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quadpack.f90
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!--------------------------------------------------------------------------------
!
! Copyright (C) 2017 L. J. Allen, H. G. Brown, A. J. D’Alfonso, S.D. Findlay, B. D. Forbes
!
! This program is free software: you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation, either version 3 of the License, or
! (at your option) any later version.
!
! This program is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License
! along with this program. If not, see <http://www.gnu.org/licenses/>.
!
!--------------------------------------------------------------------------------
!This module is adapted from the quadpack library for numerical integration,
!available at <http://people.sc.fsu.edu/~jburkardt/f_src/quadpack/quadpack.html>.
!The only modifications were to make make the real variables double precision (64-bit)
!and to place the routines within a module. The original MuSTEM codes used the
!Romberg integration routine described in the book "Numerical Recipes in FORTRAN: The
!Art of Scientific Computing" by Press et al. (1986), however the copyright conditions
!of that software package prevented them being included in this open source implementation
!of MuSTEM.
module quadpack
contains
subroutine aaaa
!*****************************************************************************80
!
!! AAAA is a dummy subroutine with QUADPACK documentation in its comments.
!
! 1. introduction
!
! quadpack is a fortran subroutine package for the numerical
! computation of definite one-dimensional integrals. it originated
! from a joint project of r. piessens and e. de doncker (appl.
! math. and progr. div.- k.u.leuven, belgium), c. ueberhuber (inst.
! fuer math.- techn.u.wien, austria), and d. kahaner (nation. bur.
! of standards- washington d.c., u.s.a.).
!
! 2. survey
!
! - qags : is an integrator based on globally adaptive interval
! subdivision in connection with extrapolation (de doncker,
! 1978) by the epsilon algorithm (wynn, 1956).
!
! - qagp : serves the same purposes as qags, but also allows
! for eventual user-supplied information, i.e. the
! abscissae of internal singularities, discontinuities
! and other difficulties of the integrand function.
! the algorithm is a modification of that in qags.
!
! - qagi : handles integration over infinite intervals. the
! infinite range is mapped onto a finite interval and
! then the same strategy as in qags is applied.
!
! - qawo : is a routine for the integration of cos(omega*x)*f(x)
! or sin(omega*x)*f(x) over a finite interval (a,b).
! omega is is specified by the user
! the rule evaluation component is based on the
! modified clenshaw-curtis technique.
! an adaptive subdivision scheme is used connected with
! an extrapolation procedure, which is a modification
! of that in qags and provides the possibility to deal
! even with singularities in f.
!
! - qawf : calculates the fourier cosine or fourier sine
! transform of f(x), for user-supplied interval (a,
! infinity), omega, and f. the procedure of qawo is
! used on successive finite intervals, and convergence
! acceleration by means of the epsilon algorithm (wynn,
! 1956) is applied to the series of the integral
! contributions.
!
! - qaws : integrates w(x)*f(x) over (a,b) with a < b finite,
! and w(x) = ((x-a)**alfa)*((b-x)**beta)*v(x)
! where v(x) = 1 or log(x-a) or log(b-x)
! or log(x-a)*log(b-x)
! and -1 < alfa, -1 < beta.
! the user specifies a, b, alfa, beta and the type of
! the function v.
! a globally adaptive subdivision strategy is applied,
! with modified clenshaw-curtis integration on the
! subintervals which contain a or b.
!
! - qawc : computes the cauchy principal value of f(x)/(x-c)
! over a finite interval (a,b) and for
! user-determined c.
! the strategy is globally adaptive, and modified
! clenshaw-curtis integration is used on the subranges
! which contain the point x = c.
!
! each of the routines above also has a "more detailed" version
! with a name ending in e, as qage. these provide more
! information and control than the easier versions.
!
!
! the preceeding routines are all automatic. that is, the user
! inputs his problem and an error tolerance. the routine
! attempts to perform the integration to within the requested
! absolute or relative error.
! there are, in addition, a number of non-automatic integrators.
! these are most useful when the problem is such that the
! user knows that a fixed rule will provide the accuracy
! required. typically they return an error estimate but make
! no attempt to satisfy any particular input error request.
!
! qk15
! qk21
! qk31
! qk41
! qk51
! qk61
! estimate the integral on [a,b] using 15, 21,..., 61
! point rule and return an error estimate.
! qk15i 15 point rule for (semi)infinite interval.
! qk15w 15 point rule for special singular weight functions.
