subroutine qng(f,a,b,epsabs,epsrel,result,abserr,neval,ier) c***begin prologue qng c***date written 800101 (yymmdd) c***revision date 830518 (yymmdd) c***category no. h2a1a1 c***keywords automatic integrator, smooth integrand, c non-adaptive, gauss-kronrod(patterson) c***author piessens,robert,appl. math. & progr. div. - k.u.leuven c de doncker,elise,appl math & progr. div. - k.u.leuven c kahaner,david,nbs - modified (2/82) c***purpose the routine calculates an approximation result to a c given definite integral i = integral of f over (a,b), c hopefully satisfying following claim for accuracy c abs(i-result).le.max(epsabs,epsrel*abs(i)). c***description c c non-adaptive integration c standard fortran subroutine c real version c c f - real version c function subprogram defining the integrand function c f(x). the actual name for f needs to be declared c e x t e r n a l in the driver program. c c a - real version c lower limit of integration c c b - real version c upper limit of integration c c epsabs - real c absolute accuracy requested c epsrel - real c relative accuracy requested c if epsabs.le.0 c and epsrel.lt.max(50*rel.mach.acc.,0.5d-28), c the routine will end with ier = 6. c c on return c result - real c approximation to the integral i c result is obtained by applying the 21-point c gauss-kronrod rule (res21) obtained by optimal c addition of abscissae to the 10-point gauss rule c (res10), or by applying the 43-point rule (res43) c obtained by optimal addition of abscissae to the c 21-point gauss-kronrod rule, or by applying the c 87-point rule (res87) obtained by optimal addition c of abscissae to the 43-point rule. c c abserr - real c estimate of the modulus of the absolute error, c which should equal or exceed abs(i-result) c c neval - integer c number of integrand evaluations c c ier - ier = 0 normal and reliable termination of the c routine. it is assumed that the requested c accuracy has been achieved. c ier.gt.0 abnormal termination of the routine. it is c assumed that the requested accuracy has c not been achieved. c error messages c ier = 1 the maximum number of steps has been c executed. the integral is probably too c difficult to be calculated by dqng. c = 6 the input is invalid, because c epsabs.le.0 and c epsrel.lt.max(50*rel.mach.acc.,0.5d-28). c result, abserr and neval are set to zero. c c***references (none) c***routines called r1mach,xerror c***end prologue qng c real a,absc,abserr,b,centr,dhlgth,epmach,epsabs,epsrel,f,fcentr, * fval,fval1,fval2,fv1,fv2,fv3,fv4,hlgth,result,res10,res21,res43, * res87,resabs,resasc,reskh,r1mach,savfun,uflow,w10,w21a,w43a, * w43b,w87a,w87b,x1,x2,x3,x4 integer ier,ipx,k,l,neval external f c dimension fv1(5),fv2(5),fv3(5),fv4(5),x1(5),x2(5),x3(11),x4(22), * w10(5),w21a(5),w21b(6),w43a(10),w43b(12),w87a(21),w87b(23), * savfun(21) c c the following data statements contain the c abscissae and weights of the integration rules used. c c x1 abscissae common to the 10-, 21-, 43- c and 87-point rule c x2 abscissae common to the 21-, 43- and c 87-point rule c x3 abscissae common to the 43- and 87-point c rule c x4 abscissae of the 87-point rule c w10 weights of the 10-point formula c w21a weights of the 21-point formula for c abscissae x1 c w21b weights of the 21-point formula for c abscissae x2 c w43a weights of the 43-point formula for c abscissae x1, x3 c w43b weights of the 43-point formula for c abscissae x3 c w87a weights of the 87-point formula for c abscissae x1, x2, x3 c w87b weights of the 87-point formula