dqk15_2d_outer.f
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! fvn comment :
! Modified version of the dqk15 quadpack routine from http://www.netlib.org/quadpack
!
! + The external 'f' function is a 2 parameters function f(x,y). The routine
! takes two more parameters 'g' and 'h' which are two external functions :
! g represent the lower bound of the integral for y parameter
! h represent the higher bound of the integral for y parameter
! The routine compute the double integral of function f with x between a and b
! and y between g(x) and h(x)
subroutine dqk15_2d_outer(f,a,b,g,h,result,abserr,resabs, &
resasc,epsabs,epsrel,key,limit)
!***begin prologue dqk15
!***date written 800101 (yymmdd)
!***revision date 830518 (yymmdd)
!***category no. h2a1a2
!***keywords 15-point gauss-kronrod rules
!***author piessens,robert,appl. math. & progr. div. - k.u.leuven
! de doncker,elise,appl. math. & progr. div - k.u.leuven
!***purpose to compute i = integral of f over (a,b), with error
! estimate
! j = integral of abs(f) over (a,b)
!***description
!
! integration rules
! standard fortran subroutine
! double precision version
!
! parameters
! on entry
! f - double precision
! function subprogram defining the integrand
! function f(x). the actual name for f needs to be
! declared e x t e r n a l in the calling program.
!
! a - double precision
! lower limit of integration
!
! b - double precision
! upper limit of integration
!
! on return
! result - double precision
! approximation to the integral i
! result is computed by applying the 15-point
! kronrod rule (resk) obtained by optimal addition
! of abscissae to the7-point gauss rule(resg).
!
! abserr - double precision
! estimate of the modulus of the absolute error,
! which should not exceed abs(i-result)
!
! resabs - double precision
! approximation to the integral j
!
! resasc - double precision
! approximation to the integral of abs(f-i/(b-a))
! over (a,b)
!
!***references (none)
!***routines called d1mach
!***end prologue dqk15
!
double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1, &
epmach,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc, &
resg,resk,reskh,result,uflow,wg,wgk,xgk
integer j,jtw,jtwm1
double precision,external :: f,g,h
double precision :: epsabs,epsrel,eval_abserr
double precision :: eval_res
integer :: limit,key,eval_ier
!
dimension fv1(7),fv2(7),wg(4),wgk(8),xgk(8)
!
! the abscissae and weights are given for the interval (-1,1).
! because of symmetry only the positive abscissae and their
! corresponding weights are given.
!
! xgk - abscissae of the 15-point kronrod rule
! xgk(2), xgk(4), ... abscissae of the 7-point
! gauss rule
! xgk(1), xgk(3), ... abscissae which are optimally
! added to the 7-point gauss rule
!
! wgk - weights of the 15-point kronrod rule
!
! wg - weights of the 7-point gauss rule
!
!
! gauss quadrature weights and kronron quadrature abscissae and weights
! as evaluated with 80 decimal digit arithmetic by l. w. fullerton,
! bell labs, nov. 1981.
!
data wg ( 1) / 0.129484966168869693270611432679082d0 /
data wg ( 2) / 0.279705391489276667901467771423780d0 /
data wg ( 3) / 0.381830050505118944950369775488975d0 /
data wg ( 4) / 0.417959183673469387755102040816327d0 /
!
data xgk ( 1) / 0.991455371120812639206854697526329d0 /
data xgk ( 2) / 0.949107912342758524526189684047851d0 /
data xgk ( 3) / 0.864864423359769072789712788640926d0 /
data xgk ( 4) / 0.741531185599394439863864773280788d0 /
data xgk ( 5) / 0.586087235467691130294144838258730d0 /
data xgk ( 6) / 0.405845151377397166906606412076961d0 /
data xgk ( 7) / 0.207784955007898467600689403773245d0 /
data xgk ( 8) / 0.000000000000000000000000000000000d0 /
!
data wgk ( 1) / 0.022935322010529224963732008058970d0 /
data wgk ( 2) / 0.063092092629978553290700663189204d0 /
data wgk ( 3) / 0.104790010322250183839876322541518d0 /
data wgk ( 4) / 0.140653259715525918745189590510238d0 /
data wgk ( 5) / 0.169004726639267902826583426598550d0 /
data wgk ( 6) / 0.190350578064785409913256402421014d0 /
data wgk ( 7) / 0.204432940075298892414161999234649d0 /
data wgk ( 8) / 0.209482141084727828012999174891714d0 /
!