! qc25c 25 point rule for cauchy principal values
! qc25o 25 point rule for sin/cos integrand.
! qmomo integrates k-th degree chebychev polynomial times
! function with various explicit singularities.
!
! 3. guidelines for the use of quadpack
!
! here it is not our purpose to investigate the question when
! automatic quadrature should be used. we shall rather attempt
! to help the user who already made the decision to use quadpack,
! with selecting an appropriate routine or a combination of
! several routines for handling his problem.
!
! for both quadrature over finite and over infinite intervals,
! one of the first questions to be answered by the user is
! related to the amount of computer time he wants to spend,
! versus his -own- time which would be needed, for example, for
! manual subdivision of the interval or other analytic
! manipulations.
!
! (1) the user may not care about computer time, or not be
! willing to do any analysis of the problem. especially when
! only one or a few integrals must be calculated, this attitude
! can be perfectly reasonable. in this case it is clear that
! either the most sophisticated of the routines for finite
! intervals, qags, must be used, or its analogue for infinite
! intervals, qagi. these routines are able to cope with
! rather difficult, even with improper integrals.
! this way of proceeding may be expensive. but the integrator
! is supposed to give you an answer in return, with additional
! information in the case of a failure, through its error
! estimate and flag. yet it must be stressed that the programs
! cannot be totally reliable.
!
! (2) the user may want to examine the integrand function.
! if bad local difficulties occur, such as a discontinuity, a
! singularity, derivative singularity or high peak at one or
! more points within the interval, the first advice is to
! split up the interval at these points. the integrand must
! then be examinated over each of the subintervals separately,
! so that a suitable integrator can be selected for each of
! them. if this yields problems involving relative accuracies
! to be imposed on -finite- subintervals, one can make use of
! qagp, which must be provided with the positions of the local
! difficulties. however, if strong singularities are present
! and a high accuracy is requested, application of qags on the
! subintervals may yield a better result.
!
! for quadrature over finite intervals we thus dispose of qags
! and
! - qng for well-behaved integrands,
! - qag for functions with an oscillating behavior of a non
! specific type,
! - qawo for functions, eventually singular, containing a
! factor cos(omega*x) or sin(omega*x) where omega is known,
! - qaws for integrands with algebraico-logarithmic end point
! singularities of known type,
! - qawc for cauchy principal values.
!
! remark
!
! on return, the work arrays in the argument lists of the
! adaptive integrators contain information about the interval
! subdivision process and hence about the integrand behavior:
! the end points of the subintervals, the local integral
! contributions and error estimates, and eventually other
! characteristics. for this reason, and because of its simple
! globally adaptive nature, the routine qag in particular is
! well-suited for integrand examination. difficult spots can
! be located by investigating the error estimates on the
! subintervals.
!
! for infinite intervals we provide only one general-purpose
! routine, qagi. it is based on the qags algorithm applied
! after a transformation of the original interval into (0,1).
! yet it may eventuate that another type of transformation is
! more appropriate, or one might prefer to break up the
! original interval and use qagi only on the infinite part
! and so on. these kinds of actions suggest a combined use of
! different quadpack integrators. note that, when the only
! difficulty is an integrand singularity at the finite
! integration limit, it will in general not be necessary to
! break up the interval, as qagi deals with several types of
! singularity at the boundary point of the integration range.
! it also handles slowly convergent improper integrals, on
! the condition that the integrand does not oscillate over
! the entire infinite interval. if it does we would advise
! to sum succeeding positive and negative contributions to
! the integral -e.g. integrate between the zeros- with one
! or more of the finite-range integrators, and apply
! convergence acceleration eventually by means of quadpack
! subroutine qelg which implements the epsilon algorithm.
! such quadrature problems include the fourier transform as
! a special case. yet for the latter we have an automatic
! integrator available, qawf.
!
return
end subroutine
subroutine qag ( f, a, b, epsabs, epsrel, key, result, abserr, neval, ier )
!*****************************************************************************80
!
!! QAG approximates an integral over a finite interval.
!
! Discussion:
!
! The routine calculates an approximation RESULT to a definite integral
! I = integral of F over (A,B),
! hopefully satisfying
! || I - RESULT || <= max ( EPSABS, EPSREL * ||I|| ).
!
! QAG is a simple globally adaptive integrator using the strategy of
! Aind (Piessens, 1973). It is possible to choose between 6 pairs of
! Gauss-Kronrod quadrature formulae for the rule evaluation component.
