs((1 - x**2) * (1 - 2 * x**2)) / x**4) / x**5 for x in (1./3., +Inf) with two singularities at [1 / sqrt(2), 1]"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int7_type: an algebraic integrand of the form f(x) = -log(abs((1 - x**2) * (1 - 2 * x**2)) / x**4) / x**5 for x in (1./3., +Inf) with two singularities at [1 / sqrt(2), 1]
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977292221649683846, 0.168283512274674281393657004006737318E-20, 0.224447384036677384031379327649902600E-22 (unbiased)? TRUE
numFuncEval
+10143
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977306706202548049, 0.169942945536630143394077258382982232E-20, 0.147090002482391233646949135657205724E-20 (unbiased)? TRUE
numFuncEval
+9681
 
 
! Assisted adaptive global quadrature by specifying points of difficulties of the integrand.
 
break = integrand%break
break
+0.707106781186547524400844362104849088, +1.00000000000000000000000000000000000
if (isFailedQuad(getFunc, lb, ub, break, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977291997199544774, 0.550213208024888852525213619612864128E-21, 0.275503508172616333885316422295596819E-27 (unbiased)? TRUE
numFuncEval
+2499
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compare integration via:
!       the Adaptive Global Gauss-Kronrod Quadrature method without extrapolation on a semi-infinite range `(-Inf, ub)`,
! with,
!       the Adaptive Global Gauss-Kronrod Quadrature method with extrapolation on a semi-infinite range `(-Inf, ub)`.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
integrand%desc
"int8_type: an algebraic integrand of the form f(x) = log(abs((1 - x**2) * (1 - 2 * x**2)) / x**4) / x**5 for x in (-Inf, -1./3.) with two singularities at [-1, -1 / sqrt(2)]"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int8_type: an algebraic integrand of the form f(x) = log(abs((1 - x**2) * (1 - 2 * x**2)) / x**4) / x**5 for x in (-Inf, -1./3.) with two singularities at [-1, -1 / sqrt(2)]
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977292221649683846, 0.168283512274674281393657004006737318E-20, 0.224447384036677384031379327649902600E-22 (unbiased)? TRUE
numFuncEval
+10143
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977306706202548049, 0.169942945536630143394077258382982232E-20, 0.147090002482391233646949135657205724E-20 (unbiased)? TRUE
numFuncEval
+9681
 
 
! Assisted adaptive global quadrature by specifying points of difficulties of the integrand.
 
break = integrand%break
break
-1.00000000000000000000000000000000000, -0.707106781186547524400844362104849088
if (isFailedQuad(getFunc, lb, ub, break, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
52.7407483834714449977291997202299809, 52.7407483834714449977291997199544774, 0.550213208024888852525213619612864128E-21, 0.275503508172616333885316422295596819E-27 (unbiased)? TRUE
numFuncEval
+2499
 
integrand%desc
"intDoncker2_type: f(x) = exp(x) / sqrt(-x) for x in (lb, ub <= 0) with a square-root singularity at 0"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intDoncker2_type: f(x) = exp(x) / sqrt(-x) for x in (lb, ub <= 0) with a square-root singularity at 0
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.77245385090551602729816748334114514, 1.77245385090551600687177993068626502, 0.574360549312137963665679582228894917E-15, 0.204263875526548801162109406012147667E-16 (unbiased)? TRUE
numFuncEval
+4431
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.77245385090551602729816748334114514, 1.77245385090551602729816748322058404, 0.316674820796741424790461948518443858E-23, 0.120561095995702138441865857630079607E-27 (unbiased)? TRUE
numFuncEval
+1575
 
