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fvn_integ/dqk41.f
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!
! fvn comment :
! Unmodified quadpack routine from http://www.netlib.org/quadpack
!
subroutine dqk41(f,a,b,result,abserr,resabs,resasc)
!***begin prologue dqk41
!***date written 800101 (yymmdd)
!***revision date 830518 (yymmdd)
!***category no. h2a1a2
!***keywords 41-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 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 41-point
! gauss-kronrod rule (resk) obtained by optimal
! addition of abscissae to the 20-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 integal of abs(f-i/(b-a))
! over (a,b)
!
!***references (none)
!***routines called d1mach
!***end prologue dqk41
!
double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1, &
epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc, &
resg,resk,reskh,result,uflow,wg,wgk,xgk
integer j,jtw,jtwm1
external f
!
dimension fv1(20),fv2(20),xgk(21),wgk(21),wg(10)
!
! 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 41-point gauss-kronrod rule
! xgk(2), xgk(4), ... abscissae of the 20-point
! gauss rule
! xgk(1), xgk(3), ... abscissae which are optimally
! added to the 20-point gauss rule
!
! wgk - weights of the 41-point gauss-kronrod rule
!
! wg - weights of the 20-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) / 0.017614007139152118311861962351853d0 /
data wg ( 2) / 0.040601429800386941331039952274932d0 /
data wg ( 3) / 0.062672048334109063569506535187042d0 /
data wg ( 4) / 0.083276741576704748724758143222046d0 /
data wg ( 5) / 0.101930119817240435036750135480350d0 /
data wg ( 6) / 0.118194531961518417312377377711382d0 /
data wg ( 7) / 0.131688638449176626898494499748163d0 /
data wg ( 8) / 0.142096109318382051329298325067165d0 /
data wg ( 9) / 0.149172986472603746787828737001969d0 /
data wg ( 10) / 0.152753387130725850698084331955098d0 /
!
data xgk ( 1) / 0.998859031588277663838315576545863d0 /
data xgk ( 2) / 0.993128599185094924786122388471320d0 /
data xgk ( 3) / 0.981507877450250259193342994720217d0 /
data xgk ( 4) / 0.963971927277913791267666131197277d0 /
data xgk ( 5) / 0.940822633831754753519982722212443d0 /
data xgk ( 6) / 0.912234428251325905867752441203298d0 /
data xgk ( 7) / 0.878276811252281976077442995113078d0 /
data xgk ( 8) / 0.839116971822218823394529061701521d0 /
data xgk ( 9) / 0.795041428837551198350638833272788d0 /
data xgk ( 10) / 0.746331906460150792614305070355642d0 /
data xgk ( 11) / 0.693237656334751384805490711845932d0 /
data xgk ( 12) / 0.636053680726515025452836696226286d0 /
data xgk ( 13) / 0.575140446819710315342946036586425d0 /
data xgk ( 14) / 0.510867001950827098004364050955251d0 /
data xgk ( 15) / 0.443593175238725103199992213492640d0 /
data xgk ( 16) / 0.373706088715419560672548177024927d0 /
data xgk ( 17) / 0.301627868114913004320555356858592d0 /
data xgk ( 18) / 0.227785851141645078080496195368575d0 /
data xgk ( 19) / 0.152605465240922675505220241022678d0 /
data xgk ( 20) / 0.076526521133497333754640409398838d0 /
data xgk ( 21) / 0.000000000000000000000000000000000d0 /
!
data wgk ( 1) / 0.003073583718520531501218293246031d0 /
data wgk ( 2) / 0.008600269855642942198661787950102d0 /
data wgk ( 3) / 0.014626169256971252983787960308868d0 /
data wgk ( 4) / 0.020388373461266523598010231432755d0 /
data wgk ( 5) / 0.025882133604951158834505067096153d0 /
data wgk ( 6) / 0.031287306777032798958543119323801d0 /
data wgk ( 7) / 0.036600169758200798030557240707211d0 /
data wgk ( 8) / 0.041668873327973686263788305936895d0 /
data wgk ( 9) / 0.046434821867497674720231880926108d0 /
data wgk ( 10) / 0.050944573923728691932707670050345d0 /
data wgk ( 11) / 0.055195105348285994744832372419777d0 /
data wgk ( 12) / 0.059111400880639572374967220648594d0 /
data wgk ( 13) / 0.062653237554781168025870122174255d0 /
data wgk ( 14) / 0.065834597133618422111563556969398d0 /
data wgk ( 15) / 0.068648672928521619345623411885368d0 /
data wgk ( 16) / 0.071054423553444068305790361723210d0 /
data wgk ( 17) / 0.073030690332786667495189417658913d0 /
data wgk ( 18) / 0.074582875400499188986581418362488d0 /
data wgk ( 19) / 0.075704497684556674659542775376617d0 /
data wgk ( 20) / 0.076377867672080736705502835038061d0 /
data wgk ( 21) / 0.076600711917999656445049901530102d0 /
!
!
! 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 20-point gauss formula
! resk - result of the 41-point kronrod formula
! reskh - approximation to 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 dqk41
epmach = d1mach(4)
uflow = d1mach(1)
!
centr = 0.5d+00*(a+b)
hlgth = 0.5d+00*(b-a)
dhlgth = dabs(hlgth)
!
! compute the 41-point gauss-kronrod approximation to
! the integral, and estimate the absolute error.
!
resg = 0.0d+00
fc = f(centr)
resk = wgk(21)*fc
resabs = dabs(resk)
do 10 j=1,10
jtw = j*2
absc = hlgth*xgk(jtw)
fval1 = f(centr-absc)
fval2 = f(centr+absc)
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 = 1,10
jtwm1 = j*2-1
absc = hlgth*xgk(jtwm1)
fval1 = f(centr-absc)
fval2 = f(centr+absc)
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(21)*dabs(fc-reskh)
do 20 j=1,20
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.d+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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