! The pairs of high degree of precision are suitable for handling
! integration difficulties due to a strongly oscillating integrand.
!
! Author:
!
! Robert Piessens, Elise de Doncker-Kapenger,
! Christian Ueberhuber, David Kahaner
!
! Reference:
!
! Robert Piessens, Elise de Doncker-Kapenger,
! Christian Ueberhuber, David Kahaner,
! QUADPACK, a Subroutine Package for Automatic Integration,
! Springer Verlag, 1983
!
! Parameters:
!
! Input, external real ( kind = 8 ) F, the name of the function routine, of the form
! function f ( x )
! real ( kind = 8 ) f
! real ( kind = 8 ) x
! which evaluates the integrand function.
!
! Input, real ( kind = 8 ) A, B, the limits of integration.
!
! Input, real ( kind = 8 ) EPSABS, EPSREL, the absolute and relative accuracy requested.
!
! Input, integer ( kind = 4 ) KEY, chooses the order of the local integration rule:
! 1, 7 Gauss points, 15 Gauss-Kronrod points,
! 2, 10 Gauss points, 21 Gauss-Kronrod points,
! 3, 15 Gauss points, 31 Gauss-Kronrod points,
! 4, 20 Gauss points, 41 Gauss-Kronrod points,
! 5, 25 Gauss points, 51 Gauss-Kronrod points,
! 6, 30 Gauss points, 61 Gauss-Kronrod points.
!
! Output, real ( kind = 8 ) RESULT, the estimated value of the integral.
!
! Output, real ( kind = 8 ) ABSERR, an estimate of || I - RESULT ||.
!
! Output, integer ( kind = 4 ) NEVAL, the number of times the integral was evaluated.
!
! Output, integer ( kind = 4 ) IER, return code.
! 0, normal and reliable termination of the routine. It is assumed that the
! requested accuracy has been achieved.
! 1, maximum number of subdivisions allowed has been achieved. One can
! allow more subdivisions by increasing the value of LIMIT in QAG.
! However, if this yields no improvement it is advised to analyze the
! integrand to determine the integration difficulties. If the position
! of a local difficulty can be determined, such as a singularity or
! discontinuity within the interval) one will probably gain from
! splitting up the interval at this point and calling the integrator
! on the subranges. If possible, an appropriate special-purpose
! integrator should be used which is designed for handling the type
! of difficulty involved.
! 2, the occurrence of roundoff error is detected, which prevents the
! requested tolerance from being achieved.
! 3, extremely bad integrand behavior occurs at some points of the
! integration interval.
! 6, the input is invalid, because EPSABS < 0 and EPSREL < 0.
!
! Local parameters:
!
! LIMIT is the maximum number of subintervals allowed in
! the subdivision process of QAGE.
!
implicit none
integer ( kind = 4 ), parameter :: limit = 500
real ( kind = 8 ),intent(in):: a
real ( kind = 8 ) abserr
real ( kind = 8 ) alist(limit)
real ( kind = 8 ),intent(in):: b
real ( kind = 8 ) blist(limit)
real ( kind = 8 ) elist(limit)
real ( kind = 8 ) epsabs
real ( kind = 8 ) epsrel
real ( kind = 8 ), external :: f
integer ( kind = 4 ) ier
integer ( kind = 4 ) iord(limit)
integer ( kind = 4 ) key
integer ( kind = 4 ) last
integer ( kind = 4 ) neval
real ( kind = 8 ) result
real ( kind = 8 ) rlist(limit)
call qage ( f, a, b, epsabs, epsrel, key, limit, result, abserr, neval, &
ier, alist, blist, rlist, elist, iord, last )
return
end subroutine
subroutine qage ( f, a, b, epsabs, epsrel, key, limit, result, abserr, neval, &
ier, alist, blist, rlist, elist, iord, last )
!*****************************************************************************80
!
!! QAGE estimates a definite integral.
!
! Discussion:
!
! The routine calculates an approximation RESULT to a definite integral
! I = integral of F over (A,B),
! hopefully satisfying
! || I - RESULT || <= max ( EPSABS, EPSREL * ||I|| ).
!
! Author:
!
! Robert Piessens, Elise de Doncker-Kapenger,
! Christian Ueberhuber, David Kahaner
!
! Reference:
!
