def _RD(x, y, z): # used by testElliptic.py
'''Degenerate symmetric integral of the third kind C{_RD}.
@return: C{_RD(x, y, z) = _RJ(x, y, z, z)}.
@see: U{C{_RD} definition<https://DLMF.NIST.gov/19.16.E5>}.
'''
# Carlson, eqs 2.28 - 2.34
m = 1.0
S = Fsum()
A = fsum_(x, y, z, z, z) * 0.2
T = (A, x, y, z)
Q = _Q(A, T, _TolRD)
for _ in range(_TRIPS):
if Q < abs(m * T[0]): # max 7 trips
break
t = T[3] # z0
r, s, T = _rsT(T)
S += 1.0 / (m * s[2] * (t + r))
m *= 4
else:
raise _convergenceError(_RD, x, y, z)
m *= T[0] # An
x = (x - A) / m
y = (y - A) / m
z = (x + y) / 3.0
z2 = z**2
xy = x * y
return _Horner(3 * S.fsum(), m * sqrt(T[0]),
xy - 6 * z2,
(xy * 3 - 8 * z2) * z,
(xy - z2) * 3 * z2,
xy * z2 * z)
def _RG(x, y, z=None):
'''Symmetric integral of the second kind C{_RG}.
@return: C{_RG(x, y, z)}.
@see: U{C{_RG} definition<https://DLMF.NIST.gov/19.16.E3>}
and in Carlson, eq. 1.5.
'''
if z is None:
# Carlson, eqs 2.36 - 2.39
a, b = sqrt(x), sqrt(y)
if a < b:
a, b = b, a
S = Fsum(0.25 * (a + b)**2)
m = -0.25 # note, negative
while abs(a - b) > (_tolRG0 * a): # max 4 trips
b, a = sqrt(a * b), (a + b) * 0.5
m *= 2
S.fadd_(m * (a - b)**2)
return S.fsum() * PI_2 / (a + b)
if not z:
y, z = z, y
# Carlson, eq 1.7
return fsum_(
_RF(x, y, z) * z,
_RD_3(x, y, z) * (x - z) * (z - y), sqrt(x * y / z)) * 0.5
def _RG_(x, y):
'''Symmetric integral of the second kind C{_RG}.
@return: C{_RG(x, y)}.
@see: U{C{_RG} definition<https://DLMF.NIST.gov/19.16.E3>}
and in Carlson, eq. 1.5.
'''
# Carlson, eqs 2.36 - 2.39
a, b = sqrt(x), sqrt(y)
if a < b:
a, b = b, a
m = 0.25
S = Fsum(m * (a + b)**2)
while abs(a - b) > (_TolRG0 * a): # max 4 trips
b, a = sqrt(a * b), (a + b) * 0.5
m *= 2
S -= m * (a - b)**2
return S.fsum() * PI_2 / (a + b)
def _RJ(x, y, z, p): # used by testElliptic.py
'''Symmetric integral of the third kind C{_RJ(x, y, z, p)}.
@return: C{_RJ(x, y, z, p)}.
@see: U{C{_RJ} definition<https://DLMF.NIST.gov/19.16.E2>} and
U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
'''
def _xyzp(x, y, z, p):
return (x + p) * (y + p) * (z + p)
# Carlson, eqs 2.17 - 2.25
m = m3 = _1_0
S = Fsum()
D = neg(_xyzp(x, y, z, -p))
A = fsum_(x, y, z, 2 * p) * _0_2
T = (A, x, y, z, p)
Q = _Q(A, T, _TolRD)
for _ in range(_TRIPS):
if Q < abs(m * T[0]): # max 7 trips
break
_, s, T = _rsT(T)
d = _xyzp(*s)
e = D / (m3 * d**2)
S += _RC(1, 1 + e) / (m * d)
m *= _4_0
m3 *= 64
else:
raise _convergenceError(_RJ, x, y, z, p)
m *= T[0] # An
x = (A - x) / m
y = (A - y) / m
z = (A - z) / m
xyz = x * y * z
p = neg(x + y + z) * _0_5
p2 = p**2
e2 = fsum_(x * y, x * z, y * z, -3 * p2)
return _horner(6 * S.fsum(), m * sqrt(T[0]), e2,
fsum_(xyz, 2 * p * e2, 4 * p * p2),
fsum_(xyz * 2, p * e2, 3 * p * p2) * p,
p2 * xyz)
def _RJ(x, y, z, p):
'''Symmetric integral of the third kind C{_RJ}.
@return: C{_RJ(x, y, z, p)}.
@see: U{C{_RJ} definition<https://DLMF.NIST.gov/19.16.E2>}.
'''
def _xyzp(x, y, z, p):
return (x + p) * (y + p) * (z + p)
# Carlson, eqs 2.17 - 2.25
m = m3 = 1.0
S = Fsum()
D = -_xyzp(x, y, z, -p)
A = fsum_(x, y, z, 2 * p) * 0.2
T = [A, x, y, z, p]
Q = _Q(A, T, _tolRD)
for _ in range(_TRIPS):
if Q < abs(m * T[0]): # max 7 trips
break
_, s, T = _rsT(T)
d = _xyzp(*s)
e = D / (m3 * d**2)
S.fadd_(_RC(1, 1 + e) / (m * d))
m *= 4
m3 *= 64
else:
raise EllipticError('no %s convergence' % ('RJ', ))
S *= 6
m *= T[0] # An
x = (A - x) / m
y = (A - y) / m
z = (A - z) / m
xyz = x * y * z
p = -(x + y + z) * 0.5
p2 = p**2
e2 = fsum_(x * y, x * z, y * z, -3 * p2)
return _Hf(S.fsum(), m * sqrt(T[0]), e2, fsum_(xyz, 2 * p * e2,
4 * p * p2),
fsum_(xyz * 2, p * e2, 3 * p * p2) * p, p2 * xyz)
def _RG_(x, y):
'''Symmetric integral of the second kind C{_RG}.
@return: C{_RG(x, y)}.
@see: U{C{_RG} definition<https://DLMF.NIST.gov/19.16.E3>} and
U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
'''
# Carlson, eqs 2.36 - 2.39
a, b = sqrt(x), sqrt(y)
if a < b:
a, b = b, a
m = 0.25
S = Fsum(m * (a + b)**2)
for _ in range(_TRIPS): # max 4 trips
if abs(a - b) <= (_TolRG0 * a):
return S.fsum() * PI_2 / (a + b)
b, a = sqrt(a * b), (a + b) * 0.5
m *= 2
S -= m * (a - b)**2
raise _convergenceError(_RG_, x, y)