integrand%desc
"intDoncker2_type: f(x) = exp(x) / sqrt(-x) for x in (lb, ub <= 0) with a square-root singularity at 0"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intDoncker2_type: f(x) = exp(x) / sqrt(-x) for x in (lb, ub <= 0) with a square-root singularity at 0
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.278805585280661976499232611077439155, 0.278805585280661976499232611077447244, 0.136847162139771908855491210817521382E-24, 0.808890576642639058283510735885669474E-32 (unbiased)? TRUE
numFuncEval
+441
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
0.278805585280661976499232611077439155, 0.278805585280661976499232611077447292, 0.136847162139771908864182157536569162E-24, 0.813705401503607147916150680742131792E-32 (unbiased)? TRUE
numFuncEval
+441
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compare integration via:
!       the Adaptive Global Gauss-Kronrod Quadrature method without extrapolation on a full-infinite range `(-Inf, +Inf)`,
! with,
!       the Adaptive Global Gauss-Kronrod Quadrature method with extrapolation on a full-infinite range `(-Inf, +Inf)`.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
integrand%desc
"int9_type: an algebraic piecewise integrand of the form f(x) = log(abs((1 - x**2) * (1 - 2 * x**2)) / x**4) / x**5 for x in (1./3., +Inf) and 1 / (acos(-1) * sqrt(-(x+10) * (x+9))) for x in (-10, 9), otherwise 0, with two singularities at [-10, -9, 1/3., 1 / sqrt(2), 1]"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! int9_type: an algebraic piecewise integrand of the form f(x) = log(abs((1 - x**2) * (1 - 2 * x**2)) / x**4) / x**5 for x in (1./3., +Inf) and 1 / (acos(-1) * sqrt(-(x+10) * (x+9))) for x in (-10, 9), otherwise 0, with two singularities at [-10, -9, 1/3., 1 / sqrt(2), 1]
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
53.7407483834714449977291997202299809, 53.7407483834714446968169568664147327, 0.297103187254200474475915711578675253E-14, 0.300912242853815248241860288447560029E-15 (unbiased)? TRUE
numFuncEval
+19593
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
53.7407483834714449977291997202299809, 53.7407483834714786408368937861800291, 0.597341489232741586490157071452958427E-12, 0.336431076940659500481100214330126955E-13 (unbiased)? TRUE
numFuncEval
+16653
 
 
! Assisted adaptive global quadrature by specifying points of difficulties of the integrand.
 
break = integrand%break
break
-10.0000000000000000000000000000000000, -9.00000000000000000000000000000000000, +0.333333333333333333333333333333333317, +0.707106781186547524400844362104849088, +1.00000000000000000000000000000000000
if (isFailedQuad(getFunc, lb, ub, break, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
53.7407483834714449977291997202299809, 53.7407483834714449977291998296210022, 0.511241450110203608013944520330859404E-22, 0.109391021252254961020398451633034320E-24 (unbiased)? TRUE
numFuncEval
+5796
 
integrand%desc
"intNormPDF_type: f(x) = exp(-0.5 * (log(x) - mu)**2 / sigma**2) / (sigma * sqrt(2 * acos(-1.))) for x in (lb, ub)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intNormPDF_type: f(x) = exp(-0.5 * (log(x) - mu)**2 / sigma**2) / (sigma * sqrt(2 * acos(-1.))) for x in (lb, ub)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.00000000000000000000000000000000000, 0.999999999999999999999999999999999133, 0.386431949672064481044988439592369946E-24, 0.866668474974256133875190074163217293E-33 (unbiased)? TRUE
numFuncEval
+483
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.00000000000000000000000000000000000, 0.999999999999999999999999999999999133, 0.386431949672064466119465327815932357E-24, 0.866668474974256133875190074163217293E-33 (unbiased)? TRUE
numFuncEval
+483
 
integrand%desc
"intGenExpGammaPDF_type: f(x) = GenExpGamma(x; kappa = 1., omega = 1., logSigma = 0.) for x in (-Inf, Inf)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intGenExpGammaPDF_type: f(x) = GenExpGamma(x; kappa = 1., omega = 1., logSigma = 0.) for x in (-Inf, Inf)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.00000000000000000000000000000000000, 1.00000000000000000000000000000013597, 0.148409338409973194577277535014189193E-22, 0.135970654073738851225752042746495869E-30 (unbiased)? TRUE
numFuncEval
+651
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.00000000000000000000000000000000000, 1.00000000000000000000000000000013597, 0.148409338409973194495421223865688864E-22, 0.135970654073738851225752042746495869E-30 (unbiased)? TRUE
numFuncEval
+651
 
integrand%desc
"intGenExpGammaPDF_type: f(x) = GenExpGamma(x; kappa = 5., omega = 10., logSigma = 2.) for x in (-Inf, Inf)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intGenExpGammaPDF_type: f(x) = GenExpGamma(x; kappa = 5., omega = 10., logSigma = 2.) for x in (-Inf, Inf)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.00000000000000000000000000000000000, 1.00000000000000000000000000000000578, 0.132417436079502091744367222811402828E-22, 0.577778983316170755916793382775478196E-32 (unbiased)? TRUE
numFuncEval
+903
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
1.00000000000000000000000000000000000, 1.00000000000000000000000000000000578, 0.132417436079839015749459180202287325E-22, 0.577778983316170755916793382775478196E-32 (unbiased)? TRUE
numFuncEval
+903
 
integrand%desc
"intPentaGammaInf_type: f(x) = sum of five Gamma PDFs with five break points in the integration range x in (-Inf, +Inf)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intPentaGammaInf_type: f(x) = sum of five Gamma PDFs with five break points in the integration range x in (-Inf, +Inf)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
5.00000000000000000000000000000000000, 4.99999999999999999999972410297847334, 0.200738257888944093088270474559436884E-19, 0.275897021526661457740995046287992818E-21 (unbiased)? TRUE
numFuncEval
+21945
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
5.00000000000000000000000000000000000, 4.99999999999999999999999459738972784, 0.120968281919312175939861440033309516E-21, 0.540261027215530879807578922336193197E-23 (unbiased)? TRUE
numFuncEval
+11025
 