! Robert Piessens, Elise de Doncker-Kapenger,
! Christian Ueberhuber, David Kahaner,
! QUADPACK, a Subroutine Package for Automatic Integration,
! Springer Verlag, 1983
!
! Parameters:
!
! Input, external real ( kind = 8 ) F, the name of the function routine, of the form
! function f ( x )
! real ( kind = 8 ) f
! real ( kind = 8 ) x
! which evaluates the integrand function.
!
! Input, real ( kind = 8 ) A, B, the limits of integration.
!
! Input, real ( kind = 8 ) EPSABS, EPSREL, the absolute and relative accuracy requested.
!
! Input, integer ( kind = 4 ) KEY, chooses the order of the local integration rule:
! 1, 7 Gauss points, 15 Gauss-Kronrod points,
! 2, 10 Gauss points, 21 Gauss-Kronrod points,
! 3, 15 Gauss points, 31 Gauss-Kronrod points,
! 4, 20 Gauss points, 41 Gauss-Kronrod points,
! 5, 25 Gauss points, 51 Gauss-Kronrod points,
! 6, 30 Gauss points, 61 Gauss-Kronrod points.
!
! Input, integer ( kind = 4 ) LIMIT, the maximum number of subintervals that
! can be used.
!
! Output, real ( kind = 8 ) RESULT, the estimated value of the integral.
!
! Output, real ( kind = 8 ) ABSERR, an estimate of || I - RESULT ||.
!
! Output, integer ( kind = 4 ) NEVAL, the number of times the integral was evaluated.
!
! Output, integer ( kind = 4 ) IER, return code.
! 0, normal and reliable termination of the routine. It is assumed that the
! requested accuracy has been achieved.
! 1, maximum number of subdivisions allowed has been achieved. One can
! allow more subdivisions by increasing the value of LIMIT in QAG.
! However, if this yields no improvement it is advised to analyze the
! integrand to determine the integration difficulties. If the position
! of a local difficulty can be determined, such as a singularity or
! discontinuity within the interval) one will probably gain from
! splitting up the interval at this point and calling the integrator
! on the subranges. If possible, an appropriate special-purpose
! integrator should be used which is designed for handling the type
! of difficulty involved.
! 2, the occurrence of roundoff error is detected, which prevents the
! requested tolerance from being achieved.
! 3, extremely bad integrand behavior occurs at some points of the
! integration interval.
! 6, the input is invalid, because EPSABS < 0 and EPSREL < 0.
!
! Workspace, real ( kind = 8 ) ALIST(LIMIT), BLIST(LIMIT), contains in entries 1
! through LAST the left and right ends of the partition subintervals.
!
! Workspace, real ( kind = 8 ) RLIST(LIMIT), contains in entries 1 through LAST
! the integral approximations on the subintervals.
!
! Workspace, real ( kind = 8 ) ELIST(LIMIT), contains in entries 1 through LAST
! the absolute error estimates on the subintervals.
!
! Output, integer ( kind = 4 ) IORD(LIMIT), the first K elements of which are pointers
! to the error estimates over the subintervals, such that
! elist(iord(1)), ..., elist(iord(k)) form a decreasing sequence, with
! k = last if last <= (limit/2+2), and k = limit+1-last otherwise.
!
! Output, integer ( kind = 4 ) LAST, the number of subintervals actually produced
! in the subdivision process.
!
! Local parameters:
!
! alist - list of left end points of all subintervals
! considered up to now
! blist - list of right end points of all subintervals
! considered up to now
! elist(i) - error estimate applying to rlist(i)
! maxerr - pointer to the interval with largest error estimate
! errmax - elist(maxerr)
! area - sum of the integrals over the subintervals
! errsum - sum of the errors over the subintervals
! errbnd - requested accuracy max(epsabs,epsrel*abs(result))
! *****1 - variable for the left subinterval
! *****2 - variable for the right subinterval
! last - index for subdivision
!
