subroutine dqk41r(f,a,b,result,abserr,resabs,resasc,i)
c***begin prologue dqk41
c***date written 800101 (yymmdd)
c***revision date 830518 (yymmdd)
c***category no. h2a1a2
c***keywords 41-point gauss-kronrod rules
c***author piessens,robert,appl. math. & progr. div. - k.u.leuven
c de doncker,elise,appl. math. & progr. div. - k.u.leuven
c***purpose to compute i = integral of f over (a,b), with error
c estimate
c j = integral of abs(f) over (a,b)
c***description
c
c integration rules
c standard fortran subroutine
c double precision version
c
c parameters
c on entry
c f - double precision
c function subprogram defining the integrand
c function f(x). the actual name for f needs to be
c declared e x t e r n a l in the calling program.
c
c a - double precision
c lower limit of integration
c
c b - double precision
c upper limit of integration
c
c on return
c result - double precision
c approximation to the integral i
c result is computed by applying the 41-point
c gauss-kronrod rule (resk) obtained by optimal
c addition of abscissae to the 20-point gauss
c rule (resg).
c
c abserr - double precision
c estimate of the modulus of the absolute error,
c which should not exceed abs(i-result)
c
c resabs - double precision
c approximation to the integral j
c
c resasc - double precision
c approximation to the integal of abs(f-i/(b-a))
c over (a,b)
c
c***references (none)
c***routines called d1mach
c***end prologue dqk41
c
double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1,
* d1mach,epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc,
* resg,resk,reskh,result,uflow,wg,wgk,xgk
integer j,jtw,jtwm1,i
external f
c
dimension fv1(20),fv2(20),xgk(21),wgk(21),wg(10)
c
c the abscissae and weights are given for the interval (-1,1).
c because of symmetry only the positive abscissae and their
c corresponding weights are given.
c
c xgk - abscissae of the 41-point gauss-kronrod rule
c xgk(2), xgk(4), ... abscissae of the 20-point
c gauss rule
c xgk(1), xgk(3), ... abscissae which are optimally
c added to the 20-point gauss rule
c
c wgk - weights of the 41-point gauss-kronrod rule
c
c wg - weights of the 20-point gauss rule
c
c
c gauss quadrature weights and kronron quadrature abscissae and weights
c as evaluated with 80 decimal digit arithmetic by l. w. fullerton,
c bell labs, nov. 1981.
c
data wg ( 1) / 0.0176140071 3915211831 1861962351 853 d0 /
data wg ( 2) / 0.0406014298 0038694133 1039952274 932 d0 /
data wg ( 3) / 0.0626720483 3410906356 9506535187 042 d0 /
data wg ( 4) / 0.0832767415 7670474872 4758143222 046 d0 /
data wg ( 5) / 0.1019301198 1724043503 6750135480 350 d0 /
data wg ( 6) / 0.1181945319 6151841731 2377377711 382 d0 /
data wg ( 7) / 0.1316886384 4917662689 8494499748 163 d0 /
data wg ( 8) / 0.1420961093 1838205132 9298325067 165 d0 /
data wg ( 9) / 0.1491729864 7260374678 7828737001 969 d0 /
data wg ( 10) / 0.1527533871 3072585069 8084331955 098 d0 /
c
data xgk ( 1) / 0.9988590315 8827766383 8315576545 863 d0 /
data xgk ( 2) / 0.9931285991 8509492478 6122388471 320 d0 /
data xgk ( 3) / 0.9815078774 5025025919 3342994720 217 d0 /
data xgk ( 4) / 0.9639719272 7791379126 7666131197 277 d0 /
data xgk ( 5) / 0.9408226338 3175475351 9982722212 443 d0 /
data xgk ( 6) / 0.9122344282 5132590586 7752441203 298 d0 /
