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SUBROUTINE gratio(a,x,ans,qans,ind)
C ----------------------------------------------------------------------
C EVALUATION OF THE INCOMPLETE GAMMA RATIO FUNCTIONS
C P(A,X) AND Q(A,X)
C
C ----------
C
C IT IS ASSUMED THAT A AND X ARE NONNEGATIVE, WHERE A AND X
C ARE NOT BOTH 0.
C
C ANS AND QANS ARE VARIABLES. GRATIO ASSIGNS ANS THE VALUE
C P(A,X) AND QANS THE VALUE Q(A,X). IND MAY BE ANY INTEGER.
C IF IND = 0 THEN THE USER IS REQUESTING AS MUCH ACCURACY AS
C POSSIBLE (UP TO 14 SIGNIFICANT DIGITS). OTHERWISE, IF
C IND = 1 THEN ACCURACY IS REQUESTED TO WITHIN 1 UNIT OF THE
C 6-TH SIGNIFICANT DIGIT, AND IF IND .NE. 0,1 THEN ACCURACY
C IS REQUESTED TO WITHIN 1 UNIT OF THE 3RD SIGNIFICANT DIGIT.
C
C ERROR RETURN ...
C ANS IS ASSIGNED THE VALUE 2 WHEN A OR X IS NEGATIVE,
C WHEN A*X = 0, OR WHEN P(A,X) AND Q(A,X) ARE INDETERMINANT.
C P(A,X) AND Q(A,X) ARE COMPUTATIONALLY INDETERMINANT WHEN
C X IS EXCEEDINGLY CLOSE TO A AND A IS EXTREMELY LARGE.
C ----------------------------------------------------------------------
C WRITTEN BY ALFRED H. MORRIS, JR.
C NAVAL SURFACE WEAPONS CENTER
C DAHLGREN, VIRGINIA
C --------------------
C .. Scalar Arguments ..
DOUBLE PRECISION a,ans,qans,x
INTEGER ind
C ..
C .. Local Scalars ..
DOUBLE PRECISION a2n,a2nm1,acc,alog10,am0,amn,an,an0,apn,b2n,
+ b2nm1,c,c0,c1,c2,c3,c4,c5,c6,cma,d10,d20,d30,d40,
+ d50,d60,d70,e,e0,g,h,j,l,r,rt2pin,rta,rtpi,rtx,s,
+ sum,t,t1,third,tol,twoa,u,w,x0,y,z
INTEGER i,iop,m,max,n
C ..
C .. Local Arrays ..
DOUBLE PRECISION acc0(3),big(3),d0(13),d1(12),d2(10),d3(8),d4(6),
+ d5(4),d6(2),e00(3),wk(20),x00(3)
C ..
C .. External Functions ..
DOUBLE PRECISION erf,erfc1,gam1,gamma,rexp,rlog,spmpar
EXTERNAL erf,erfc1,gam1,gamma,rexp,rlog,spmpar
C ..
C .. Intrinsic Functions ..
INTRINSIC abs,dble,dlog,dmax1,exp,int,sqrt
C ..
C .. Data statements ..
C --------------------
C --------------------
C ALOG10 = LN(10)
C RT2PIN = 1/SQRT(2*PI)
C RTPI = SQRT(PI)
C --------------------
C --------------------
C --------------------
C --------------------
C --------------------
C --------------------
C --------------------
C --------------------
C --------------------
DATA acc0(1)/5.D-15/,acc0(2)/5.D-7/,acc0(3)/5.D-4/
DATA big(1)/20.0D0/,big(2)/14.0D0/,big(3)/10.0D0/
DATA e00(1)/.25D-3/,e00(2)/.25D-1/,e00(3)/.14D0/
DATA x00(1)/31.0D0/,x00(2)/17.0D0/,x00(3)/9.7D0/
DATA alog10/2.30258509299405D0/
DATA rt2pin/.398942280401433D0/
DATA rtpi/1.77245385090552D0/
DATA third/.333333333333333D0/
DATA d0(1)/.833333333333333D-01/,d0(2)/-.148148148148148D-01/,
