subroutine qk41(f,a,b,result,abserr,resabs,resasc) c***begin prologue qk41 c***date written 800101 (yymmdd) c***revision date 830518 (yymmdd) c***category no. h2a1a2 c***keywords 41-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 41-point c gauss-kronrod rule (resk) obtained by optimal c addition of abscissae to the 20-point gauss c 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 integal of abs(f-i/(b-a)) c over (a,b) c c***references (none) c***routines called r1mach c***end prologue qk41 c 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(20),fv2(20),xgk(21),wgk(21),wg(10) 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 41-point gauss-kronrod rule c xgk(2), xgk(4), ... abscissae of the 20-point c gauss rule c xgk(1), xgk(3), ... abscissae which are optimally c added to the 20-point gauss rule c c wgk - weights of the 41-point gauss-kronrod rule c c wg - weights of the 20-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),xgk(17),xgk(18),xgk(19),xgk(20),xgk(21)/ * 0.9988590315882777e+00, 0.9931285991850949e+00, * 0.9815078774502503e+00, 0.9639719272779138e+00, * 0.9408226338317548e+00, 0.9122344282513259e+00, * 0.8782768112522820e+00, 0.8391169718222188e+00, * 0.7950414288375512e+00, 0.7463319064601508e+00, * 0.6932376563347514e+00, 0.6360536807265150e+00, * 0.5751404468197103e+00, 0.5108670019508271e+00, * 0.4435931752387251e+00, 0.3737060887154196e+00, * 0.3016278681149130e+00, 0.2277858511416451e+00, * 0.1526054652409227e+00, 0.7652652113349733e-01, * 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), * wgk(17),wgk(18),wgk(19),wgk(20),wgk(21)/ * 0.3073583718520532e-02, 0.8600269855642942e-02, * 0.1462616925697125e-01, 0.2038837346126652e-01, * 0.2588213360495116e-01, 0.3128730677703280e-01, * 0.3660016975820080e-01, 0.4166887332797369e-01, * 0.4643482186749767e-01, 0.5094457392372869e-01, * 0.5519510534828599e-01, 0.5911140088063957e-01, * 0.6265323755478117e-01, 0.6583459713361842e-01, * 0.6864867292852162e-01, 0.7105442355344407e-01, * 0.7303069033278667e-01, 0.7458287540049919e-01, * 0.7570449768455667e-01, 0.7637786767208074e-01, * 0.7660071191799966e-01/ data wg(1),wg(2),wg(3),wg(4),wg(5),wg(6),wg(7),wg(8),wg(9),wg(10)/ * 0.1761400713915212e-01, 0.4060142980038694e-01, * 0.6267204833410906e-01, 0.8327674157670475e-01, * 0.1019301198172404e+00, 0.1181945319615184e+00, * 0.1316886384491766e+00, 0.1420961093183821e+00, * 0.1491729864726037e+00, 0.1527533871307259e+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 20-point gauss formula c resk - result of the 41-point kronrod formula c reskh - approximation to mean value of f over (a,b), i.e. c to i/(b-a) 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 qk41 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 41-point gauss-kronrod approximation to c the integral, and estimate the absolute error. c resg = 0.0e+00 fc = f(centr) resk = wgk(21)*fc resabs = abs(resk) do 10 j=1,10 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 = 1,10 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(21)*abs(fc-reskh) do 20 j=1,20 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.e+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