implicit none
integer ( kind = 4 ) limit
real ( kind = 8 ),intent(in):: a
real ( kind = 8 ) abserr
real ( kind = 8 ) alist(limit)
real ( kind = 8 ) area
real ( kind = 8 ) area1
real ( kind = 8 ) area12
real ( kind = 8 ) area2
real ( kind = 8 ) a1
real ( kind = 8 ) a2
real ( kind = 8 ),intent(in):: b
real ( kind = 8 ) blist(limit)
real ( kind = 8 ) b1
real ( kind = 8 ) b2
real ( kind = 8 ) c
real ( kind = 8 ) defabs
real ( kind = 8 ) defab1
real ( kind = 8 ) defab2
real ( kind = 8 ) elist(limit)
real ( kind = 8 ) epsabs
real ( kind = 8 ) epsrel
real ( kind = 8 ) errbnd
real ( kind = 8 ) errmax
real ( kind = 8 ) error1
real ( kind = 8 ) error2
real ( kind = 8 ) erro12
real ( kind = 8 ) errsum
real ( kind = 8 ), external :: f
integer ( kind = 4 ) ier
integer ( kind = 4 ) iord(limit)
integer ( kind = 4 ) iroff1
integer ( kind = 4 ) iroff2
integer ( kind = 4 ) key
integer ( kind = 4 ) keyf
integer ( kind = 4 ) last
integer ( kind = 4 ) maxerr
integer ( kind = 4 ) neval
integer ( kind = 4 ) nrmax
real ( kind = 8 ) resabs
real ( kind = 8 ) result
real ( kind = 8 ) rlist(limit)
!
! Test on validity of parameters.
!
ier = 0
neval = 0
last = 0
result = 0.0E+00
abserr = 0.0E+00
alist(1) = a
blist(1) = b
rlist(1) = 0.0E+00
elist(1) = 0.0E+00
iord(1) = 0
if ( epsabs < 0.0E+00 .and. epsrel < 0.0E+00 ) then
ier = 6
return
end if
!
! First approximation to the integral.
!
keyf = key
keyf = max ( keyf, 1 )
keyf = min ( keyf, 6 )
c = keyf
neval = 0
if ( keyf == 1 ) then
call qk15 ( f, a, b, result, abserr, defabs, resabs )
else if ( keyf == 2 ) then
call qk21 ( f, a, b, result, abserr, defabs, resabs )
else if ( keyf == 3 ) then
call qk31 ( f, a, b, result, abserr, defabs, resabs )
else if ( keyf == 4 ) then
call qk41 ( f, a, b, result, abserr, defabs, resabs )
else if ( keyf == 5 ) then
call qk51 ( f, a, b, result, abserr, defabs, resabs )
else if ( keyf == 6 ) then
call qk61 ( f, a, b, result, abserr, defabs, resabs )
end if
last = 1
rlist(1) = result
elist(1) = abserr
iord(1) = 1
!
! Test on accuracy.
!
errbnd = max ( epsabs, epsrel * abs ( result ) )
if ( abserr <= 5.0E+01 * epsilon ( defabs ) * defabs .and. &
errbnd < abserr ) then
ier = 2
end if
if ( limit == 1 ) then
ier = 1
end if
if ( ier /= 0 .or. &
( abserr <= errbnd .and. abserr /= resabs ) .or. &
abserr == 0.0E+00 ) then
if ( keyf /= 1 ) then
neval = (10*keyf+1) * (2*neval+1)
else
neval = 30 * neval + 15
end if
return
end if
!
! Initialization.
!
errmax = abserr
maxerr = 1
area = result
errsum = abserr
nrmax = 1
iroff1 = 0
iroff2 = 0
do last = 2, limit
!
! Bisect the subinterval with the largest error estimate.
!
a1 = alist(maxerr)
b1 = 0.5E+00 * ( alist(maxerr) + blist(maxerr) )
a2 = b1
b2 = blist(maxerr)
if ( keyf == 1 ) then
call qk15 ( f, a1, b1, area1, error1, resabs, defab1 )
else if ( keyf == 2 ) then
call qk21 ( f, a1, b1, area1, error1, resabs, defab1 )
else if ( keyf == 3 ) then
call qk31 ( f, a1, b1, area1, error1, resabs, defab1 )
else if ( keyf == 4 ) then
call qk41 ( f, a1, b1, area1, error1, resabs, defab1)
else if ( keyf == 5 ) then
call qk51 ( f, a1, b1, area1, error1, resabs, defab1 )
else if ( keyf == 6 ) then
call qk61 ( f, a1, b1, area1, error1, resabs, defab1 )
end if
if ( keyf == 1 ) then
call qk15 ( f, a2, b2, area2, error2, resabs, defab2 )
else if ( keyf == 2 ) then
call qk21 ( f, a2, b2, area2, error2, resabs, defab2 )
else if ( keyf == 3 ) then
call qk31 ( f, a2, b2, area2, error2, resabs, defab2 )
else if ( keyf == 4 ) then
call qk41 ( f, a2, b2, area2, error2, resabs, defab2 )
else if ( keyf == 5 ) then
call qk51 ( f, a2, b2, area2, error2, resabs, defab2 )
else if ( keyf == 6 ) then
call qk61 ( f, a2, b2, area2, error2, resabs, defab2 )
end if
!
