subroutine dqk15r(f,a,b,result,abserr,resabs,resasc,i) c***begin prologue dqk15 c***date written 800101 (yymmdd) c***revision date 830518 (yymmdd) c***category no. h2a1a2 c***keywords 15-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 calling 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 15-point c kronrod rule (resk) obtained by optimal addition c of abscissae to the7-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 dqk15 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(7),fv2(7),wg(4),wgk(8),xgk(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 15-point kronrod rule c xgk(2), xgk(4), ... abscissae of the 7-point c gauss rule c xgk(1), xgk(3), ... abscissae which are optimally c added to the 7-point gauss rule c c wgk - weights of the 15-point kronrod rule c c wg - weights of the 7-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) / 0.1294849661 6886969327 0611432679 082 d0 / data wg ( 2) / 0.2797053914 8927666790 1467771423 780 d0 / data wg ( 3) / 0.3818300505 0511894495 0369775488 975 d0 / data wg ( 4) / 0.4179591836 7346938775 5102040816 327 d0 / c data xgk ( 1) / 0.9914553711 2081263920 6854697526 329 d0 / data xgk ( 2) / 0.9491079123 4275852452 6189684047 851 d0 / data xgk ( 3) / 0.8648644233 5976907278 9712788640 926 d0 / data xgk ( 4) / 0.7415311855 9939443986 3864773280 788 d0 / data xgk ( 5) / 0.5860872354 6769113029 4144838258 730 d0 / data xgk ( 6) / 0.4058451513 7739716690 6606412076 961 d0 / data xgk ( 7) / 0.2077849550 0789846760 0689403773 245 d0 / data xgk ( 8) / 0.0000000000 0000000000 0000000000 000 d0 / c data wgk ( 1) / 0.0229353220 1052922496 3732008058 970 d0 / data wgk ( 2) / 0.0630920926 2997855329 0700663189 204 d0 / data wgk ( 3) / 0.1047900103 2225018383 9876322541 518 d0 / data wgk ( 4) / 0.1406532597 1552591874 5189590510 238 d0 / data wgk ( 5) / 0.1690047266 3926790282 6583426598 550 d0 / data wgk ( 6) / 0.1903505780 6478540991 3256402421 014 d0 / data wgk ( 7) / 0.2044329400 7529889241 4161999234 649 d0 / data wgk ( 8) / 0.2094821410 8472782801 2999174891 714 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 7-point gauss formula c resk - result of the 15-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 c epmach is the largest relative spacing. c uflow is the smallest positive magnitude. c c***first executable statement dqk15 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 15-point kronrod approximation to c the integral, and estimate the absolute error. c fc = f(centr,i) resg = fc*wg(4) resk = fc*wgk(8) resabs = dabs(resk) do 10 j=1,3 jtw = j*2 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 = 1,4 jtwm1 = j*2-1 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(8)*dabs(fc-reskh) do 20 j=1,7 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