!
! list of major variables
! -----------------------
!
! centr - mid point of the interval
! hlgth - half-length of the interval
! absc - abscissa
! fval* - function value
! resg - result of the 7-point gauss formula
! resk - result of the 15-point kronrod formula
! reskh - approximation to the mean value of f over (a,b),
! i.e. to i/(b-a)
!
! machine dependent constants
! ---------------------------
!
! epmach is the largest relative spacing.
! uflow is the smallest positive magnitude.
!
!***first executable statement dqk15
epmach = d1mach(4)
uflow = d1mach(1)
!
centr = 0.5d+00*(a+b)
hlgth = 0.5d+00*(b-a)
dhlgth = dabs(hlgth)
!
! compute the 15-point kronrod approximation to
! the integral, and estimate the absolute error.
!
! fc = f(centr)
call fvn_d_integ_2_inner_gk(f,centr,g(centr),h(centr),epsabs, &
epsrel,key,eval_res,eval_abserr,eval_ier,limit)
fc=eval_res
resg = fc*wg(4)
resk = fc*wgk(8)
resabs = dabs(resk)
do 10 j=1,3
jtw = j*2
absc = hlgth*xgk(jtw)
! fval1 = f(centr-absc)
call fvn_d_integ_2_inner_gk(f,centr-absc,g(centr-absc), &
h(centr-absc),epsabs,epsrel,key,eval_res,eval_abserr, &
eval_ier,limit)
fval1=eval_res
! fval2 = f(centr+absc)
call fvn_d_integ_2_inner_gk(f,centr+absc,g(centr+absc), &
h(centr+absc),epsabs,epsrel,key,eval_res,eval_abserr, &
eval_ier,limit)
fval2=eval_res
fv1(jtw) = fval1
fv2(jtw) = fval2
fsum = fval1+fval2
resg = resg+wg(j)*fsum
resk = resk+wgk(jtw)*fsum
resabs = resabs+wgk(jtw)*(dabs(fval1)+dabs(fval2))
10 continue
do 15 j = 1,4
jtwm1 = j*2-1
absc = hlgth*xgk(jtwm1)
! fval1 = f(centr-absc)
call fvn_d_integ_2_inner_gk(f,centr-absc,g(centr-absc), &
h(centr-absc),epsabs,epsrel,key,eval_res,eval_abserr, &
eval_ier,limit)
fval1=eval_res
! fval2 = f(centr+absc)
call fvn_d_integ_2_inner_gk(f,centr+absc,g(centr+absc), &
h(centr+absc),epsabs,epsrel,key,eval_res,eval_abserr, &
eval_ier,limit)
fval2=eval_res
fv1(jtwm1) = fval1
fv2(jtwm1) = fval2
fsum = fval1+fval2
resk = resk+wgk(jtwm1)*fsum
resabs = resabs+wgk(jtwm1)*(dabs(fval1)+dabs(fval2))
15 continue
reskh = resk*0.5d+00
resasc = wgk(8)*dabs(fc-reskh)
do 20 j=1,7
resasc = resasc+wgk(j)*(dabs(fv1(j)-reskh)+dabs(fv2(j)-reskh))
20 continue
result = resk*hlgth
resabs = resabs*dhlgth
resasc = resasc*dhlgth
abserr = dabs((resk-resg)*hlgth)
if(resasc.ne.0.0d+00.and.abserr.ne.0.0d+00) &
abserr = resasc*dmin1(0.1d+01,(0.2d+03*abserr/resasc)**1.5d+00)
if(resabs.gt.uflow/(0.5d+02*epmach)) abserr = dmax1 &
((epmach*0.5d+02)*resabs,abserr)
return
end subroutine