subroutine dqk61r(f,a,b,result,abserr,resabs,resasc,i) c***begin prologue dqk61 c***date written 800101 (yymmdd) c***revision date 830518 (yymmdd) c***category no. h2a1a2 c***keywords 61-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 dabs(f) over (a,b) c***description c c integration rule c standard fortran subroutine c double precision version c 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 61-point c kronrod rule (resk) obtained by optimal addition of c abscissae to the 30-point gauss rule (resg). c c abserr - double precision c estimate of the modulus of the absolute error, c which should equal or exceed dabs(i-result) c c resabs - double precision c approximation to the integral j c c resasc - double precision c approximation to the integral of dabs(f-i/(b-a)) c c c***references (none) c***routines called d1mach c***end prologue dqk61 c double precision a,dabsc,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(30),fv2(30),xgk(31),wgk(31),wg(15) c c the abscissae and weights are given for the c interval (-1,1). because of symmetry only the positive c abscissae and their corresponding weights are given. c c xgk - abscissae of the 61-point kronrod rule c xgk(2), xgk(4) ... abscissae of the 30-point c gauss rule c xgk(1), xgk(3) ... optimally added abscissae c to the 30-point gauss rule c c wgk - weights of the 61-point kronrod rule c c wg - weigths of the 30-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) / 0.0079681924 9616660561 5465883474 674 d0 / data wg ( 2) / 0.0184664683 1109095914 2302131912 047 d0 / data wg ( 3) / 0.0287847078 8332336934 9719179611 292 d0 / data wg ( 4) / 0.0387991925 6962704959 6801936446 348 d0 / data wg ( 5) / 0.0484026728 3059405290 2938140422 808 d0 / data wg ( 6) / 0.0574931562 1761906648 1721689402 056 d0 / data wg ( 7) / 0.0659742298 8218049512 8128515115 962 d0 / data wg ( 8) / 0.0737559747 3770520626 8243850022 191 d0 / data wg ( 9) / 0.0807558952 2942021535 4694938460 530 d0 / data wg ( 10) / 0.0868997872 0108297980 2387530715 126 d0 / data wg ( 11) / 0.0921225222 3778612871 7632707087 619 d0 / data wg ( 12) / 0.0963687371 7464425963 9468626351 810 d0 / data wg ( 13) / 0.0995934205 8679526706 2780282103 569 d0 / data wg ( 14) / 0.1017623897 4840550459 6428952168 554 d0 / data wg ( 15) / 0.1028526528 9355884034 1285636705 415 d0 / c data xgk ( 1) / 0.9994844100 5049063757 1325895705 811 d0 / data xgk ( 2) / 0.9968934840 7464954027 1630050918 695 d0 / data xgk ( 3) / 0.9916309968 7040459485 8628366109 486 d0 / data xgk ( 4) / 0.9836681232 7974720997 0032581605 663 d0 / data xgk ( 5) / 0.9731163225 0112626837 4693868423 707 d0 / data xgk ( 6) / 0.9600218649 6830751221 6871025581 798 d0 / data xgk ( 7) / 0.9443744447 4855997941 5831324037 439 d0 / data xgk ( 8) / 0.9262000474 2927432587 9324277080 474 d0 / data xgk ( 9) / 0.9055733076 9990779854 6522558925 958 d0 / data xgk ( 10) / 0.8825605357 9205268154 3116462530 226 d0 / data xgk ( 11) / 0.8572052335 4606109895 8658510658 944 d0 / data xgk ( 12) / 0.8295657623 8276839744 2898119732 502 d0 / data xgk ( 13) / 0.7997278358 2183908301 3668942322 683 d0 / data xgk ( 14) / 0.7677774321 0482619491 7977340974 503 d0 / data xgk ( 15) / 0.7337900624 5322680472 6171131369 528 d0 / data xgk ( 16) / 0.6978504947 9331579693 2292388026 640 d0 / data xgk ( 17) / 0.6600610641 2662696137 0053668149 271 d0 / data xgk ( 18) / 0.6205261829 8924286114 0477556431 189 d0 / data xgk ( 19) / 0.5793452358 2636169175 6024932172 540 d0 / data xgk ( 20) / 0.5366241481 4201989926 4169793311 073 d0 / data xgk ( 21) / 0.4924804678 6177857499 