for c abscissae x4 c data x1(1),x1(2),x1(3),x1(4),x1(5)/ * 0.9739065285171717e+00, 0.8650633666889845e+00, * 0.6794095682990244e+00, 0.4333953941292472e+00, * 0.1488743389816312e+00/ data x2(1),x2(2),x2(3),x2(4),x2(5)/ * 0.9956571630258081e+00, 0.9301574913557082e+00, * 0.7808177265864169e+00, 0.5627571346686047e+00, * 0.2943928627014602e+00/ data x3(1),x3(2),x3(3),x3(4),x3(5),x3(6),x3(7),x3(8), * x3(9),x3(10),x3(11)/ * 0.9993333609019321e+00, 0.9874334029080889e+00, * 0.9548079348142663e+00, 0.9001486957483283e+00, * 0.8251983149831142e+00, 0.7321483889893050e+00, * 0.6228479705377252e+00, 0.4994795740710565e+00, * 0.3649016613465808e+00, 0.2222549197766013e+00, * 0.7465061746138332e-01/ data x4(1),x4(2),x4(3),x4(4),x4(5),x4(6),x4(7),x4(8),x4(9), * x4(10),x4(11),x4(12),x4(13),x4(14),x4(15),x4(16),x4(17),x4(18), * x4(19),x4(20),x4(21),x4(22)/ 0.9999029772627292e+00, * 0.9979898959866787e+00, 0.9921754978606872e+00, * 0.9813581635727128e+00, 0.9650576238583846e+00, * 0.9431676131336706e+00, 0.9158064146855072e+00, * 0.8832216577713165e+00, 0.8457107484624157e+00, * 0.8035576580352310e+00, 0.7570057306854956e+00, * 0.7062732097873218e+00, 0.6515894665011779e+00, * 0.5932233740579611e+00, 0.5314936059708319e+00, * 0.4667636230420228e+00, 0.3994248478592188e+00, * 0.3298748771061883e+00, 0.2585035592021616e+00, * 0.1856953965683467e+00, 0.1118422131799075e+00, * 0.3735212339461987e-01/ data w10(1),w10(2),w10(3),w10(4),w10(5)/ * 0.6667134430868814e-01, 0.1494513491505806e+00, * 0.2190863625159820e+00, 0.2692667193099964e+00, * 0.2955242247147529e+00/ data w21a(1),w21a(2),w21a(3),w21a(4),w21a(5)/ * 0.3255816230796473e-01, 0.7503967481091995e-01, * 0.1093871588022976e+00, 0.1347092173114733e+00, * 0.1477391049013385e+00/ data w21b(1),w21b(2),w21b(3),w21b(4),w21b(5),w21b(6)/ * 0.1169463886737187e-01, 0.5475589657435200e-01, * 0.9312545458369761e-01, 0.1234919762620659e+00, * 0.1427759385770601e+00, 0.1494455540029169e+00/ data w43a(1),w43a(2),w43a(3),w43a(4),w43a(5),w43a(6),w43a(7), * w43a(8),w43a(9),w43a(10)/ 0.1629673428966656e-01, * 0.3752287612086950e-01, 0.5469490205825544e-01, * 0.6735541460947809e-01, 0.7387019963239395e-01, * 0.5768556059769796e-02, 0.2737189059324884e-01, * 0.4656082691042883e-01, 0.6174499520144256e-01, * 0.7138726726869340e-01/ data w43b(1),w43b(2),w43b(3),w43b(4),w43b(5),w43b(6), * w43b(7),w43b(8),w43b(9),w43b(10),w43b(11),w43b(12)/ * 0.1844477640212414e-02, 0.1079868958589165e-01, * 0.2189536386779543e-01, 0.3259746397534569e-01, * 0.4216313793519181e-01, 0.5074193960018458e-01, * 0.5837939554261925e-01, 0.6474640495144589e-01, * 0.6956619791235648e-01, 0.7282444147183321e-01, * 0.7450775101417512e-01, 0.7472214751740301e-01/ data w87a(1),w87a(2),w87a(3),w87a(4),w87a(5),w87a(6), * w87a(7),w87a(8),w87a(9),w87a(10),w87a(11),w87a(12), * w87a(13),w87a(14),w87a(15),w87a(16),w87a(17),w87a(18), * w87a(19),w87a(20),w87a(21)/ * 0.8148377384149173e-02, 0.1876143820156282e-01, * 0.2734745105005229e-01, 0.3367770731163793e-01, * 0.3693509982042791e-01, 0.2884872430211531e-02, * 0.1368594602271270e-01, 0.2328041350288831e-01, * 0.3087249761171336e-01, 0.3569363363941877e-01, * 0.9152833452022414e-03, 0.5399280219300471e-02, * 0.1094767960111893e-01, 0.1629873169678734e-01, * 0.2108156888920384e-01, 0.2537096976925383e-01, * 0.2918969775647575e-01, 0.3237320246720279e-01, * 0.3478309895036514e-01, 0.3641222073135179e-01, * 0.3725387550304771e-01/ data