+ d0(3)/.115740740740741D-02/,d0(4)/.352733686067019D-03/,
+ d0(5)/-.178755144032922D-03/,d0(6)/.391926317852244D-04/,
+ d0(7)/-.218544851067999D-05/,d0(8)/-.185406221071516D-05/,
+ d0(9)/.829671134095309D-06/,d0(10)/-.176659527368261D-06/,
+ d0(11)/.670785354340150D-08/,d0(12)/.102618097842403D-07/,
+ d0(13)/-.438203601845335D-08/
DATA d10/-.185185185185185D-02/,d1(1)/-.347222222222222D-02/,
+ d1(2)/.264550264550265D-02/,d1(3)/-.990226337448560D-03/,
+ d1(4)/.205761316872428D-03/,d1(5)/-.401877572016461D-06/,
+ d1(6)/-.180985503344900D-04/,d1(7)/.764916091608111D-05/,
+ d1(8)/-.161209008945634D-05/,d1(9)/.464712780280743D-08/,
+ d1(10)/.137863344691572D-06/,d1(11)/-.575254560351770D-07/,
+ d1(12)/.119516285997781D-07/
DATA d20/.413359788359788D-02/,d2(1)/-.268132716049383D-02/,
+ d2(2)/.771604938271605D-03/,d2(3)/.200938786008230D-05/,
+ d2(4)/-.107366532263652D-03/,d2(5)/.529234488291201D-04/,
+ d2(6)/-.127606351886187D-04/,d2(7)/.342357873409614D-07/,
+ d2(8)/.137219573090629D-05/,d2(9)/-.629899213838006D-06/,
+ d2(10)/.142806142060642D-06/
DATA d30/.649434156378601D-03/,d3(1)/.229472093621399D-03/,
+ d3(2)/-.469189494395256D-03/,d3(3)/.267720632062839D-03/,
+ d3(4)/-.756180167188398D-04/,d3(5)/-.239650511386730D-06/,
+ d3(6)/.110826541153473D-04/,d3(7)/-.567495282699160D-05/,
+ d3(8)/.142309007324359D-05/
DATA d40/-.861888290916712D-03/,d4(1)/.784039221720067D-03/,
+ d4(2)/-.299072480303190D-03/,d4(3)/-.146384525788434D-05/,
+ d4(4)/.664149821546512D-04/,d4(5)/-.396836504717943D-04/,
+ d4(6)/.113757269706784D-04/
DATA d50/-.336798553366358D-03/,d5(1)/-.697281375836586D-04/,
+ d5(2)/.277275324495939D-03/,d5(3)/-.199325705161888D-03/,
+ d5(4)/.679778047793721D-04/
DATA d60/.531307936463992D-03/,d6(1)/-.592166437353694D-03/,
+ d6(2)/.270878209671804D-03/
DATA d70/.344367606892378D-03/
C ..
C .. Executable Statements ..
C --------------------
C ****** E IS A MACHINE DEPENDENT CONSTANT. E IS THE SMALLEST
C FLOATING POINT NUMBER FOR WHICH 1.0 + E .GT. 1.0 .
C
e = spmpar(1)
C
C --------------------
IF (a.LT.0.0D0 .OR. x.LT.0.0D0) GO TO 430
IF (a.EQ.0.0D0 .AND. x.EQ.0.0D0) GO TO 430
IF (a*x.EQ.0.0D0) GO TO 420
C
iop = ind + 1
IF (iop.NE.1 .AND. iop.NE.2) iop = 3
acc = dmax1(acc0(iop),e)
e0 = e00(iop)
x0 = x00(iop)
C
C SELECT THE APPROPRIATE ALGORITHM
C
IF (a.GE.1.0D0) GO TO 10
IF (a.EQ.0.5D0) GO TO 390
IF (x.LT.1.1D0) GO TO 160
t1 = a*dlog(x) - x
u = a*exp(t1)
IF (u.EQ.0.0D0) GO TO 380
r = u* (1.0D0+gam1(a))
GO TO 250
C
10 IF (a.GE.big(iop)) GO TO 30
IF (a.GT.x .OR. x.GE.x0) GO TO 20
twoa = a + a
m = int(twoa)
IF (twoa.NE.dble(m)) GO TO 20
i = m/2
