1D integration of cos(omega*x)*f(x) or sin(omega*x)*f(x)
over a finite interval, adaptive subdivision with extrapolation
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 function |
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| real(kind=wp), | intent(in) | :: | a |
lower limit of integration |
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| real(kind=wp), | intent(in) | :: | b |
upper limit of integration |
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| real(kind=wp), | intent(in) | :: | Omega |
parameter in the integrand weight function |
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| integer, | intent(in) | :: | Integr |
indicates which of the weight functions is used
if |
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| real(kind=wp), | intent(in) | :: | Epsabs |
absolute accuracy requested |
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| real(kind=wp), | intent(in) | :: | Epsrel |
relative accuracy requested
if |
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| real(kind=wp), | intent(out) | :: | Result |
approximation to the integral |
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| real(kind=wp), | intent(out) | :: | Abserr |
estimate of the modulus of the absolute error,
which should equal or exceed |
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| integer, | intent(out) | :: | Neval |
number of integrand evaluations |
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| integer, | intent(out) | :: | Ier |
error messages:
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| integer, | intent(in) | :: | Leniw |
dimensioning parameter for |
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| 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 |
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| integer, | intent(in) | :: | Lenw |
dimensioning parameter for |
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| integer, | intent(out) | :: | Last |
on return, |
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| integer | :: | Iwork(Leniw) |
vector of dimension at least leniw
on return, the first |
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| real(kind=wp) | :: | Work(Lenw) |
vector of dimension at least
note that |
subroutine dqawo(f, a, b, Omega, Integr, Epsabs, Epsrel, Result, Abserr, & Neval, Ier, Leniw, Maxp1, Lenw, Last, Iwork, Work) implicit none procedure(func) :: f !! function subprogram defining the 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 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. 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 !! `(= leniw/2)` has been achieved. one can !! allow more subdivisions by increasing the !! value of leniw (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. !! * 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 interior 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`. !! * ier = 6 the input is invalid, because !! `(epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28))` !! or `(integr/=1 and integr/=2)`, !! or `leniw<2` or `maxp1<1` or !! `lenw<leniw*2+maxp1*25`. !! `result`, `abserr`, `neval`, `last` are set to !! zero. except when `leniw`, `maxp1` or `lenw` are !! invalid, `work(limit*2+1)`, `work(limit*3+1)`, !! `iwork(1)`, `iwork(limit+1)` are set to zero, !! `work(1)` is set to `a` and `work(limit+1)` to !! `b`. integer, intent(in) :: Leniw !! dimensioning parameter for `iwork`. !! `leniw/2` equals the maximum number of subintervals !! allowed in the partition of the given integration !! interval `(a,b)`, `leniw>=2`. !! if `leniw<2`, 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. integer, intent(in) :: Lenw !! dimensioning parameter for `work` !! `lenw` must be at least `leniw*2+maxp1*25`. !! if `lenw<(leniw*2+maxp1*25)`, the routine will !! end with ier = 6. integer, intent(out) :: Last !! on return, `last` equals the number of subintervals !! produced in the subdivision process, which !! determines the number of significant elements !! actually in the work arrays. integer :: Iwork(Leniw) !! vector of dimension at least leniw !! on return, the first `k` elements of which contain !! pointers to the error estimates over the !! subintervals, such that !! `work(limit*3+iwork(1)), .., work(limit*3+iwork(k))` !! form a decreasing !! sequence, with `limit = lenw/2` , and `k = last` !! if `last<=(limit/2+2)`, and `k = limit+1-last` !! otherwise. !! furthermore, `iwork(limit+1), ..., iwork(limit+last)` !! indicate the subdivision levels of the !! subintervals, such that `iwork(limit+i) = l` means !! that the subinterval numbered `i` is of length !! `abs(b-a)*2**(1-l)`. real(wp) :: Work(Lenw) !! vector of dimension at least `lenw`. !! on return: !! !! * `work(1), ..., work(last)` contain the left !! end points of the subintervals in the !! partition of `(a,b)`, !! * `work(limit+1), ..., work(limit+last)` contain !! the right end points, !! * `work(limit*2+1), ..., work(limit*2+last)` contain !! the integral approximations over the !! subintervals, !! * `work(limit*3+1), ..., work(limit*3+last)` !! contain the error estimates. !! * `work(limit*4+1), ..., work(limit*4+maxp1*25)` !! provide space for storing the chebyshev moments. !! !! note that `limit = lenw/2`. integer :: limit, lvl, l1, l2, l3, l4, momcom ! check validity of leniw, maxp1 and lenw. Ier = 6 Neval = 0 Last = 0 Result = 0.0_wp Abserr = 0.0_wp if (Leniw >= 2 .and. Maxp1 >= 1 .and. Lenw >= (Leniw*2 + Maxp1*25)) then ! prepare call for dqawoe limit = Leniw/2 l1 = limit + 1 l2 = limit + l1 l3 = limit + l2 l4 = limit + l3 call dqawoe(f, a, b, Omega, Integr, Epsabs, Epsrel, limit, 1, Maxp1, & Result, Abserr, Neval, Ier, Last, Work(1), Work(l1), & Work(l2), Work(l3), Iwork(1), Iwork(l1), momcom, & Work(l4)) ! call error handler if necessary lvl = 0 end if if (Ier == 6) lvl = 1 if (Ier /= 0) call xerror('abnormal return from dqawo', Ier, lvl) end subroutine dqawo