3693061207 709 d0 / data xgk ( 22) / 0.4470337695 3808917678 0609900322 854 d0 / data xgk ( 23) / 0.4004012548 3039439253 5476211542 661 d0 / data xgk ( 24) / 0.3527047255 3087811347 1037207089 374 d0 / data xgk ( 25) / 0.3040732022 7362507737 2677107199 257 d0 / data xgk ( 26) / 0.2546369261 6788984643 9805129817 805 d0 / data xgk ( 27) / 0.2045251166 8230989143 8957671002 025 d0 / data xgk ( 28) / 0.1538699136 0858354696 3794672743 256 d0 / data xgk ( 29) / 0.1028069379 6673703014 7096751318 001 d0 / data xgk ( 30) / 0.0514718425 5531769583 3025213166 723 d0 / data xgk ( 31) / 0.0000000000 0000000000 0000000000 000 d0 / c data wgk ( 1) / 0.0013890136 9867700762 4551591226 760 d0 / data wgk ( 2) / 0.0038904611 2709988405 1267201844 516 d0 / data wgk ( 3) / 0.0066307039 1593129217 3319826369 750 d0 / data wgk ( 4) / 0.0092732796 5951776342 8441146892 024 d0 / data wgk ( 5) / 0.0118230152 5349634174 2232898853 251 d0 / data wgk ( 6) / 0.0143697295 0704580481 2451432443 580 d0 / data wgk ( 7) / 0.0169208891 8905327262 7572289420 322 d0 / data wgk ( 8) / 0.0194141411 9394238117 3408951050 128 d0 / data wgk ( 9) / 0.0218280358 2160919229 7167485738 339 d0 / data wgk ( 10) / 0.0241911620 7808060136 5686370725 232 d0 / data wgk ( 11) / 0.0265099548 8233310161 0601709335 075 d0 / data wgk ( 12) / 0.0287540487 6504129284 3978785354 334 d0 / data wgk ( 13) / 0.0309072575 6238776247 2884252943 092 d0 / data wgk ( 14) / 0.0329814470 5748372603 1814191016 854 d0 / data wgk ( 15) / 0.0349793380 2806002413 7499670731 468 d0 / data wgk ( 16) / 0.0368823646 5182122922 3911065617 136 d0 / data wgk ( 17) / 0.0386789456 2472759295 0348651532 281 d0 / data wgk ( 18) / 0.0403745389 5153595911 1995279752 468 d0 / data wgk ( 19) / 0.0419698102 1516424614 7147541285 970 d0 / data wgk ( 20) / 0.0434525397 0135606931 6831728117 073 d0 / data wgk ( 21) / 0.0448148001 3316266319 2355551616 723 d0 / data wgk ( 22) / 0.0460592382 7100698811 6271735559 374 d0 / data wgk ( 23) / 0.0471855465 6929915394 5261478181 099 d0 / data wgk ( 24) / 0.0481858617 5708712914 0779492298 305 d0 / data wgk ( 25) / 0.0490554345 5502977888 7528165367 238 d0 / data wgk ( 26) / 0.0497956834 2707420635 7811569379 942 d0 / data wgk ( 27) / 0.0504059214 0278234684 0893085653 585 d0 / data wgk ( 28) / 0.0508817958 9874960649 2297473049 805 d0 / data wgk ( 29) / 0.0512215478 4925877217 0656282604 944 d0 / data wgk ( 30) / 0.0514261285 3745902593 3862879215 781 d0 / data wgk ( 31) / 0.0514947294 2945156755 8340433647 099 d0 / c c list of major variables c ----------------------- c c centr - mid point of the interval c hlgth - half-length of the interval c dabsc - abscissa c fval* - function value c resg - result of the 30-point gauss rule c resk - result of the 61-point kronrod rule c reskh - approximation to the mean value of f c over (a,b), 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 epmach = d1mach(4) uflow = d1mach(1) c centr = 0.5d+00*(b+a) hlgth = 0.5d+00*(b-a) dhlgth = dabs(hlgth) c c compute the 61-point kronrod approximation to the c integral, and estimate the absolute error. c c***first executable statement dqk61 resg = 0.0d+00 fc = f(centr,i) resk = wgk(31)*fc resabs = dabs(resk) do 10 j=1,15 jtw = j*2 dabsc = hlgth*xgk(jtw) fval1 = f(centr-dabsc,i) fval2 = f(centr+dabsc,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=1,15 jtwm1 = j*2-1 dabsc = hlgth*xgk(jtwm1) fval1 = f(centr-dabsc,i) fval2 = f(centr+dabsc,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(31)*dabs(fc-reskh) do 20 j=1,30 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