! Improve previous approximations to integral and error and
! test for accuracy.
!
neval = neval + 1
area12 = area1 + area2
erro12 = error1 + error2
errsum = errsum + erro12 - errmax
area = area + area12 - rlist(maxerr)
if ( defab1 /= error1 .and. defab2 /= error2 ) then
if ( abs ( rlist(maxerr) - area12 ) <= 1.0E-05 * abs ( area12 ) &
.and. 9.9E-01 * errmax <= erro12 ) then
iroff1 = iroff1 + 1
end if
if ( 10 < last .and. errmax < erro12 ) then
iroff2 = iroff2 + 1
end if
end if
rlist(maxerr) = area1
rlist(last) = area2
errbnd = max ( epsabs, epsrel * abs ( area ) )
!
! Test for roundoff error and eventually set error flag.
!
if ( errbnd < errsum ) then
if ( 6 <= iroff1 .or. 20 <= iroff2 ) then
ier = 2
end if
!
! Set error flag in the case that the number of subintervals
! equals limit.
!
if ( last == limit ) then
ier = 1
end if
!
! Set error flag in the case of bad integrand behavior
! at a point of the integration range.
!
if ( max ( abs ( a1 ), abs ( b2 ) ) <= ( 1.0E+00 + c * 1.0E+03 * &
epsilon ( a1 ) ) * ( abs ( a2 ) + 1.0E+04 * tiny ( a2 ) ) ) then
ier = 3
end if
end if
!
! Append the newly-created intervals to the list.
!
if ( error2 <= error1 ) then
alist(last) = a2
blist(maxerr) = b1
blist(last) = b2
elist(maxerr) = error1
elist(last) = error2
else
alist(maxerr) = a2
alist(last) = a1
blist(last) = b1
rlist(maxerr) = area2
rlist(last) = area1
elist(maxerr) = error2
elist(last) = error1
end if
!
! Call QSORT to maintain the descending ordering
! in the list of error estimates and select the subinterval
! with the largest error estimate (to be bisected next).
!
call qsort ( limit, last, maxerr, errmax, elist, iord, nrmax )
if ( ier /= 0 .or. errsum <= errbnd ) then
exit
end if
end do
!
! Compute final result.
!
result = sum ( rlist(1:last) )
abserr = errsum
if ( keyf /= 1 ) then
neval = ( 10 * keyf + 1 ) * ( 2 * neval + 1 )
else
neval = 30 * neval + 15
end if
return
end subroutine
subroutine qagi ( f, bound, inf, epsabs, epsrel, result, abserr, neval, ier )
!*****************************************************************************80
!
!! QAGI estimates an integral over a semi-infinite or infinite interval.
!
! Discussion:
!
! The routine calculates an approximation RESULT to a definite integral
! I = integral of F over (A, +Infinity),
! or
! I = integral of F over (-Infinity,A)
! or
! I = integral of F over (-Infinity,+Infinity),
! hopefully satisfying
! || I - RESULT || <= max ( EPSABS, EPSREL * ||I|| ).
!
! Author:
!
! Robert Piessens, Elise de Doncker-Kapenger,
! Christian Ueberhuber, David Kahaner
!
! Reference:
!
! Robert Piessens, Elise de Doncker-Kapenger,
! Christian Ueberhuber, David Kahaner,
! QUADPACK, a Subroutine Package for Automatic Integration,
! Springer Verlag, 1983
!
! Parameters:
!
! Input, external real ( kind = 8 ) F, the name of the function routine, of the form
! function f ( x )
! real ( kind = 8 ) f
! real ( kind = 8 ) x
! which evaluates the integrand function.
!
! Input, real ( kind = 8 ) BOUND, the value of the finite endpoint of the integration
! range, if any, that is, if INF is 1 or -1.
!
! Input, integer ( kind = 4 ) INF, indicates the type of integration range.
! 1: ( BOUND, +Infinity),
! -1: ( -Infinity, BOUND),
! 2: ( -Infinity, +Infinity).
!
