dqk31_2d_inner.f
7.95 KB
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! fvn comment :
! Modified version of the dqk31 quadpack routine from http://www.netlib.org/quadpack
!
! + The external 'f' function is a 2 parameters function f(x,y). The routine
! takes one more parameter 'x' and evaluate the integral of f against y between a and b
! for a given x
subroutine dqk31_2d_inner(f,x,a,b,result,abserr,resabs,resasc)
!***begin prologue dqk31
!***date written 800101 (yymmdd)
!***revision date 830518 (yymmdd)
!***category no. h2a1a2
!***keywords 31-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 31-point
! gauss-kronrod rule (resk), obtained by optimal
! addition of abscissae to the 15-point gauss
! rule (resg).
!
! abserr - double precison
! estimate of the modulus of the modulus,
! 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 dqk31
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,x
integer j,jtw,jtwm1
external f
!
dimension fv1(15),fv2(15),xgk(16),wgk(16),wg(8)
!
! 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 31-point kronrod rule
! xgk(2), xgk(4), ... abscissae of the 15-point
! gauss rule
! xgk(1), xgk(3), ... abscissae which are optimally
! added to the 15-point gauss rule
!
! wgk - weights of the 31-point kronrod rule
!
! wg - weights of the 15-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) / 0.030753241996117268354628393577204d0 /
data wg ( 2) / 0.070366047488108124709267416450667d0 /
data wg ( 3) / 0.107159220467171935011869546685869d0 /
data wg ( 4) / 0.139570677926154314447804794511028d0 /
data wg ( 5) / 0.166269205816993933553200860481209d0 /
data wg ( 6) / 0.186161000015562211026800561866423d0 /
data wg ( 7) / 0.198431485327111576456118326443839d0 /
data wg ( 8) / 0.202578241925561272880620199967519d0 /
!
data xgk ( 1) / 0.998002298693397060285172840152271d0 /
data xgk ( 2) / 0.987992518020485428489565718586613d0 /
data xgk ( 3) / 0.967739075679139134257347978784337d0 /
data xgk ( 4) / 0.937273392400705904307758947710209d0 /
data xgk ( 5) / 0.897264532344081900882509656454496d0 /
data xgk ( 6) / 0.848206583410427216200648320774217d0 /
data xgk ( 7) / 0.790418501442465932967649294817947d0 /
data xgk ( 8) / 0.724417731360170047416186054613938d0 /
data xgk ( 9) / 0.650996741297416970533735895313275d0 /
data xgk ( 10) / 0.570972172608538847537226737253911d0 /
data xgk ( 11) / 0.485081863640239680693655740232351d0 /
data xgk ( 12) / 0.394151347077563369897207370981045d0 /
data xgk ( 13) / 0.299180007153168812166780024266389d0 /
data xgk ( 14) / 0.201194093997434522300628303394596d0 /
data xgk ( 15) / 0.101142066918717499027074231447392d0 /
data xgk ( 16) / 0.000000000000000000000000000000000d0 /
!
data wgk ( 1) / 0.005377479872923348987792051430128d0 /
data wgk ( 2) / 0.015007947329316122538374763075807d0 /
data wgk ( 3) / 0.025460847326715320186874001019653d0 /
data wgk ( 4) / 0.035346360791375846222037948478360d0 /
data wgk ( 5) / 0.044589751324764876608227299373280d0 /
data wgk ( 6) / 0.053481524690928087265343147239430d0 /
data wgk ( 7) / 0.062009567800670640285139230960803d0 /
data wgk ( 8) / 0.069854121318728258709520077099147d0 /
data wgk ( 9) / 0.076849680757720378894432777482659d0 /
data wgk ( 10) / 0.083080502823133021038289247286104d0 /
data wgk ( 11) / 0.088564443056211770647275443693774d0 /
data wgk ( 12) / 0.093126598170825321225486872747346d0 /
data wgk ( 13) / 0.096642726983623678505179907627589d0 /
data wgk ( 14) / 0.099173598721791959332393173484603d0 /
data wgk ( 15) / 0.100769845523875595044946662617570d0 /
data wgk ( 16) / 0.101330007014791549017374792767493d0 /
!
!
! 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 15-point gauss formula
! resk - result of the 31-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 dqk31
epmach = d1mach(4)
uflow = d1mach(1)
!
centr = 0.5d+00*(a+b)
hlgth = 0.5d+00*(b-a)
dhlgth = dabs(hlgth)
!
! compute the 31-point kronrod approximation to
! the integral, and estimate the absolute error.
!
fc = f(x,centr)
resg = wg(8)*fc
resk = wgk(16)*fc
resabs = dabs(resk)
do 10 j=1,7
jtw = j*2
absc = hlgth*xgk(jtw)
fval1 = f(x,centr-absc)
fval2 = f(x,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 = 1,8
jtwm1 = j*2-1
absc = hlgth*xgk(jtwm1)
fval1 = f(x,centr-absc)
fval2 = f(x,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(16)*dabs(fc-reskh)
do 20 j=1,15
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