subroutine qk31(f,a,b,result,abserr,resabs,resasc) c***begin prologue qk31 c***date written 800101 (yymmdd) c***revision date 830518 (yymmdd) c***category no. h2a1a2 c***keywords 31-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 calling 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 31-point c gauss-kronrod rule (resk), obtained by optimal c addition of abscissae to the 15-point gauss c rule (resg). c c abserr - real c estimate of the modulus of the modulus, 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 qk31 real a,absc,abserr,b,centr,dhlgth,epmach,f,fc,fsum,fval1,fval2, * fv1,fv2,hlgth,resabs,resasc,resg,resk,reskh,result,r1mach,uflow, * wg,wgk,xgk integer j,jtw,jtwm1 external f c dimension fv1(15),fv2(15),xgk(16),wgk(16),wg(8) 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 31-point kronrod rule c xgk(2), xgk(4), ... abscissae of the 15-point c gauss rule c xgk(1), xgk(3), ... abscissae which are optimally c added to the 15-point gauss rule c c wgk - weights of the 31-point kronrod rule c c wg - weights of the 15-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),xgk(12),xgk(13),xgk(14),xgk(15), * xgk(16)/ * 0.9980022986933971e+00, 0.9879925180204854e+00, * 0.9677390756791391e+00, 0.9372733924007059e+00, * 0.8972645323440819e+00, 0.8482065834104272e+00, * 0.7904185014424659e+00, 0.7244177313601700e+00, * 0.6509967412974170e+00, 0.5709721726085388e+00, * 0.4850818636402397e+00, 0.3941513470775634e+00, * 0.2991800071531688e+00, 0.2011940939974345e+00, * 0.1011420669187175e+00, 0.0e+00 / data wgk(1),wgk(2),wgk(3),wgk(4),wgk(5),wgk(6),wgk(7),wgk(8), * wgk(9),wgk(10),wgk(11),wgk(12),wgk(13),wgk(14),wgk(15), * wgk(16)/ * 0.5377479872923349e-02, 0.1500794732931612e-01, * 0.2546084732671532e-01, 0.3534636079137585e-01, * 0.4458975132476488e-01, 0.5348152469092809e-01, * 0.6200956780067064e-01, 0.6985412131872826e-01, * 0.7684968075772038e-01, 0.8308050282313302e-01, * 0.8856444305621177e-01, 0.9312659817082532e-01, * 0.9664272698362368e-01, 0.9917359872179196e-01, * 0.1007698455238756e+00, 0.1013300070147915e+00/ data wg(1),wg(2),wg(3),wg(4),wg(5),wg(6),wg(7),wg(8)/ * 0.3075324199611727e-01, 0.7036604748810812e-01, * 0.1071592204671719e+00, 0.1395706779261543e+00, * 0.1662692058169939e+00, 0.1861610000155622e+00, * 0.1984314853271116e+00, 0.2025782419255613e+00/ c c c list of major variables 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 15-point gauss formula c resk - result of the 31-point kronrod formula c reskh - approximation to the mean value of f over (a,b), c i.e. to i/(b-a) c c machine dependent constants c --------------------------- c epmach is the largest relative spacing. c uflow is the smallest positive magnitude. c c***first executable statement qk31 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 31-point kronrod approximation to c the integral, and estimate the absolute error. c fc = f(centr) resg = wg(8)*fc resk = wgk(16)*fc resabs = abs(resk) do 10 j=1,7 jtw = j*2 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 = 1,8 jtwm1 = j*2-1 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(16)*abs(fc-reskh) do 20 j=1,15 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