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cdgamma.f
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c-----------------------------------------------------------------------
c This code was modified from a subroutine from the gamerf package at
c http://momonga.t.u-tokyo.ac.jp/~ooura/gamerf.html .
c The original copyright statement states:
c Copyright(C) 1996 Takuya OOURA (email: [email protected]).
c You may use, copy, modify this code for any purpose and
c without fee. You may distribute this ORIGINAL package.
c
c Permission to distribute this modified gamma function code
c with the FFTLog package has been granted
c (email from Takuya Ooura to Andrew Hamilton dated 16 March 1999).
c
c Original gamerf2a.doc documentation states:
c
c Gamma(z)=sqrt(2*pi)*(z+r)^(z-1/2)*exp(-z-r)
c *(a_0
c +a_1*(z-1)/z
c +a_2*(z-1)*(z-2)/z/(z+1)
c +a_3*(z-1)*(z-2)*(z-3)/z/(z+1)/(z+2)
c +...)
c a_n= f_n*(2*n)*(2*n-1)*(2*n-2)*...*(n+1)/1/2/3/.../n
c -a_0*(2*n)*(2*n-1)*(2*n-2)*...*(n+1)/1/2/3/.../n
c -a_1*(2*n)*(2*n-1)*(2*n-2)*...*(n+2)/1/2/3/.../(n-1)
c -...
c -a_(n-1)*(2*n)/1
c f_n=1/sqrt(2*pi)*(1*2*3*...*n)*(n+1+r)^(-n-1/2)*exp(n+1+r)
c C.Lanczos,A Precision Approximation of the Gamma Function,
c J.SIAM Numer.Anal.Ser.B,Vol.1,1964
c
c Modified 28 Oct 98 by Andrew J S Hamilton
c http://casa.colorado.edu/~ajsh/
c (1) to return ln[Gamma(x)] with the correct phase,
c as well as Gamma(x), and
c (2) to remain accurate for large absolute values of input x,
c and for x near 0 or negative integers.
c-----------------------------------------------------------------------
complex*16 function cdgamma(x,l)
integer l
complex*16 x
c *
c * Complex Gamma function in double precision.
c *
c * l = 0: Gamma(x)
c * 1: ln[Gamma(x)]
c *
real*8 pi,pv,pu,pr,p1,p2,p3,p4,p5,p6,q1,q2,q3,q4,q5,q6
parameter (
& pi = 3.14159265358979324d+00,
& pv = 7.31790632447016203d+00,
& pu = 3.48064577727581257d+00,
& pr = 3.27673720261526849d-02,
& p1 = 1.05400280458730808d+01,
& p2 = 4.73821439163096063d+01,
& p3 = 9.11395751189899762d+01,
& p4 = 6.62756400966213521d+01,
& p5 = 1.32280130755055088d+01,
& p6 = 2.93729529320536228d-01)
parameter (
& q1 = 9.99999999999975753d-01,
& q2 = 2.00000000000603851d+00,
& q3 = 2.99999999944915534d+00,
& q4 = 4.00000003016801681d+00,
& q5 = 4.99999857982434025d+00,
& q6 = 6.00009857740312429d+00)
real*8 big
parameter (big=1.d20)
real*8 t,ui,ur,vi,vr,wr,wi,xi,xr,yi,yr,zero
data zero /0.d0/
c
xr = dble(x)
xi = dimag(x)
c---x = 0, -1, -2
if (xr .eq. aint(xr) .and. xr .le. 0.d0 .and. xi .eq. 0.d0) then
c...Gamma
if (l .eq. 0) then
wr = xr / 2.d0
c +Infinity at even negative integers
if (wr .eq. aint(wr)) then
yr = 1.d0 / zero
c -Infinity at odd negative integers
else
yr = -1.d0 / zero
endif
yi = 0.d0
c...lnGamma
elseif (l .eq. 1) then
c real part is +Infinity
yr = 1.d0 / zero
yi = pi * aint(xr)
endif
goto 200
endif
c---Re(x) < 1/2 : use reflection formula
if (xr .lt. .5d0) then
wr = 1.d0 - xr
wi = -xi
else
wr = xr
wi = xi
end if
c---large |x|: keep only leading term of rational function
t = wr * wr + wi * wi
if (t .gt. big * big) then
c Rational function v
vr = wr / t + pr
vi = - wi / t
c ln(overall factor)
c u = ln(x + pv) - 1
yr = wr + pv
ur = yr
if (ur .lt. 0.d0) ur = - ur
ui = wi
if (ui .lt. 0.d0) ui = - ui
if (ur.ge.ui) then
t = wi / yr
ur = log(ur) + log(1.d0 + t * t) / 2.d0 - 1.d0
else
t = yr / wi
ur = log(ui) + log(1.d0 + t * t) / 2.d0 - 1.d0
endif
ui = atan2(wi, yr)
c---not large |x|