IF (a.EQ.dble(i)) GO TO 210
GO TO 220
20 t1 = a*dlog(x) - x
r = exp(t1)/gamma(a)
GO TO 40
C
30 l = x/a
IF (l.EQ.0.0D0) GO TO 370
s = 0.5D0 + (0.5D0-l)
z = rlog(l)
IF (z.GE.700.0D0/a) GO TO 410
y = a*z
rta = sqrt(a)
IF (abs(s).LE.e0/rta) GO TO 330
IF (abs(s).LE.0.4D0) GO TO 270
C
t = (1.0D0/a)**2
t1 = (((0.75D0*t-1.0D0)*t+3.5D0)*t-105.0D0)/ (a*1260.0D0)
t1 = t1 - y
r = rt2pin*rta*exp(t1)
C
40 IF (r.EQ.0.0D0) GO TO 420
IF (x.LE.dmax1(a,alog10)) GO TO 50
IF (x.LT.x0) GO TO 250
GO TO 100
C
C TAYLOR SERIES FOR P/R
C
50 apn = a + 1.0D0
t = x/apn
wk(1) = t
DO 60 n = 2,20
apn = apn + 1.0D0
t = t* (x/apn)
IF (t.LE.1.D-3) GO TO 70
wk(n) = t
60 CONTINUE
n = 20
C
70 sum = t
tol = 0.5D0*acc
80 apn = apn + 1.0D0
t = t* (x/apn)
sum = sum + t
IF (t.GT.tol) GO TO 80
C
max = n - 1
DO 90 m = 1,max
n = n - 1
sum = sum + wk(n)
90 CONTINUE
ans = (r/a)* (1.0D0+sum)
qans = 0.5D0 + (0.5D0-ans)
RETURN
C
C ASYMPTOTIC EXPANSION
C
100 amn = a - 1.0D0
t = amn/x
wk(1) = t
DO 110 n = 2,20
amn = amn - 1.0D0
t = t* (amn/x)
IF (abs(t).LE.1.D-3) GO TO 120
wk(n) = t
110 CONTINUE
n = 20
C
120 sum = t
130 IF (abs(t).LE.acc) GO TO 140
amn = amn - 1.0D0
t = t* (amn/x)
sum = sum + t
GO TO 130
C
140 max = n - 1
DO 150 m = 1,max
n = n - 1
sum = sum + wk(n)
150 CONTINUE
qans = (r/x)* (1.0D0+sum)
ans = 0.5D0 + (0.5D0-qans)
RETURN
C
C TAYLOR SERIES FOR P(A,X)/X**A
C
160 an = 3.0D0
c = x
sum = x/ (a+3.0D0)
tol = 3.0D0*acc/ (a+1.0D0)
170 an = an + 1.0D0
c = -c* (x/an)
t = c/ (a+an)
sum = sum + t
IF (abs(t).GT.tol) GO TO 170
j = a*x* ((sum/6.0D0-0.5D0/ (a+2.0D0))*x+1.0D0/ (a+1.0D0))
C
z = a*dlog(x)
h = gam1(a)
g = 1.0D0 + h
IF (x.LT.0.25D0) GO TO 180
IF (a.LT.x/2.59D0) GO TO 200
GO TO 190
180 IF (z.GT.-.13394D0) GO TO 200
C
190 w = exp(z)
ans = w*g* (0.5D0+ (0.5D0-j))
qans = 0.5D0 + (0.5D0-ans)
RETURN
C
200 l = rexp(z)
w = 0.5D0 + (0.5D0+l)
qans = (w*j-l)*g - h
IF (qans.LT.0.0D0) GO TO 380
ans = 0.5D0 + (0.5D0-qans)
RETURN
C
C FINITE SUMS FOR Q WHEN A .GE. 1
C AND 2*A IS AN INTEGER
C
210 sum = exp(-x)
t = sum
n = 1
c = 0.0D0
GO TO 230
C
220 rtx = sqrt(x)
sum = erfc1(0,rtx)
t = exp(-x)/ (rtpi*rtx)
n = 0
c = -0.5D0
C
230 IF (n.EQ.i) GO TO 240
n = n + 1
c = c + 1.0D0
t = (x*t)/c
sum = sum + t
GO TO 230
240 qans = sum
ans = 0.5D0 + (0.5D0-qans)
RETURN
C
C CONTINUED FRACTION EXPANSION
C
250 tol = dmax1(5.0D0*e,acc)
a2nm1 = 1.0D0
a2n = 1.0D0
b2nm1 = x
b2n = x + (1.0D0-a)
c = 1.0D0
260 a2nm1 = x*a2n + c*a2nm1
b2nm1 = x*b2n + c*b2nm1
am0 = a2nm1/b2nm1
c = c + 1.0D0
cma = c - a
a2n = a2nm1 + cma*a2n
b2n = b2nm1 + cma*b2n
an0 = a2n/b2n
IF (abs(an0-am0).GE.tol*an0) GO TO 260
C
qans = r*an0
ans = 0.5D0 + (0.5D0-qans)
