subroutine dqk31(f,a,b,result,abserr,resabs,resasc) c***begin prologue dqk31 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 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 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 - double precison c estimate of the modulus of the modulus, 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 dqk31 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(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 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) / 0.0307532419 9611726835 4628393577 204 d0 / data wg ( 2) / 0.0703660474 8810812470 9267416450 667 d0 / data wg ( 3) / 0.1071592204 6717193501 1869546685 869 d0 / data wg ( 4) / 0.1395706779 2615431444 7804794511 028 d0 / data wg ( 5) / 0.1662692058 1699393355 3200860481 209 d0 / data wg ( 6) / 0.1861610000 1556221102 6800561866 423 d0 / data wg ( 7) / 0.1984314853 2711157645 6118326443 839 d0 / data wg ( 8) / 0.2025782419 2556127288 0620199967 519 d0 / c data xgk ( 1) / 0.9980022986 9339706028 5172840152 271 d0 / data xgk ( 2) / 0.9879925180 2048542848 9565718586 613 d0 / data xgk ( 3) / 0.9677390756 7913913425 7347978784 337 d0 / data xgk ( 4) / 0.9372733924 0070590430 7758947710 209 d0 / data xgk ( 5) / 0.8972645323 4408190088 2509656454 496 d0 / data xgk ( 6) / 0.8482065834 1042721620 0648320774 217 d0 / data xgk ( 7) / 0.7904185014 4246593296 7649294817 947 d0 / data xgk ( 8) / 0.7244177313 6017004741 6186054613 938 d0 / data xgk ( 9) / 0.6509967412 9741697053 3735895313 275 d0 / data xgk ( 10) / 0.5709721726 0853884753 7226737253 911 d0 / data xgk ( 11) / 0.4850818636 4023968069 3655740232 351 d0 / data xgk ( 12) / 0.3941513470 7756336989 7207370981 045 d0 / data xgk ( 13) / 0.2991800071 5316881216 6780024266 389 d0 / data xgk ( 14) / 0.2011940939 9743452230 0628303394 596 d0 / data xgk ( 15) / 0.1011420669 1871749902 7074231447 392 d0 / data xgk ( 16) / 0.0000000000 0000000000 0000000000 000 d0 / c data wgk ( 1) / 0.0053774798 7292334898 7792051430 128 d0 / data wgk ( 2) / 0.0150079473 2931612253 8374763075 807 d0 / data wgk ( 3) / 0.0254608473 2671532018 6874001019 653 d0 / data wgk ( 4) / 0.0353463607 9137584622 2037948478 360 d0 / data wgk ( 5) / 0.0445897513 2476487660 8227299373 280 d0 / data wgk ( 6) / 0.0534815246 9092808726 5343147239 430 d0 / data wgk ( 7) / 0.0620095678 0067064028 5139230960 803 d0 / data wgk ( 8) / 0.0698541213 1872825870 9520077099 147 d0 / data wgk ( 9) / 0.0768496807 5772037889 4432777482 659 d0 / data wgk ( 10) / 0.0830805028 2313302103 8289247286 104 d0 / data wgk ( 11) / 0.0885644430 5621177064 7275443693 774 d0 / data wgk ( 12) / 0.0931265981 7082532122 5486872747 346 d0 / data wgk ( 13) / 0.0966427269 8362367850 5179907627 589 d0 / data wgk ( 14) / 0.0991735987 2179195933 2393173484 603 d0 / data wgk ( 15) / 0.1007698455 2387559504 4946662617 570 d0 / data wgk ( 16) / 0.1013300070 1479154901 7374792767 493 d0 / 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***first executable statement dqk31 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 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 = dabs(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)*(dabs(fval1)+dabs(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)*(dabs(fval1)+dabs(fval2)) 15 continue reskh = resk*0.5d+00 resasc = wgk(16)*dabs(fc-reskh) do 20 j=1,15 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