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