RETURN
C
C GENERAL TEMME EXPANSION
C
270 IF (abs(s).LE.2.0D0*e .AND. a*e*e.GT.3.28D-3) GO TO 430
c = exp(-y)
w = 0.5D0*erfc1(1,sqrt(y))
u = 1.0D0/a
z = sqrt(z+z)
IF (l.LT.1.0D0) z = -z
IF (iop.lt.2) GO TO 280
IF (iop.eq.2) GO TO 290
GO TO 300
C
280 IF (abs(s).LE.1.D-3) GO TO 340
c0 = ((((((((((((d0(13)*z+d0(12))*z+d0(11))*z+d0(10))*z+d0(9))*z+
+ d0(8))*z+d0(7))*z+d0(6))*z+d0(5))*z+d0(4))*z+d0(3))*z+d0(2))*
+ z+d0(1))*z - third
c1 = (((((((((((d1(12)*z+d1(11))*z+d1(10))*z+d1(9))*z+d1(8))*z+
+ d1(7))*z+d1(6))*z+d1(5))*z+d1(4))*z+d1(3))*z+d1(2))*z+d1(1))*
+ z + d10
c2 = (((((((((d2(10)*z+d2(9))*z+d2(8))*z+d2(7))*z+d2(6))*z+
+ d2(5))*z+d2(4))*z+d2(3))*z+d2(2))*z+d2(1))*z + d20
c3 = (((((((d3(8)*z+d3(7))*z+d3(6))*z+d3(5))*z+d3(4))*z+d3(3))*z+
+ d3(2))*z+d3(1))*z + d30
c4 = (((((d4(6)*z+d4(5))*z+d4(4))*z+d4(3))*z+d4(2))*z+d4(1))*z +
+ d40
c5 = (((d5(4)*z+d5(3))*z+d5(2))*z+d5(1))*z + d50
c6 = (d6(2)*z+d6(1))*z + d60
t = ((((((d70*u+c6)*u+c5)*u+c4)*u+c3)*u+c2)*u+c1)*u + c0
GO TO 310
C
290 c0 = (((((d0(6)*z+d0(5))*z+d0(4))*z+d0(3))*z+d0(2))*z+d0(1))*z -
+ third
c1 = (((d1(4)*z+d1(3))*z+d1(2))*z+d1(1))*z + d10
c2 = d2(1)*z + d20
t = (c2*u+c1)*u + c0
GO TO 310
C
300 t = ((d0(3)*z+d0(2))*z+d0(1))*z - third
C
310 IF (l.LT.1.0D0) GO TO 320
qans = c* (w+rt2pin*t/rta)
ans = 0.5D0 + (0.5D0-qans)
RETURN
320 ans = c* (w-rt2pin*t/rta)
qans = 0.5D0 + (0.5D0-ans)
RETURN
C
C TEMME EXPANSION FOR L = 1
C
330 IF (a*e*e.GT.3.28D-3) GO TO 430
c = 0.5D0 + (0.5D0-y)
w = (0.5D0-sqrt(y)* (0.5D0+ (0.5D0-y/3.0D0))/rtpi)/c
u = 1.0D0/a
z = sqrt(z+z)
IF (l.LT.1.0D0) z = -z
IF (iop.lt.2) GO TO 340
IF (iop.eq.2) GO TO 350
GO TO 360
C
340 c0 = ((((((d0(7)*z+d0(6))*z+d0(5))*z+d0(4))*z+d0(3))*z+d0(2))*z+
+ d0(1))*z - third
c1 = (((((d1(6)*z+d1(5))*z+d1(4))*z+d1(3))*z+d1(2))*z+d1(1))*z +
+ d10
c2 = ((((d2(5)*z+d2(4))*z+d2(3))*z+d2(2))*z+d2(1))*z + d20
c3 = (((d3(4)*z+d3(3))*z+d3(2))*z+d3(1))*z + d30
c4 = (d4(2)*z+d4(1))*z + d40
c5 = (d5(2)*z+d5(1))*z + d50
c6 = d6(1)*z + d60
t = ((((((d70*u+c6)*u+c5)*u+c4)*u+c3)*u+c2)*u+c1)*u + c0
GO TO 310
C
350 c0 = (d0(2)*z+d0(1))*z - third
c1 = d1(1)*z + d10
t = (d20*u+c1)*u + c0
GO TO 310
C
360 t = d0(1)*z - third
GO TO 310
C
C SPECIAL CASES
C
370 ans = 0.0D0
qans = 1.0D0
RETURN
C
380 ans = 1.0D0
qans = 0.0D0
RETURN
C
390 IF (x.GE.0.25D0) GO TO 400
ans = erf(sqrt(x))
qans = 0.5D0 + (0.5D0-ans)
RETURN
400 qans = erfc1(0,sqrt(x))
ans = 0.5D0 + (0.5D0-qans)
RETURN
C
410 IF (abs(s).LE.2.0D0*e) GO TO 430
420 IF (x.LE.a) GO TO 370
GO TO 380
C
C ERROR RETURN
C
430 ans = 2.0D0
RETURN
END
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