! Input, real ( kind = 8 ) EPSABS, EPSREL, the absolute and relative accuracy requested.
!
! Output, real ( kind = 8 ) RESULT, the estimated value of the integral.
!
! Output, real ( kind = 8 ) ABSERR, an estimate of || I - RESULT ||.
!
! Output, integer ( kind = 4 ) NEVAL, the number of times the integral was evaluated.
!
! Output, integer ( kind = 4 ) IER, error indicator.
! 0, normal and reliable termination of the routine. It is assumed that
! the requested accuracy has been achieved.
! > 0, abnormal termination of the routine. The estimates for result
! and error are less reliable. It is assumed that the requested
! accuracy has not been achieved.
! 1, maximum number of subdivisions allowed has been achieved. One can
! allow more subdivisions by increasing the data value of LIMIT in QAGI
! (and taking the according dimension adjustments into account).
! However, if this yields no improvement it is advised to analyze the
! integrand in order to determine the integration difficulties. If the
! position of a local difficulty can be determined (e.g. singularity,
! discontinuity within the interval) one will probably gain from
! splitting up the interval at this point and calling the integrator
! on the subranges. If possible, an appropriate special-purpose
! integrator should be used, which is designed for handling the type
! of difficulty involved.
! 2, the occurrence of roundoff error is detected, which prevents the
! requested tolerance from being achieved. The error may be
! under-estimated.
! 3, extremely bad integrand behavior occurs at some points of the
! integration interval.
! 4, the algorithm does not converge. Roundoff error is detected in the
! extrapolation table. It is assumed that the requested tolerance
! cannot be achieved, and that the returned result is the best which
! can be obtained.
! 5, the integral is probably divergent, or slowly convergent. It must
! be noted that divergence can occur with any other value of IER.
! 6, the input is invalid, because INF /= 1 and INF /= -1 and INF /= 2, or
! epsabs < 0 and epsrel < 0. result, abserr, neval are set to zero.
!
! Local parameters:
!
! the dimension of rlist2 is determined by the value of
! limexp in QEXTR.
!
! alist - list of left end points of all subintervals
! considered up to now
! blist - list of right end points of all subintervals
! considered up to now
! rlist(i) - approximation to the integral over
! (alist(i),blist(i))
! rlist2 - array of dimension at least (limexp+2),
! containing the part of the epsilon table
! which is still needed for further computations
! elist(i) - error estimate applying to rlist(i)
! maxerr - pointer to the interval with largest error
! estimate
! errmax - elist(maxerr)
! erlast - error on the interval currently subdivided
! (before that subdivision has taken place)
! area - sum of the integrals over the subintervals
! errsum - sum of the errors over the subintervals
! errbnd - requested accuracy max(epsabs,epsrel*
! abs(result))
! *****1 - variable for the left subinterval
! *****2 - variable for the right subinterval
! last - index for subdivision
! nres - number of calls to the extrapolation routine
! numrl2 - number of elements currently in rlist2. if an
! appropriate approximation to the compounded
! integral has been obtained, it is put in
! rlist2(numrl2) after numrl2 has been increased
! by one.
! small - length of the smallest interval considered up
! to now, multiplied by 1.5
! erlarg - sum of the errors over the intervals larger
! than the smallest interval considered up to now
! extrap - logical variable denoting that the routine
! is attempting to perform extrapolation. i.e.
! before subdividing the smallest interval we
! try to decrease the value of erlarg.
! noext - logical variable denoting that extrapolation
! is no longer allowed (true-value)
!
