subroutine dqk21r(f,a,b,result,abserr,resabs,resasc,i)
c***begin prologue dqk21
c***date written 800101 (yymmdd)
c***revision date 830518 (yymmdd)
c***category no. h2a1a2
c***keywords 21-point gauss-kronrod rules
c***author piessens,robert,appl. math. & progr. div. - k.u.leuven
c de doncker,elise,appl. math. & progr. div. - k.u.leuven
c***purpose to compute i = integral of f over (a,b), with error
c estimate
c j = integral of abs(f) over (a,b)
c***description
c
c integration rules
c standard fortran subroutine
c double precision version
c
c parameters
c on entry
c f - double precision
c function subprogram defining the integrand
c function f(x). the actual name for f needs to be
c declared e x t e r n a l in the driver program.
c
c a - double precision
c lower limit of integration
c
c b - double precision
c upper limit of integration
c
c on return
c result - double precision
c approximation to the integral i
c result is computed by applying the 21-point
c kronrod rule (resk) obtained by optimal addition
c of abscissae to the 10-point gauss rule (resg).
c
c abserr - double precision
c estimate of the modulus of the absolute error,
c which should not exceed abs(i-result)
c
c resabs - double precision
c approximation to the integral j
c
c resasc - double precision
c approximation to the integral of abs(f-i/(b-a))
c over (a,b)
c
c***references (none)
c***routines called d1mach
c***end prologue dqk21
c
double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1,
* d1mach,epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc,
* resg,resk,reskh,result,uflow,wg,wgk,xgk
integer j,jtw,jtwm1,i
external f
c
dimension fv1(10),fv2(10),wg(5),wgk(11),xgk(11)
c
c the abscissae and weights are given for the interval (-1,1).
c because of symmetry only the positive abscissae and their
c corresponding weights are given.
c
c xgk - abscissae of the 21-point kronrod rule
c xgk(2), xgk(4), ... abscissae of the 10-point
c gauss rule
c xgk(1), xgk(3), ... abscissae which are optimally
c added to the 10-point gauss rule
c
c wgk - weights of the 21-point kronrod rule
c
c wg - weights of the 10-point gauss rule
c
c
c gauss quadrature weights and kronron quadrature abscissae and weights
c as evaluated with 80 decimal digit arithmetic by l. w. fullerton,
c bell labs, nov. 1981.
c
data wg ( 1) / 0.0666713443 0868813759 3568809893 332 d0 /
data wg ( 2) / 0.1494513491 5058059314 5776339657 697 d0 /
data wg ( 3) / 0.2190863625 1598204399 5534934228 163 d0 /
data wg ( 4) / 0.2692667193 0999635509 1226921569 469 d0 /
data wg ( 5) / 0.2955242247 1475287017 3892994651 338 d0 /
c
data xgk ( 1) / 0.9956571630 2580808073 5527280689 003 d0 /
data xgk ( 2) / 0.9739065285 1717172007 7964012084 452 d0 /
data xgk ( 3) / 0.9301574913 5570822600 1207180059 508 d0 /
data xgk ( 4) / 0.8650633666 8898451073 2096688423 493 d0 /
data xgk ( 5) / 0.7808177265 8641689706 3717578345 042 d0 /
data xgk ( 6) / 0.6794095682 9902440623 4327365114 874 d0 /
data xgk ( 7) / 0.5627571346 6860468333 9000099272 694 d0 /
data xgk ( 8) / 0.4333953941 2924719079 9265943165 784 d0 /
data xgk ( 9) / 0.2943928627 0146019813 1126603103 866 d0 /
data xgk ( 10) / 0.1488743389 8163121088 4826001129 720 d0 /
data xgk ( 11) / 0.0000000000 0000000000 0000000000 000 d0 /
c
data wgk ( 1) / 0.0116946388 6737187427 8064396062 192 d0 /
data wgk ( 2) / 0.0325581623 0796472747 8818972459 390 d0 /
data wgk ( 3) / 0.0547558965 7435199603 1381300244 580 d0 /
data wgk ( 4) / 0.0750396748 1091995276 7043140916 190 d0 /
data wgk ( 5) / 0.0931254545 8369760553 5065465083 366 d0 /
data wgk ( 6) / 0.1093871588 0229764189 9210590325 805 d0 /
data wgk ( 7) / 0.1234919762 6206585107 7958109831 074 d0 /
data wgk ( 8) / 0.1347092173 1147332592 8054001771 707 d0 /
data wgk ( 9) / 0.1427759385 7706008079 7094273138 717 d0 /
data wgk ( 10) / 0.1477391049 0133849137 4841515972 068 d0 /
data wgk ( 11) / 0.1494455540 0291690566 4936468389 821 d0 /
c
c
c list of major variables
c -----------------------
c
c centr - mid point of the interval
c hlgth - half-length of the interval
c absc - abscissa
c fval* - function value
c resg - result of the 10-point gauss formula
c resk - result of the 21-point kronrod formula
c reskh - approximation to the mean value of f over (a,b),
c i.e. to i/(b-a)
c
c
c machine dependent constants
c ---------------------------
c
c epmach is the largest relative spacing.
c uflow is the smallest positive magnitude.
c
c***first executable statement dqk21
epmach = d1mach(4)
uflow = d1mach(1)
c
centr = 0.5d+00*(a+b)
hlgth = 0.5d+00*(b-a)
dhlgth = dabs(hlgth)
c
c compute the 21-point kronrod approximation to
c the integral, and estimate the absolute error.
c
resg = 0.0d+00
fc = f(centr,i)
resk = wgk(11)*fc
resabs = dabs(resk)
do 10 j=1,5
jtw = 2*j
absc = hlgth*xgk(jtw)
fval1 = f(centr-absc,i)
fval2 = f(centr+absc,i)
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 = 1,5
jtwm1 = 2*j-1
absc = hlgth*xgk(jtwm1)
fval1 = f(centr-absc,i)
fval2 = f(centr+absc,i)
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(11)*dabs(fc-reskh)
do 20 j=1,10
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
