!   fvn comment :
!   Modified version of the dqk51 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 dqk51_2d_outer(f,a,b,g,h,result,abserr,resabs, &
      resasc,epsabs,epsrel,key,limit)
!***begin prologue  dqk51
!***date written   800101   (yymmdd)
!***revision date  830518   (yymmdd)
!***category no.  h2a1a2
!***keywords  51-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 subroutine 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 calling 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 51-point
!                       kronrod rule (resk) obtained by optimal addition
!                       of abscissae to the 25-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  dqk51
!
      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(25),fv2(25),xgk(26),wgk(26),wg(13)
!
!           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 51-point kronrod rule
!                    xgk(2), xgk(4), ...  abscissae of the 25-point
!                    gauss rule
!                    xgk(1), xgk(3), ...  abscissae which are optimally
!                    added to the 25-point gauss rule
!
!           wgk    - weights of the 51-point kronrod rule
!
!           wg     - weights of the 25-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) / 0.011393798501026287947902964113235d0 /
      data wg  (  2) / 0.026354986615032137261901815295299d0 /
      data wg  (  3) / 0.040939156701306312655623487711646d0 /
      data wg  (  4) / 0.054904695975835191925936891540473d0 /
      data wg  (  5) / 0.068038333812356917207187185656708d0 /
      data wg  (  6) / 0.080140700335001018013234959669111d0 /
      data wg  (  7) / 0.091028261982963649811497220702892d0 /
      data wg  (  8) / 0.100535949067050644202206890392686d0 /
      data wg  (  9) / 0.108519624474263653116093957050117d0 /
      data wg  ( 10) / 0.114858259145711648339325545869556d0 /
      data wg  ( 11) / 0.119455763535784772228178126512901d0 /
      data wg  ( 12) / 0.122242442990310041688959518945852d0 /
      data wg  ( 13) / 0.123176053726715451203902873079050d0 /
!
      data xgk (  1) / 0.999262104992609834193457486540341d0 /
      data xgk (  2) / 0.995556969790498097908784946893902d0 /
      data xgk (  3) / 0.988035794534077247637331014577406d0 /
      data xgk (  4) / 0.976663921459517511498315386479594d0 /
      data xgk (  5) / 0.961614986425842512418130033660167d0 /
      data xgk (  6) / 0.942974571228974339414011169658471d0 /
      data xgk (  7) / 0.920747115281701561746346084546331d0 /
      data xgk (  8) / 0.894991997878275368851042006782805d0 /
      data xgk (  9) / 0.865847065293275595448996969588340d0 /
      data xgk ( 10) / 0.833442628760834001421021108693570d0 /
      data xgk ( 11) / 0.797873797998500059410410904994307d0 /
      data xgk ( 12) / 0.759259263037357630577282865204361d0 /
      data xgk ( 13) / 0.717766406813084388186654079773298d0 /
      data xgk ( 14) / 0.673566368473468364485120633247622d0 /
      data xgk ( 15) / 0.626810099010317412788122681624518d0 /
      data xgk ( 16) / 0.577662930241222967723689841612654d0 /
      data xgk ( 17) / 0.526325284334719182599623778158010d0 /
      data xgk ( 18) / 0.473002731445714960522182115009192d0 /
      data xgk ( 19) / 0.417885382193037748851814394594572d0 /
      data xgk ( 20) / 0.361172305809387837735821730127641d0 /
      data xgk ( 21) / 0.303089538931107830167478909980339d0 /
      data xgk ( 22) / 0.243866883720988432045190362797452d0 /
      data xgk ( 23) / 0.183718939421048892015969888759528d0 /
      data xgk ( 24) / 0.122864692610710396387359818808037d0 /
      data xgk ( 25) / 0.061544483005685078886546392366797d0 /
      data xgk ( 26) / 0.000000000000000000000000000000000d0 /
!
      data wgk (  1) / 0.001987383892330315926507851882843d0 /
      data wgk (  2) / 0.005561932135356713758040236901066d0 /
      data wgk (  3) / 0.009473973386174151607207710523655d0 /
      data wgk (  4) / 0.013236229195571674813656405846976d0 /
      data wgk (  5) / 0.016847817709128298231516667536336d0 /
      data wgk (  6) / 0.020435371145882835456568292235939d0 /
      data wgk (  7) / 0.024009945606953216220092489164881d0 /
      data wgk (  8) / 0.027475317587851737802948455517811d0 /
      data wgk (  9) / 0.030792300167387488891109020215229d0 /
      data wgk ( 10) / 0.034002130274329337836748795229551d0 /
      data wgk ( 11) / 0.037116271483415543560330625367620d0 /
      data wgk ( 12) / 0.040083825504032382074839284467076d0 /
      data wgk ( 13) / 0.042872845020170049476895792439495d0 /
      data wgk ( 14) / 0.045502913049921788909870584752660d0 /
      data wgk ( 15) / 0.047982537138836713906392255756915d0 /
      data wgk ( 16) / 0.050277679080715671963325259433440d0 /
      data wgk ( 17) / 0.052362885806407475864366712137873d0 /
      data wgk ( 18) / 0.054251129888545490144543370459876d0 /
      data wgk ( 19) / 0.055950811220412317308240686382747d0 /
      data wgk ( 20) / 0.057437116361567832853582693939506d0 /
      data wgk ( 21) / 0.058689680022394207961974175856788d0 /
      data wgk ( 22) / 0.059720340324174059979099291932562d0 /
      data wgk ( 23) / 0.060539455376045862945360267517565d0 /
      data wgk ( 24) / 0.061128509717053048305859030416293d0 /
      data wgk ( 25) / 0.061471189871425316661544131965264d0 /
!       note: wgk (26) was calculated from the values of wgk(1..25)
      data wgk ( 26) / 0.061580818067832935078759824240066d0 /
!
!
!           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 25-point gauss formula
!           resk   - result of the 51-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  dqk51
      epmach = d1mach(4)
      uflow = d1mach(1)
!
      centr = 0.5d+00*(a+b)
      hlgth = 0.5d+00*(b-a)
      dhlgth = dabs(hlgth)
!
!           compute the 51-point kronrod approximation to
!           the integral, and estimate the absolute error.
!
      !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
      resg = wg(13)*fc
      resk = wgk(26)*fc
      resabs = dabs(resk)
      do 10 j=1,12
        jtw = j*2
        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 = 1,13
        jtwm1 = j*2-1
        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(26)*dabs(fc-reskh)
      do 20 j=1,25
        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