else
c u = u(x) = x + q6 = O(x)
ur = wr + q6
c v = v(x,u) = (x + q5) u = (x + q5)(x + q6) = O(x^2)
vr = ur * (wr + q5) - wi * wi
vi = wi * (wr + q5) + ur * wi
c y = y(x,u,v) = p6 + p5 u + p4 v = p4 x^2 + ... = O(x^2)
yr = p6 + (p5 * ur + p4 * vr)
yi = p5 * wi + p4 * vi
c u = u(x,v) = (x + q4) v = (x + q4)(x + q5)(x + q6) = O(x^3)
ur = vr * (wr + q4) - vi * wi
ui = vi * (wr + q4) + vr * wi
c v = v(x,u) = (x + q3) u = (x + q3)(x + q4)...(x + q6) = O(x^4)
vr = ur * (wr + q3) - ui * wi
vi = ui * (wr + q3) + ur * wi
c y = y(y,u,v) = y + p3 u + p2 v = p2 x^4 + ... = O(x^4)
yr = yr + (p3 * ur + p2 * vr)
yi = yi + (p3 * ui + p2 * vi)
c u = u(x,v) = (x + q2) v = (x + q2)(x + q3)...(x + q6) = O(x^5)
ur = vr * (wr + q2) - vi * wi
ui = vi * (wr + q2) + vr * wi
c v = v(x,u) = (x + q1) u = (x + q1)(x + q2)...(x + q6) = O(x^6)
vr = ur * (wr + q1) - ui * wi
vi = ui * (wr + q1) + ur * wi
c Numerator
c y = y(y,u,v) = y + p1 u + v = x^6 + ... = O(x^6)
c = (x+q1)...(x+q6) + p1 (x+q2)...(x+q6) + ... + p5 (x+q6) + p6
yr = yr + (p1 * ur + vr)
yi = yi + (p1 * ui + vi)
c Denominator
c u = x v = x(x + q1)(x + q2)...(x + q6) = O(x^7)
ur = vr * wr - vi * wi
ui = vi * wr + vr * wi
c t = |u|^2
t = ur * ur + ui * ui
c Rational function v = y u*/|u|^2 + pr = y/u + pr = pr + 1/x + ...
c = pr + 1/x ( 1 + 1/(x+q1) ( p1 + 1/(x+q2) ( p2 + ...
vr = (yr * ur + yi * ui) / t + pr
vi = (yi * ur - yr * ui) / t
c Overall factor
c u = ln(x + pv) - 1
yr = wr + pv
ur = log(yr * yr + wi * wi) / 2.d0 - 1.d0
ui = atan2(wi, yr)
endif
c---lnGamma
c y = u(x - .5) - pu
c = (x - .5) [ln(x + pv) - 1] - pu
yr = ur * (wr - 0.5d0) - ui * wi - pu
yi = ui * (wr - 0.5d0) + ur * wi
c y = y + ln(v)
c = (x - .5) [ln(x + pv) - 1] - pu + ln(Rational)
c = lnGamma(x)
yr = yr + log(vr * vr + vi * vi) / 2.d0
yi = yi + atan2(vi, vr)
c---Reflection formula Gamma(x) Gamma(1-x) = pi/sin(pi x)
c sign of Gamma
t = 1.d0
if (xr .lt. .5d0) then
wi = anint(xr)
wr = xr - wi
if (wi .gt. xr) wi = wi - 1.d0
c case of real x
if (xi .eq. 0.d0) then
c w = ln[sin(pi x)]
wr = log(sin(pi * abs(wr)))
if (l .eq. 0) then
if (wi .ne. 2.d0 * aint(wi / 2)) t = -1.d0
wi = 0.d0
elseif (l .eq. 1) then
wi = - pi * wi
endif
c case where imaginary part of x is < 1 in absolute value
elseif (abs(xi) .lt. 1.d0) then
if (l .eq. 0) then
if (wi .ne. 2.d0 * aint(wi / 2.d0)) t = -1.d0
ui = 0.d0
elseif (l .eq. 1) then
ui = -pi * wi
if (xi .lt. 0.d0) ui = -ui
endif
wr = pi * wr
wi = pi * xi
vr = sin(wr) * cosh(wi)
vi = cos(wr) * sinh(wi)
if (wr .lt. 0.d0) then
vr = -vr
vi = -vi
endif
c w = ln[sin(pi x)]
wr = log(vr * vr + vi * vi) / 2.d0
wi = ui + atan2(vi, vr)
c case where imaginary part of x is >= 1 in absolute value
else
if (l .eq. 0) then
if (wi .ne. 2.d0 * aint(wi / 2)) t = -1.d0
if (wr .ge. 0.d0) then
ui = pi * (.5d0 - wr)
else
ui = pi * (- .5d0 - wr)
endif
elseif (l .eq. 1) then
ui = pi * (.5d0 - xr)
endif
wi = exp(- 2.d0 * pi * abs(xi))
wr = 2.d0 * pi * wr
vr = (1.d0 - cos(wr) * wi) / 2.d0
vi = - sin(wr) * wi / 2.d0
ur = pi * xi
c w = ln[sin(pi x)]
if (xi .gt. 0.d0) then
wr = ur + log(vr * vr + vi * vi) / 2.d0
wi = ui + atan2(vi, vr)
elseif (xi .lt. 0.d0) then
wr = - ur + log(vr * vr + vi * vi) / 2.d0
wi = - ui - atan2(vi, vr)
endif
endif
c y = ln[Gamma(x)]
yr = log(pi) - yr - wr
yi = - yi - wi
endif
c---Gamma
if (l .eq. 0) then
ur = exp(yr)
if (xi .eq. 0.d0) then
yr = t * ur
yi = 0.d0
else
yr = t * ur * cos(yi)
yi = ur * sin(yi)
endif
endif
c---finish
200 cdgamma = dcmplx(yr, yi)
end
c