 
! Assisted adaptive global quadrature by specifying points of difficulties of the integrand.
 
break = integrand%break
break
-9.00000000000000000000000000000000000, -5.00000000000000000000000000000000000, +2.00000000000000000000000000000000000, +5.00000000000000000000000000000000000, +7.00000000000000000000000000000000000
if (isFailedQuad(getFunc, lb, ub, break, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
5.00000000000000000000000000000000000, 4.99999999999999999999999999996475779, 0.454086489790053140200838733525734823E-22, 0.352422068663531514279007291757730680E-28 (unbiased)? TRUE
numFuncEval
+5502
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the Cauchy Principal Value over a finite interval.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
integrand%desc
"intCauchy1_type: an integrand of the form w(x) * f(x) with Cauchy weight w(x) 1 / (x - cs) and f(x) = 1 ~,~ x \in (lb < cs, cs < ub)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intCauchy1_type: an integrand of the form w(x) * f(x) with Cauchy weight w(x) 1 / (x - cs) and f(x) = 1 ~,~ x \in (lb < cs, cs < ub)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.405465108108164381978013115464349228, -2.04356971055286136189443548589605091, 7.24146052036062818462419495305868050, 1.63810460244469697991642237043170168 (unbiased)? TRUE
numFuncEval
+4389
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.405465108108164381978013115464349228, -0.405465108108164381978013115464718525, 0.948013123403926992764805620933964846E-25, 0.369297066836252474823483770490659813E-30 (unbiased)? TRUE
numFuncEval
+1155
 
 
! Assisted adaptive global quadrature by specifying Cauchy weight of the integrand.
 
integrand%wcauchy%cs
+1.00000000000000000000000000000000000
if (isFailedQuad(getFuncUnweighted, lb, ub, integrand%wcauchy, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.405465108108164381978013115464349228, -0.405465108108164381978013115464349036, 0.192592994438723585305597794258492732E-33, 0.192592994438723585305597794258492732E-33 (unbiased)? TRUE
numFuncEval
+25
 
integrand%desc
"intCauchy2_type: an integrand of the form w(x) * f(x) = 1 / (x + 2.) / (x - 3.) for x in (-3., 2.)"
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! intCauchy2_type: an integrand of the form w(x) * f(x) = 1 / (x + 2.) / (x - 3.) for x in (-3., 2.)
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
! Regular adaptive global quadrature.
 
if (isFailedQuad(getFunc, lb, ub, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.635610766069589123929388320259411106, -0.313635651749756707119425316680125425, 1.43562881981449143882504794728653286, 0.321975114319832416809963003579285681 (unbiased)? TRUE
numFuncEval
+4347
 
 
! Assisted adaptive global quadrature by the Wynn Epsilon extrapolation.
 
if (isFailedQuad(getFunc, lb, ub, weps, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
 
       ******** Integration did NOT converge. ********
 
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.635610766069589123929388320259411106, -0.635610766069589123929388320259460313, 0.597574167278168521319843806854314588E-24, 0.492075100790938760455802364330448930E-31 (unbiased)? TRUE
numFuncEval
+1281
 
 
! Assisted adaptive global quadrature by specifying Cauchy weight of the integrand.
 
integrand%wcauchy%cs
-2.00000000000000000000000000000000000
if (isFailedQuad(getFuncUnweighted, lb, ub, integrand%wcauchy, integral, abserr, neval = numFuncEval)) call disp%show('       ******** Integration did NOT converge. ********', tmsize = 1_IK, bmsize = 1_IK)
getStr([truth, integral, abserr, abs(integral - truth)])//SK_' (unbiased)? '//getStr(abs(integral - truth) <= abserr)
-0.635610766069589123929388320259411106, -0.635610766069589123929388320259411106, 0.485640976828224618892203332811773371E-24, 0.00000000000000000000000000000000000 (unbiased)? TRUE
numFuncEval
+771
 
 

Test:

test_pm_quadPack

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.

1. 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.
   
2. 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 4989 of file pm_quadPack.F90.

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

• src/fortran/main/pm_quadPack.F90