same as dqawo but provides more information and control
the routine calculates an approximation result to a given
definite integral
i = integral of f(x)*w(x)
over (a,b)
where w(x) = cos(omega*x)
or w(x)=sin(omega*x)
,
hopefully satisfying following claim for accuracy
abs(i-result)<=max(epsabs,epsrel*abs(i))
.
Type | Intent | Optional | Attributes | Name | ||
---|---|---|---|---|---|---|
procedure(func) | :: | f |
function subprogram defining the integrand function |
|||
real(kind=wp), | intent(in) | :: | a |
lower limit of integration |
||
real(kind=wp), | intent(in) | :: | b |
upper limit of integration |
||
real(kind=wp), | intent(in) | :: | Omega |
parameter in the integrand weight function |
||
integer, | intent(in) | :: | Integr |
indicates which of the weight functions is to be used:
if |
||
real(kind=wp), | intent(in) | :: | Epsabs |
absolute accuracy requested |
||
real(kind=wp), | intent(in) | :: | Epsrel |
relative accuracy requested.
if |
||
integer, | intent(in) | :: | Limit |
gives an upper bound on the number of subdivisions
in the partition of |
||
integer, | intent(in) | :: | Icall |
if dqawoe is to be used only once, icall must
be set to 1. assume that during this call, the
chebyshev moments (for clenshaw-curtis integration
of degree 24) have been computed for intervals of
lengths |
||
integer, | intent(in) | :: | Maxp1 |
gives an upper bound on the number of chebyshev
moments which can be stored, i.e. for the
intervals of lengths |
||
real(kind=wp), | intent(out) | :: | Result |
approximation to the integral |
||
real(kind=wp), | intent(out) | :: | Abserr |
estimate of the modulus of the absolute error,
which should equal or exceed |
||
integer, | intent(out) | :: | Neval |
number of integrand evaluations |
||
integer, | intent(out) | :: | Ier |
error messages:
|
||
integer, | intent(out) | :: | Last |
on return, |
||
real(kind=wp), | intent(out) | :: | Alist(Limit) |
vector of dimension at least |
||
real(kind=wp), | intent(out) | :: | Blist(Limit) |
vector of dimension at least |
||
real(kind=wp), | intent(out) | :: | Rlist(Limit) |
vector of dimension at least |
||
real(kind=wp), | intent(out) | :: | Elist(Limit) |
vector of dimension at least |
||
integer, | intent(out) | :: | Iord(Limit) |
vector of dimension at least |
||
integer, | intent(out) | :: | Nnlog(Limit) |
vector of dimension at least |
||
integer, | intent(inout) | :: | Momcom |
indicating that the chebyshev moments
have been computed for intervals of lengths
|
||
real(kind=wp), | intent(inout) | :: | Chebmo(Maxp1,25) |
array of dimension |
subroutine dqawoe(f, a, b, Omega, Integr, Epsabs, Epsrel, Limit, Icall, & Maxp1, Result, Abserr, Neval, Ier, Last, Alist, Blist, & Rlist, Elist, Iord, Nnlog, Momcom, Chebmo) implicit none procedure(func) :: f !! function subprogram defining the integrand function `f(x)`. real(wp), intent(in) :: a !! lower limit of integration real(wp), intent(in) :: b !! upper limit of integration real(wp), intent(in) :: Omega !! parameter in the integrand weight function integer, intent(in) :: Integr !! indicates which of the weight functions is to be !! used: !! !! * integr = 1 `w(x) = cos(omega*x)` !! * integr = 2 `w(x) = sin(omega*x)` !! !! if `integr/=1` and `integr/=2`, the routine !! will end with ier = 6. real(wp), intent(in) :: Epsabs !! absolute accuracy requested real(wp), intent(in) :: Epsrel !! relative accuracy requested. !! if `epsabs<=0` !! and `epsrel<max(50*rel.mach.acc.,0.5e-28)`, !! the routine will end with ier = 6. integer, intent(in) :: Limit !! gives an upper bound on the number of subdivisions !! in the partition of `(a,b)`, `limit>=1`. integer, intent(in) :: Icall !! if dqawoe is to be used only once, icall must !! be set to 1. assume that during this call, the !! chebyshev moments (for clenshaw-curtis integration !! of degree 24) have been computed for intervals of !! lengths `(abs(b-a))*2**(-l), l=0,1,2,...momcom-1`. !! if `icall>1` this