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fvn_integ/dqk61_2d_outer.f90
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! fvn comment :
! Modified version of the dqk61 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 dqk61_2d_outer(f,a,b,g,h,result,abserr,resabs, &
resasc,epsabs,epsrel,key,limit)
!***begin prologue dqk61
!***date written 800101 (yymmdd)
!***revision date 830518 (yymmdd)
!***category no. h2a1a2
!***keywords 61-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 dabs(f) over (a,b)
!***description
!
! integration rule
! 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 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 61-point
! kronrod rule (resk) obtained by optimal addition of
! abscissae to the 30-point gauss rule (resg).
!
! abserr - double precision
! estimate of the modulus of the absolute error,
! which should equal or exceed dabs(i-result)
!
! resabs - double precision
! approximation to the integral j
!
! resasc - double precision
! approximation to the integral of dabs(f-i/(b-a))
!
!
!***references (none)
!***routines called d1mach
!***end prologue dqk61
!
double precision a,dabsc,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(30),fv2(30),xgk(31),wgk(31),wg(15)
!
! 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 61-point kronrod rule
! xgk(2), xgk(4) ... abscissae of the 30-point
! gauss rule
! xgk(1), xgk(3) ... optimally added abscissae
! to the 30-point gauss rule
!
! wgk - weights of the 61-point kronrod rule
!
! wg - weigths of the 30-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) / 0.007968192496166605615465883474674d0 /
data wg ( 2) / 0.018466468311090959142302131912047d0 /
data wg ( 3) / 0.028784707883323369349719179611292d0 /
data wg ( 4) / 0.038799192569627049596801936446348d0 /
data wg ( 5) / 0.048402672830594052902938140422808d0 /
data wg ( 6) / 0.057493156217619066481721689402056d0 /
data wg ( 7) / 0.065974229882180495128128515115962d0 /
data wg ( 8) / 0.073755974737705206268243850022191d0 /
data wg ( 9) / 0.080755895229420215354694938460530d0 /
data wg ( 10) / 0.086899787201082979802387530715126d0 /
data wg ( 11) / 0.092122522237786128717632707087619d0 /
data wg ( 12) / 0.096368737174644259639468626351810d0 /
data wg ( 13) / 0.099593420586795267062780282103569d0 /
data wg ( 14) / 0.101762389748405504596428952168554d0 /
data wg ( 15) / 0.102852652893558840341285636705415d0 /
!
data xgk ( 1) / 0.999484410050490637571325895705811d0 /
data xgk ( 2) / 0.996893484074649540271630050918695d0 /
data xgk ( 3) / 0.991630996870404594858628366109486d0 /
data xgk ( 4) / 0.983668123279747209970032581605663d0 /
data xgk ( 5) / 0.973116322501126268374693868423707d0 /
data xgk ( 6) / 0.960021864968307512216871025581798d0 /
data xgk ( 7) / 0.944374444748559979415831324037439d0 /
data xgk ( 8) / 0.926200047429274325879324277080474d0 /
data xgk ( 9) / 0.905573307699907798546522558925958d0 /
data xgk ( 10) / 0.882560535792052681543116462530226d0 /
data xgk ( 11) / 0.857205233546061098958658510658944d0 /
data xgk ( 12) / 0.829565762382768397442898119732502d0 /
data xgk ( 13) / 0.799727835821839083013668942322683d0 /
data xgk ( 14) / 0.767777432104826194917977340974503d0 /
data xgk ( 15) / 0.733790062453226804726171131369528d0 /
data xgk ( 16) / 0.697850494793315796932292388026640d0 /
data xgk ( 17) / 0.660061064126626961370053668149271d0 /
data xgk ( 18) / 0.620526182989242861140477556431189d0 /
data xgk ( 19) / 0.579345235826361691756024932172540d0 /
data xgk ( 20) / 0.536624148142019899264169793311073d0 /
data xgk ( 21) / 0.492480467861778574993693061207709d0 /
data xgk ( 22) / 0.447033769538089176780609900322854d0 /
data xgk ( 23) / 0.400401254830394392535476211542661d0 /
data xgk ( 24) / 0.352704725530878113471037207089374d0 /
data xgk ( 25) / 0.304073202273625077372677107199257d0 /
data xgk ( 26) / 0.254636926167889846439805129817805d0 /
data xgk ( 27) / 0.204525116682309891438957671002025d0 /
data xgk ( 28) / 0.153869913608583546963794672743256d0 /
data xgk ( 29) / 0.102806937966737030147096751318001d0 /
data xgk ( 30) / 0.051471842555317695833025213166723d0 /
data xgk ( 31) / 0.000000000000000000000000000000000d0 /
!
