subroutine qk21(f,a,b,result,abserr,resabs,resasc) c***begin prologue qk21 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 real version c c parameters c on entry c f - real 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 - real c lower limit of integration c c b - real c upper limit of integration c c on return c result - real 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 - real c estimate of the modulus of the absolute error, c which should not exceed abs(i-result) c c resabs - real c approximation to the integral j c c resasc - real c approximation to the integral of abs(f-i/(b-a)) c over (a,b) c c***references (none) c***routines called r1mach c***end prologue qk21 c real a,absc,abserr,b,centr,dhlgth,epmach,f,fc,fsum,fval1,fval2, * fv1,fv2,hlgth,resabs,resg,resk,reskh,result,r1mach,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 data xgk(1),xgk(2),xgk(3),xgk(4),xgk(5),xgk(6),xgk(7), * xgk(8),xgk(9),xgk(10),xgk(11)/ * 0.9956571630258081e+00, 0.9739065285171717e+00, * 0.9301574913557082e+00, 0.8650633666889845e+00, * 0.7808177265864169e+00, 0.6794095682990244e+00, * 0.5627571346686047e+00, 0.4333953941292472e+00, * 0.2943928627014602e+00, 0.1488743389816312e+00, * 0.0000000000000000e+00/ c data wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7), * wgk(8),wgk(9),wgk(10),wgk(11)/ * 0.1169463886737187e-01, 0.3255816230796473e-01, * 0.5475589657435200e-01, 0.7503967481091995e-01, * 0.9312545458369761e-01, 0.1093871588022976e+00, * 0.1234919762620659e+00, 0.1347092173114733e+00, * 0.1427759385770601e+00, 0.1477391049013385e+00, * 0.1494455540029169e+00/ c data wg(1),wg(2),wg(3),wg(4),wg(5)/ * 0.6667134430868814e-01, 0.1494513491505806e+00, * 0.2190863625159820e+00, 0.2692667193099964e+00, * 0.2955242247147529e+00/ 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 qk21 epmach = r1mach(4) uflow = r1mach(1) c centr = 0.5e+00*(a+b) hlgth = 0.5e+00*(b-a) dhlgth = abs(hlgth) c c compute the 21-point kronrod approximation to c the integral, and estimate the absolute error. c resg = 0.0e+00 fc = f(centr) resk = wgk(11)*fc resabs = abs(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)*(abs(fval1)+abs(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)*(abs(fval1)+abs(fval2)) 15 continue reskh = resk*0.5e+00 resasc = wgk(11)*abs(fc-reskh) do 20 j=1,10 resasc = resasc+wgk(j)*(abs(fv1(j)-reskh)+abs(fv2(j)-reskh)) 20 continue result = resk*hlgth resabs = resabs*dhlgth resasc = resasc*dhlgth abserr = abs((resk-resg)*hlgth) if(resasc.ne.0.0e+00.and.abserr.ne.0.0e+00) * abserr = resasc*amin1(0.1e+01, * (0.2e+03*abserr/resasc)**1.5e+00) if(resabs.gt.uflow/(0.5e+02*epmach)) abserr = amax1 * ((epmach*0.5e+02)*resabs,abserr) return end