means that dqawoe has been !! called twice or more on intervals of the same !! length `abs(b-a)`. the chebyshev moments already !! computed are then re-used in subsequent calls. !! if `icall<1`, the routine will end with ier = 6. integer, intent(in) :: Maxp1 !! gives an upper bound on the number of chebyshev !! moments which can be stored, i.e. for the !! intervals of lengths `abs(b-a)*2**(-l)`, !! `l=0,1, ..., maxp1-2, maxp1>=1`. !! if `maxp1<1`, the routine will end with ier = 6. real(wp), intent(out) :: Result !! approximation to the integral real(wp), intent(out) :: Abserr !! estimate of the modulus of the absolute error, !! which should equal or exceed `abs(i-result)` integer, intent(out) :: Neval !! number of integrand evaluations integer, intent(out) :: Ier !! * ier = 0 normal and reliable termination of the !! routine. it is assumed that the !! requested accuracy has been achieved. !! * ier>0 abnormal termination of the routine. !! the estimates for integral and error are !! less reliable. it is assumed that the !! requested accuracy has not been achieved. !! !! error messages: !! !! * ier = 1 maximum number of subdivisions allowed !! has been achieved. one can allow more !! subdivisions by increasing the value of !! limit (and taking 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. !! * ier = 2 the occurrence of roundoff error is !! detected, which prevents the requested !! tolerance from being achieved. !! the error may be under-estimated. !! * ier = 3 extremely bad integrand behaviour occurs !! at some points of the integration !! interval. !! * ier = 4 the algorithm does not converge. !! roundoff error is detected in the !! extrapolation table. !! it is presumed that the requested !! tolerance cannot be achieved due to !! roundoff in the extrapolation table, !! and that the returned result is the !! best which can be obtained. !! * ier = 5 the integral is probably divergent, or !! slowly convergent. it must be noted that !! divergence can occur with any other value !! of ier>0. !! * ier = 6 the input is invalid, because !! `(epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28_wp))` !! or `(integr/=1 and integr/=2)` or !! `icall<1` or `maxp1<1`. !! `result`, `abserr`, `neval`, `last`, `rlist(1)`, !! `elist(1)`, `iord(1)` and `nnlog(1)` are set !! to zero. `alist(1)` and `blist(1)` are set !! to `a` and `b` respectively. integer, intent(out) :: Last !! on return, `last` equals the number of !! subintervals produces in the subdivision !! process, which determines the number of !! significant elements actually in the !! work arrays. real(wp), intent(out) :: Alist(Limit) !! vector of dimension at least `limit`, the first !! `last` elements of which are the left !! end points of the subintervals in the partition !! of the given integration range `(a,b)` real(wp), intent(out) :: Blist(Limit) !! vector of dimension at least `limit`, the first !! `last` elements of which are the right !! end points of the subintervals in the partition !! of the given integration range `(a,b)` real(wp), intent(out) :: Rlist(Limit) !! vector of dimension at least `limit`, the first !! `last` elements of which are the integral !! approximations on the subintervals real(wp), intent(out) :: Elist(Limit) !! vector of dimension at least `limit`, the first !! `last` elements of which are the moduli of the !! absolute error estimates on the subintervals integer, intent(out) :: Iord(Limit) !! vector of dimension at least `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. integer, intent(out) :: Nnlog(Limit) !! vector of dimension at least `limit`, containing the !! subdivision levels of the subintervals, i.e. !! iwork(i) = l means that the subinterval !! numbered `i` is of length `abs(b-a)*2**(1-l)` integer, intent(inout) :: Momcom !! indicating that the chebyshev moments !! have been computed for intervals of lengths !! `(abs(b-a))*2**(-l), l=0,1,2, ..., momcom-1`, !! `momcom<maxp1` real(wp), intent(inout) :: Chebmo(Maxp1, 25) !! array of dimension `(maxp1,25)` !! containing the chebyshev moments real(wp) :: rlist2(limexp + 2) !! array of dimension at least `limexp+2` !! containing the part of the epsilon table !! which is still needed for further computations integer :: maxerr !! pointer to the interval with largest error estimate real(wp) :: errmax !! `elist(maxerr)` real(wp) :: erlast !! error on the interval currently subdivided real(wp) :: area !! sum of the integrals over the subintervals real(wp) :: errsum !! sum of the errors over the subintervals real(wp) :: errbnd !! requested accuracy `max(epsabs,epsrel*abs(result))` real(wp) :: a1, area1, b1, error1 !! variable for the left subinterval real(wp) :: a2, area2, b2, error2 !! variable for the right subinterval integer :: nres !! number of calls to the extrapolation routine integer :: numrl2 !! number of elements 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 real(wp) :: small !! length of the smallest interval considered !! up to now, multiplied by 1.5 real(wp) :: erlarg !! sum of the errors over the intervals larger !! than the smallest interval considered up to now real(wp) :: area12 !! `area1 + area2` real(wp) :: erro12 !! `error1 + error2` logical :: 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 logical :: noext !! logical variable denoting that extrapolation !! is no longer allowed (true value) real(wp) :: abseps, correc, defab1, defab2, & defabs, domega, dres, ertest, resabs, & reseps, res3la(3), width integer :: id, ierro, iroff1, iroff2, iroff3, & jupbnd, k, ksgn, ktmin, nev, nrmax, nrmom logical :: extall, done, test ! test on validity of parameters Ier = 0 Neval = 0 Last = 0 Result = 0.0_wp Abserr = 0.0_wp Alist(1) = a Blist(1) = b Rlist(1) = 0.0_wp Elist(1) = 0.0_wp Iord(1) = 0 Nnlog(1) = 0 if ((Integr /= 1 .and. Integr /= 2) .or. & (Epsabs <= 0.0_wp .and. Epsrel < max(50.0_wp*epmach, 0.5e-28_wp)) & .or. Icall < 1 .or. Maxp1 < 1) Ier = 6 if (Ier == 6) return done = .false. ! first approximation to the integral domega = abs(Omega) nrmom = 0 if (Icall <= 1) Momcom = 0 call dqc25f(f, a, b, domega, Integr, nrmom, Maxp1, 0, Result, Abserr, & Neval, defabs, resabs, Momcom, Chebmo) ! test on accuracy. dres = abs(Result) errbnd = max(Epsabs, Epsrel*dres) Rlist(1) = Result Elist(1) = Abserr Iord(1) = 1 if (Abserr <= 100.0_wp*epmach*defabs .and. Abserr > errbnd) & Ier = 2 if (Limit == 1) Ier = 1 if (Ier /= 0 .or. Abserr <= errbnd) then if (Integr == 2 .and. Omega < 0.0_wp) Result = -Result return end if ! initializations errmax = Abserr maxerr = 1 area = Result errsum = Abserr Abserr = oflow nrmax = 1 extrap = .false. noext = .false. ierro = 0 iroff1 = 0 iroff2 = 0 iroff3 = 0 ktmin = 0 small = abs(b - a)*0.75_wp nres = 0 numrl2 = 0 extall = .false. if (0.5_wp*abs(b - a)*domega <= 2.0_wp) then numrl2 = 1 extall = .true. rlist2(1) = Result end if if (0.25_wp*abs(b - a)*domega <= 2.0_wp) extall = .true. ksgn = -1 if (dres >= (1.0_wp - 50.0_wp*epmach)*defabs) ksgn = 1 ! main do-loop do Last = 2, Limit ! bisect the subinterval with the nrmax-th largest ! error estimate. nrmom = Nnlog(maxerr) + 1 a1 = Alist(maxerr) b1 = 0.5_wp*(Alist(maxerr) + Blist(maxerr)) a2 = b1 b2 = Blist(maxerr) erlast = errmax call dqc25f(f, a1, b1, domega, Integr, nrmom, Maxp1, 0, area1, & error1, nev, resabs, defab1, Momcom, Chebmo) Neval = Neval + nev call dqc25f(f, a2, b2, domega, Integr, nrmom, Maxp1, 1, area2, & error2, nev, resabs, defab2, Momcom, Chebmo) Neval = Neval + nev ! 