data wgk ( 1) / 0.001389013698677007624551591226760d0 /
data wgk ( 2) / 0.003890461127099884051267201844516d0 /
data wgk ( 3) / 0.006630703915931292173319826369750d0 /
data wgk ( 4) / 0.009273279659517763428441146892024d0 /
data wgk ( 5) / 0.011823015253496341742232898853251d0 /
data wgk ( 6) / 0.014369729507045804812451432443580d0 /
data wgk ( 7) / 0.016920889189053272627572289420322d0 /
data wgk ( 8) / 0.019414141193942381173408951050128d0 /
data wgk ( 9) / 0.021828035821609192297167485738339d0 /
data wgk ( 10) / 0.024191162078080601365686370725232d0 /
data wgk ( 11) / 0.026509954882333101610601709335075d0 /
data wgk ( 12) / 0.028754048765041292843978785354334d0 /
data wgk ( 13) / 0.030907257562387762472884252943092d0 /
data wgk ( 14) / 0.032981447057483726031814191016854d0 /
data wgk ( 15) / 0.034979338028060024137499670731468d0 /
data wgk ( 16) / 0.036882364651821229223911065617136d0 /
data wgk ( 17) / 0.038678945624727592950348651532281d0 /
data wgk ( 18) / 0.040374538951535959111995279752468d0 /
data wgk ( 19) / 0.041969810215164246147147541285970d0 /
data wgk ( 20) / 0.043452539701356069316831728117073d0 /
data wgk ( 21) / 0.044814800133162663192355551616723d0 /
data wgk ( 22) / 0.046059238271006988116271735559374d0 /
data wgk ( 23) / 0.047185546569299153945261478181099d0 /
data wgk ( 24) / 0.048185861757087129140779492298305d0 /
data wgk ( 25) / 0.049055434555029778887528165367238d0 /
data wgk ( 26) / 0.049795683427074206357811569379942d0 /
data wgk ( 27) / 0.050405921402782346840893085653585d0 /
data wgk ( 28) / 0.050881795898749606492297473049805d0 /
data wgk ( 29) / 0.051221547849258772170656282604944d0 /
data wgk ( 30) / 0.051426128537459025933862879215781d0 /
data wgk ( 31) / 0.051494729429451567558340433647099d0 /
!
! list of major variables
! -----------------------
!
! centr - mid point of the interval
! hlgth - half-length of the interval
! dabsc - abscissa
! fval* - function value
! resg - result of the 30-point gauss rule
! resk - result of the 61-point kronrod rule
! 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.
!
epmach = d1mach(4)
uflow = d1mach(1)
!
centr = 0.5d+00*(b+a)
hlgth = 0.5d+00*(b-a)
dhlgth = dabs(hlgth)
!
! compute the 61-point kronrod approximation to the
! integral, and estimate the absolute error.
!
!***first executable statement dqk61
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(31)*fc
resabs = dabs(resk)
do 10 j=1,15
jtw = j*2
dabsc = hlgth*xgk(jtw)
!fval1 = f(centr-dabsc)
call fvn_d_integ_2_inner_gk(f,centr-dabsc,g(centr-dabsc), &
h(centr-dabsc),epsabs,epsrel,key,eval_res,eval_abserr, &
eval_ier,limit)
fval1=eval_res
!fval2 = f(centr+dabsc)
call fvn_d_integ_2_inner_gk(f,centr+dabsc,g(centr+dabsc), &
h(centr+dabsc),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=1,15
jtwm1 = j*2-1
dabsc = hlgth*xgk(jtwm1)
!fval1 = f(centr-dabsc)
call fvn_d_integ_2_inner_gk(f,centr-dabsc,g(centr-dabsc), &
h(centr-dabsc),epsabs,epsrel,key,eval_res,eval_abserr, &
eval_ier,limit)
fval1=eval_res
!fval2 = f(centr+dabsc)
call fvn_d_integ_2_inner_gk(f,centr+dabsc,g(centr+dabsc), &
h(centr+dabsc),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(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 subroutine
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