subroutine dqk21(f,a,b,result,abserr,resabs,resasc) 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 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) resk = wgk(11)*fc resabs = dabs(resk) do 10 j=1,5 jtw = 2*j absc = hlgth*xgk(jtw) fval1 = f(centr-absc) fval2 = f(centr+absc) 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) fval2 = f(centr+absc) 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