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) <= 0.1e-4_wp*abs(area12) & .and. erro12 >= 0.99_wp*errmax) then if (extrap) iroff2 = iroff2 + 1 if (.not. extrap) iroff1 = iroff1 + 1 end if if (Last > 10 .and. erro12 > errmax) iroff3 = iroff3 + 1 end if Rlist(maxerr) = area1 Rlist(Last) = area2 Nnlog(maxerr) = nrmom Nnlog(Last) = nrmom errbnd = max(Epsabs, Epsrel*abs(area)) ! test for roundoff error and eventually set error flag. if (iroff1 + iroff2 >= 10 .or. iroff3 >= 20) Ier = 2 if (iroff2 >= 5) ierro = 3 ! set error flag in the case that the number of ! subintervals equals limit. if (Last == Limit) Ier = 1 ! set error flag in the case of bad integrand behaviour ! at a point of the integration range. if (max(abs(a1), abs(b2)) <= (1.0_wp + 100.0_wp*epmach) & *(abs(a2) + 1000.0_wp*uflow)) Ier = 4 ! append the newly-created intervals to the list. if (error2 > error1) then Alist(maxerr) = a2 Alist(Last) = a1 Blist(Last) = b1 Rlist(maxerr) = area2 Rlist(Last) = area1 Elist(maxerr) = error2 Elist(Last) = error1 else Alist(Last) = a2 Blist(maxerr) = b1 Blist(Last) = b2 Elist(maxerr) = error1 Elist(Last) = error2 end if ! call subroutine dqpsrt to maintain the descending ordering ! in the list of error estimates and select the subinterval ! with nrmax-th largest error estimate (to bisected next). call dqpsrt(Limit, Last, maxerr, errmax, Elist, Iord, nrmax) if (errsum <= errbnd) then ! ***jump out of do-loop done = .true. exit end if if (Ier /= 0) exit if (Last == 2 .and. extall) then small = small*0.5_wp numrl2 = numrl2 + 1 rlist2(numrl2) = area else if (noext) cycle test = .true. if (extall) then erlarg = erlarg - erlast if (abs(b1 - a1) > small) erlarg = erlarg + erro12 if (extrap) test = .false. end if if (test) then ! test whether the interval to be bisected next is the ! smallest interval. width = abs(Blist(maxerr) - Alist(maxerr)) if (width > small) cycle if (extall) then extrap = .true. nrmax = 2 else ! test whether we can start with the extrapolation procedure ! (we do this if we integrate over the next interval with ! use of a gauss-kronrod rule - see subroutine dqc25f). small = small*0.5_wp if (0.25_wp*width*domega > 2.0_wp) cycle extall = .true. ertest = errbnd erlarg = errsum cycle end if end if if (ierro /= 3 .and. erlarg > ertest) then ! the smallest interval has the largest error. ! before bisecting decrease the sum of the errors over ! the larger intervals (erlarg) and perform extrapolation. jupbnd = Last if (Last > (Limit/2 + 2)) jupbnd = Limit + 3 - Last id = nrmax do k = id, jupbnd maxerr = Iord(nrmax) errmax = Elist(maxerr) if (abs(Blist(maxerr) - Alist(maxerr)) > small) & cycle nrmax = nrmax + 1 end do end if ! perform extrapolation. numrl2 = numrl2 + 1 rlist2(numrl2) = area if (numrl2 >= 3) then call dqelg(numrl2, rlist2, reseps, abseps, res3la, nres) ktmin = ktmin + 1 if (ktmin > 5 .and. Abserr < 0.1e-02_wp*errsum) Ier = 5 if (abseps < Abserr) then ktmin = 0 Abserr = abseps Result = reseps correc = erlarg ertest = max(Epsabs, Epsrel*abs(reseps)) ! ***jump out of do-loop if (Abserr <= ertest) exit end if ! prepare bisection of the smallest interval. if (numrl2 == 1) noext = .true. if (Ier == 5) exit end if maxerr = Iord(1) errmax = Elist(maxerr) nrmax = 1 extrap = .false. small = small*0.5_wp erlarg = errsum cycle end if ertest = errbnd erlarg = errsum end do final : block if (done) exit final ! set the final result. if (Abserr /= oflow .and. nres /= 0) then if (Ier + ierro /= 0) then if (ierro == 3) Abserr = Abserr + correc if (Ier == 0) Ier = 3 if (Result == 0.0_wp .or. area == 0.0_wp) then if (Abserr > errsum) exit final if (area == 0.0_wp) then if (Ier > 2) Ier = Ier - 1 if (Integr == 2 .and. Omega < 0.0_wp) Result = -Result return end if elseif (Abserr/abs(Result) > errsum/abs(area)) then exit final end if end if ! test on divergence. if (ksgn /= (-1) .or. max(abs(Result), abs(area)) & > defabs*0.01_wp) then if (0.01_wp > (Result/area) .or. (Result/area) & > 100.0_wp .or. errsum >= abs(area)) Ier = 6 end if if (Ier > 2) Ier = Ier - 1 if (Integr == 2 .and. Omega < 0.0_wp) Result = -Result return end if end block final ! compute global integral sum. Result = 0.0_wp do k = 1, Last Result = Result + Rlist(k) end do Abserr = errsum if (Ier > 2) Ier = Ier - 1 if (Integr == 2 .and. Omega < 0.0_wp) Result = -Result end subroutine dqawoe