w87b(1),w87b(2),w87b(3),w87b(4),w87b(5),w87b(6),w87b(7), * w87b(8),w87b(9),w87b(10),w87b(11),w87b(12),w87b(13),w87b(14), * w87b(15),w87b(16),w87b(17),w87b(18),w87b(19),w87b(20), * w87b(21),w87b(22),w87b(23)/ 0.2741455637620724e-03, * 0.1807124155057943e-02, 0.4096869282759165e-02, * 0.6758290051847379e-02, 0.9549957672201647e-02, * 0.1232944765224485e-01, 0.1501044734638895e-01, * 0.1754896798624319e-01, 0.1993803778644089e-01, * 0.2219493596101229e-01, 0.2433914712600081e-01, * 0.2637450541483921e-01, 0.2828691078877120e-01, * 0.3005258112809270e-01, 0.3164675137143993e-01, * 0.3305041341997850e-01, 0.3425509970422606e-01, * 0.3526241266015668e-01, 0.3607698962288870e-01, * 0.3669860449845609e-01, 0.3712054926983258e-01, * 0.3733422875193504e-01, 0.3736107376267902e-01/ c c list of major variables c ----------------------- c c centr - mid point of the integration interval c hlgth - half-length of the integration interval c fcentr - function value at mid point c absc - abscissa c fval - function value c savfun - array of function values which c have already been computed c res10 - 10-point gauss result c res21 - 21-point kronrod result c res43 - 43-point result c res87 - 87-point result c resabs - approximation to the integral of abs(f) c resasc - approximation to the integral of abs(f-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 qng epmach = r1mach(4) uflow = r1mach(1) c c test on validity of parameters c ------------------------------ c result = 0.0e+00 abserr = 0.0e+00 neval = 0 ier = 6 if(epsabs.le.0.0e+00.and.epsrel.lt.amax1(0.5e-14,0.5e+02*epmach)) * go to 80 hlgth = 0.5e+00*(b-a) dhlgth = abs(hlgth) centr = 0.5e+00*(b+a) fcentr = f(centr) neval = 21 ier = 1 c c compute the integral using the 10- and 21-point formula. c do 70 l = 1,3 go to (5,25,45),l 5 res10 = 0.0e+00 res21 = w21b(6)*fcentr resabs = w21b(6)*abs(fcentr) do 10 k=1,5 absc = hlgth*x1(k) fval1 = f(centr+absc) fval2 = f(centr-absc) fval = fval1+fval2 res10 = res10+w10(k)*fval res21 = res21+w21a(k)*fval resabs = resabs+w21a(k)*(abs(fval1)+abs(fval2)) savfun(k) = fval fv1(k) = fval1 fv2(k) = fval2 10 continue ipx = 5 do 15 k=1,5 ipx = ipx+1 absc = hlgth*x2(k) fval1 = f(centr+absc) fval2 = f(centr-absc) fval = fval1+fval2 res21 = res21+w21b(k)*fval resabs = resabs+w21b(k)*(abs(fval1)+abs(fval2)) savfun(ipx) = fval fv3(k) = fval1 fv4(k) = fval2 15 continue c c test for convergence. c result = res21*hlgth resabs = resabs*dhlgth reskh = 0.5e+00*res21 resasc = w21b(6)*abs(fcentr-reskh) do 20 k = 1,5 resasc = resasc+w21a(k)*(abs(fv1(k)-reskh)+abs(fv2(k)-reskh)) * +w21b(k)*(abs(fv3(k)-reskh)+abs(fv4(k)-reskh)) 20 continue abserr = abs((res21-res10)*hlgth) resasc = resasc*dhlgth go to 65 c c compute the integral using the 43-point formula. c 25 res43 = w43b(12)*fcentr neval = 43 do 30 k=1,10 res43 = res43+savfun(k)*w43a(k) 30 continue do 40 k=1,11 ipx = ipx+1 absc = hlgth*x3(k) fval = f(absc+centr)+f(centr-absc) res43 = res43+fval*w43b(k) savfun(ipx) = fval 40 continue c c test for convergence. c result = res43*hlgth abserr = abs((res43-res21)*hlgth) go to 65 c c compute the integral using the 87-point formula. c 45 res87 = w87b(23)*fcentr neval = 87 do 50 k=1,21 res87 = res87+savfun(k)*w87a(k) 50 continue do 60 k=1,22 absc = hlgth*x4(k) res87 = res87+w87b(k)*(f(absc+centr)+f(centr-absc)) 60 continue result = res87*hlgth abserr = abs((res87-res43)*hlgth) 65 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) if (abserr.le.amax1(epsabs,epsrel*abs(result))) ier = 0 c ***jump out of do-loop if (ier.eq.0) go to 999 70 continue 80 call xerror(26habnormal return from qng ,26,ier,0) 999 return end