! fvn comment : ! Modified version of the dqk21 quadpack routine from http://www.netlib.org/quadpack ! ! + The external 'f' function is a 2 parameters function f(x,y). The routine ! takes two more parameters 'g' and 'h' which are two external functions : ! g represent the lower bound of the integral for y parameter ! h represent the higher bound of the integral for y parameter ! The routine compute the double integral of function f with x between a and b ! and y between g(x) and h(x) subroutine dqk21_2d_outer(f,a,b,g,h,result,abserr,resabs, & resasc,epsabs,epsrel,key,limit) !***begin prologue dqk21 !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a2 !***keywords 21-point gauss-kronrod rules !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math. & progr. div. - k.u.leuven !***purpose to compute i = integral of f over (a,b), with error ! estimate ! j = integral of abs(f) over (a,b) !***description ! ! integration rules ! standard fortran subroutine ! double precision version ! ! parameters ! on entry ! f - double precision ! function subprogram defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the driver program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! on return ! result - double precision ! approximation to the integral i ! result is computed by applying the 21-point ! kronrod rule (resk) obtained by optimal addition ! of abscissae to the 10-point gauss rule (resg). ! ! abserr - double precision ! estimate of the modulus of the absolute error, ! which should not exceed abs(i-result) ! ! resabs - double precision ! approximation to the integral j ! ! resasc - double precision ! approximation to the integral of abs(f-i/(b-a)) ! over (a,b) ! !***references (none) !***routines called d1mach !***end prologue dqk21 ! double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1, & epmach,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc, & resg,resk,reskh,result,uflow,wg,wgk,xgk integer j,jtw,jtwm1 double precision,external :: f,g,h double precision :: eval_res double precision :: epsabs,epsrel,eval_abserr integer :: limit,key,eval_ier ! dimension fv1(10),fv2(10),wg(5),wgk(11),xgk(11) ! ! the abscissae and weights are given for the interval (-1,1). ! because of symmetry only the positive abscissae and their ! corresponding weights are given. ! ! xgk - abscissae of the 21-point kronrod rule ! xgk(2), xgk(4), ... abscissae of the 10-point ! gauss rule ! xgk(1), xgk(3), ... abscissae which are optimally ! added to the 10-point gauss rule ! ! wgk - weights of the 21-point kronrod rule ! ! wg - weights of the 10-point gauss rule ! ! ! gauss quadrature weights and kronron quadrature abscissae and weights ! as evaluated with 80 decimal digit arithmetic by l. w. fullerton, ! bell labs, nov. 1981. ! data wg ( 1) / 0.066671344308688137593568809893332d0 / data wg ( 2) / 0.149451349150580593145776339657697d0 / data wg ( 3) / 0.219086362515982043995534934228163d0 / data wg ( 4) / 0.269266719309996355091226921569469d0 / data wg ( 5) / 0.295524224714752870173892994651338d0 / ! data xgk ( 1) / 0.995657163025808080735527280689003d0 / data xgk ( 2) / 0.973906528517171720077964012084452d0 / data xgk ( 3) / 0.930157491355708226001207180059508d0 / data xgk ( 4) / 0.865063366688984510732096688423493d0 / data xgk ( 5) / 0.780817726586416897063717578345042d0 / data xgk ( 6) / 0.679409568299024406234327365114874d0 / data xgk ( 7) / 0.562757134668604683339000099272694d0 / data xgk ( 8) / 0.433395394129247190799265943165784d0 / data xgk ( 9) / 0.294392862701460198131126603103866d0 / data xgk ( 10) / 0.148874338981631210884826001129720d0 / data xgk ( 11) / 0.000000000000000000000000000000000d0 / ! data wgk ( 1) / 0.011694638867371874278064396062192d0 / data wgk ( 2) / 0.032558162307964727478818972459390d0 / data wgk ( 3) / 0.054755896574351996031381300244580d0 / data wgk ( 4) / 0.075039674810919952767043140916190d0 / data wgk ( 5) / 0.093125454583697605535065465083366d0 / data wgk ( 6) / 0.109387158802297641899210590325805d0 / data wgk ( 7) / 0.123491976262065851077958109831074d0 / data wgk ( 8) / 0.134709217311473325928054001771707d0 / data wgk ( 9) / 0.142775938577060080797094273138717d0 / data wgk ( 10) / 0.147739104901338491374841515972068d0 / data wgk ( 11) / 0.149445554002916905664936468389821d0 / ! ! ! list of major variables ! ----------------------- ! ! centr - mid point of the interval ! hlgth - half-length of the interval ! absc - abscissa ! fval* - function value ! resg - result of the 10-point gauss formula ! resk - result of the 21-point kronrod formula ! reskh - approximation to the mean value of f over (a,b), ! i.e. to i/(b-a) ! ! ! machine dependent constants ! --------------------------- ! ! epmach is the largest relative spacing. ! uflow is the smallest positive magnitude. ! !***first executable statement dqk21 epmach = d1mach(4) uflow = d1mach(1) ! centr = 0.5d+00*(a+b) hlgth = 0.5d+00*(b-a) dhlgth = dabs(hlgth) ! ! compute the 21-point kronrod approximation to ! the integral, and estimate the absolute error. ! resg = 0.0d+00 !fc = f(centr) call fvn_d_integ_2_inner_gk(f,centr,g(centr), & h(centr),epsabs,epsrel,key,eval_res,eval_abserr, & eval_ier,limit) fc=eval_res resk = wgk(11)*fc resabs = dabs(resk) do 10 j=1,5 jtw = 2*j absc = hlgth*xgk(jtw) !fval1 = f(centr-absc) call fvn_d_integ_2_inner_gk(f,centr-absc,g(centr-absc), & h(centr-absc),epsabs,epsrel,key,eval_res,eval_abserr, & eval_ier,limit) fval1=eval_res !fval2 = f(centr+absc) call fvn_d_integ_2_inner_gk(f,centr+absc,g(centr+absc), & h(centr+absc),epsabs,epsrel,key,eval_res,eval_abserr, & eval_ier,limit) fval2=eval_res 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 = 1,5 jtwm1 = 2*j-1 absc = hlgth*xgk(jtwm1) !fval1 = f(centr-absc) call fvn_d_integ_2_inner_gk(f,centr-absc,g(centr-absc), & h(centr-absc),epsabs,epsrel,key,eval_res,eval_abserr, & eval_ier,limit) fval1=eval_res !fval2 = f(centr+absc) call fvn_d_integ_2_inner_gk(f,centr+absc,g(centr+absc), & h(centr+absc),epsabs,epsrel,key,eval_res,eval_abserr, & eval_ier,limit) fval2=eval_res 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(11)*dabs(fc-reskh) do 20 j=1,10 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 subroutine