implicit none
integer ( kind = 4 ), parameter :: limit = 500
real ( kind = 8 ) abseps
real ( kind = 8 ) abserr
real ( kind = 8 ) alist(limit)
real ( kind = 8 ) area
real ( kind = 8 ) area1
real ( kind = 8 ) area12
real ( kind = 8 ) area2
real ( kind = 8 ) a1
real ( kind = 8 ) a2
real ( kind = 8 ) blist(limit)
real ( kind = 8 ) boun
real ( kind = 8 ) bound
real ( kind = 8 ) b1
real ( kind = 8 ) b2
real ( kind = 8 ) correc
real ( kind = 8 ) defabs
real ( kind = 8 ) defab1
real ( kind = 8 ) defab2
real ( kind = 8 ) dres
real ( kind = 8 ) elist(limit)
real ( kind = 8 ) epsabs
real ( kind = 8 ) epsrel
real ( kind = 8 ) erlarg
real ( kind = 8 ) erlast
real ( kind = 8 ) errbnd
real ( kind = 8 ) errmax
real ( kind = 8 ) error1
real ( kind = 8 ) error2
real ( kind = 8 ) erro12
real ( kind = 8 ) errsum
real ( kind = 8 ) ertest
logical extrap
real ( kind = 8 ), external :: f
integer ( kind = 4 ) id
integer ( kind = 4 ) ier
integer ( kind = 4 ) ierro
integer ( kind = 4 ) inf
integer ( kind = 4 ) iord(limit)
integer ( kind = 4 ) iroff1
integer ( kind = 4 ) iroff2
integer ( kind = 4 ) iroff3
integer ( kind = 4 ) jupbnd
integer ( kind = 4 ) k
integer ( kind = 4 ) ksgn
integer ( kind = 4 ) ktmin
integer ( kind = 4 ) last
integer ( kind = 4 ) maxerr
integer ( kind = 4 ) neval
logical noext
integer ( kind = 4 ) nres
integer ( kind = 4 ) nrmax
integer ( kind = 4 ) numrl2
real ( kind = 8 ) resabs
real ( kind = 8 ) reseps
real ( kind = 8 ) result
real ( kind = 8 ) res3la(3)
real ( kind = 8 ) rlist(limit)
real ( kind = 8 ) rlist2(52)
real ( kind = 8 ) small
!
! Test on validity of parameters.
!
ier = 0
neval = 0
last = 0
result = 0.0E+00
abserr = 0.0E+00
alist(1) = 0.0E+00
blist(1) = 1.0E+00
rlist(1) = 0.0E+00
elist(1) = 0.0E+00
iord(1) = 0
if ( epsabs < 0.0E+00 .and. epsrel < 0.0E+00 ) then
ier = 6
return
end if
!
! First approximation to the integral.
!
! Determine the interval to be mapped onto (0,1).
! If INF = 2 the integral is computed as i = i1+i2, where
! i1 = integral of f over (-infinity,0),
! i2 = integral of f over (0,+infinity).
!
if ( inf == 2 ) then
boun = 0.0E+00
else
boun = bound
end if
call qk15i ( f, boun, inf, 0.0_8, 1.0_8, result, abserr, defabs, resabs )
!
! Test on accuracy.
!
last = 1
rlist(1) = result
elist(1) = abserr
iord(1) = 1
dres = abs ( result )
errbnd = max ( epsabs, epsrel * dres )
if ( abserr <= 100.0E+00 * epsilon ( defabs ) * defabs .and. &
errbnd < abserr ) then
ier = 2
end if
if ( limit == 1 ) then
ier = 1
end if
if ( ier /= 0 .or. (abserr <= errbnd .and. abserr /= resabs ) .or. &
abserr == 0.0E+00 ) go to 130
!
! Initialization.
!
rlist2(1) = result
errmax = abserr
maxerr = 1
area = result
errsum = abserr
abserr = huge ( abserr )
nrmax = 1
nres = 0
ktmin = 0
numrl2 = 2
extrap = .false.
noext = .false.
ierro = 0
iroff1 = 0
iroff2 = 0
iroff3 = 0
if ( ( 1.0E+00 - 5.0E+01 * epsilon ( defabs ) ) * defabs <= dres ) then
ksgn = 1
else
ksgn = -1
end if
do last = 2, limit
!
! Bisect the subinterval with nrmax-th largest error estimate.
!
a1 = alist(maxerr)
b1 = 5.0E-01 * ( alist(maxerr) + blist(maxerr) )
a2 = b1
b2 = blist(maxerr)
erlast = errmax
call qk15i ( f, boun, inf, a1, b1, area1, error1, resabs, defab1 )
call qk15i ( f, boun, inf, a2, b2, area2, error2, resabs, defab2 )
!
! Improve previous approximations to integral and error
! and test for accuracy.
!
area12 = area1 + area2
erro12 = error1 + error2
errsum = errsum + erro12 - errmax
area = area + area12 - rlist(maxerr)
if ( defab1 /= error1 .and. defab2 /= error2 ) then
if ( abs ( rlist(maxerr) - area12 ) <= 1.0E-05 * abs ( area12 ) &
.and. 9.9E-01 * errmax <= erro12 ) then
if ( extrap ) then
iroff2 = iroff2 + 1
end if
if ( .not. extrap ) then