data xgk ( 7) / 0.8782768112 5228197607 7442995113 078 d0 /
data xgk ( 8) / 0.8391169718 2221882339 4529061701 521 d0 /
data xgk ( 9) / 0.7950414288 3755119835 0638833272 788 d0 /
data xgk ( 10) / 0.7463319064 6015079261 4305070355 642 d0 /
data xgk ( 11) / 0.6932376563 3475138480 5490711845 932 d0 /
data xgk ( 12) / 0.6360536807 2651502545 2836696226 286 d0 /
data xgk ( 13) / 0.5751404468 1971031534 2946036586 425 d0 /
data xgk ( 14) / 0.5108670019 5082709800 4364050955 251 d0 /
data xgk ( 15) / 0.4435931752 3872510319 9992213492 640 d0 /
data xgk ( 16) / 0.3737060887 1541956067 2548177024 927 d0 /
data xgk ( 17) / 0.3016278681 1491300432 0555356858 592 d0 /
data xgk ( 18) / 0.2277858511 4164507808 0496195368 575 d0 /
data xgk ( 19) / 0.1526054652 4092267550 5220241022 678 d0 /
data xgk ( 20) / 0.0765265211 3349733375 4640409398 838 d0 /
data xgk ( 21) / 0.0000000000 0000000000 0000000000 000 d0 /
c
data wgk ( 1) / 0.0030735837 1852053150 1218293246 031 d0 /
data wgk ( 2) / 0.0086002698 5564294219 8661787950 102 d0 /
data wgk ( 3) / 0.0146261692 5697125298 3787960308 868 d0 /
data wgk ( 4) / 0.0203883734 6126652359 8010231432 755 d0 /
data wgk ( 5) / 0.0258821336 0495115883 4505067096 153 d0 /
data wgk ( 6) / 0.0312873067 7703279895 8543119323 801 d0 /
data wgk ( 7) / 0.0366001697 5820079803 0557240707 211 d0 /
data wgk ( 8) / 0.0416688733 2797368626 3788305936 895 d0 /
data wgk ( 9) / 0.0464348218 6749767472 0231880926 108 d0 /
data wgk ( 10) / 0.0509445739 2372869193 2707670050 345 d0 /
data wgk ( 11) / 0.0551951053 4828599474 4832372419 777 d0 /
data wgk ( 12) / 0.0591114008 8063957237 4967220648 594 d0 /
data wgk ( 13) / 0.0626532375 5478116802 5870122174 255 d0 /
data wgk ( 14) / 0.0658345971 3361842211 1563556969 398 d0 /
data wgk ( 15) / 0.0686486729 2852161934 5623411885 368 d0 /
data wgk ( 16) / 0.0710544235 5344406830 5790361723 210 d0 /
data wgk ( 17) / 0.0730306903 3278666749 5189417658 913 d0 /
data wgk ( 18) / 0.0745828754 0049918898 6581418362 488 d0 /
data wgk ( 19) / 0.0757044976 8455667465 9542775376 617 d0 /
data wgk ( 20) / 0.0763778676 7208073670 5502835038 061 d0 /
data wgk ( 21) / 0.0766007119 1799965644 5049901530 102 d0 /
c
c
c list of major variables
c -----------------------
c
c centr - mid point of the interval
c hlgth - half-length of the interval
c absc - abscissa
c fval* - function value
c resg - result of the 20-point gauss formula
c resk - result of the 41-point kronrod formula
c reskh - approximation to mean value of f over (a,b), i.e.
c to i/(b-a)
c
c machine dependent constants
c ---------------------------
c
c epmach is the largest relative spacing.
c uflow is the smallest positive magnitude.
c
c***first executable statement dqk41
epmach = d1mach(4)
uflow = d1mach(1)
c
centr = 0.5d+00*(a+b)
hlgth = 0.5d+00*(b-a)
dhlgth = dabs(hlgth)
c
c compute the 41-point gauss-kronrod approximation to
c the integral, and estimate the absolute error.
c
resg = 0.0d+00
fc = f(centr,i)
resk = wgk(21)*fc
resabs = dabs(resk)
do 10 j=1,10
jtw = j*2
absc = hlgth*xgk(jtw)
fval1 = f(centr-absc,i)
fval2 = f(centr+absc,i)
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,i)
fval2 = f(centr+absc,i)
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
