def jacobi_elliptic_cn(u, m, verbose=False):
"""
Implements the jacobi elliptic cn function, using the expansion in
terms of q, from Abramowitz 16.23.2.
"""
u = convert_lossless(u)
m = convert_lossless(m)
if verbose:
print >> sys.stderr, '\nelliptic.jacobi_elliptic_cn'
print >> sys.stderr, '\tu: %1.12f' % u
print >> sys.stderr, '\tm: %1.12f' % m
zero = mpf('0')
onehalf = mpf('0.5')
one = mpf('1')
two = mpf('2')
if m == zero: # cn collapses to cos(u)
if verbose:
print >> sys.stderr, 'cn: special case, m == 0'
return cos(u)
elif m == one: # cn collapses to sech(u)
if verbose:
print >> sys.stderr, 'cn: special case, m == 1'
return sech(u)
else:
k = sqrt(m) # convert m to k
q = calculate_nome(k)
kprimesquared = one - k**2
kprime = sqrt(kprimesquared)
v = (pi * u) / (two*ellipk(k**2))
sum = zero
term = zero # series starts at zero
if verbose:
print >> sys.stderr, 'elliptic.jacobi_elliptic_cn: calculating'
while True:
factor1 = (q**(term + onehalf)) / (one + q**(two*term + one))
factor2 = cos((two*term + one)*v)
term_n = factor1*factor2
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if not factor2 == zero:
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + one
answer = (two*pi) / (sqrt(m) * ellipk(k**2)) * sum
return answer
def jacobi_elliptic_cn(u, m, verbose=False):
"""
Implements the jacobi elliptic cn function, using the expansion in
terms of q, from Abramowitz 16.23.2.
"""
u = convert_lossless(u)
m = convert_lossless(m)
if verbose:
print >> sys.stderr, '\nelliptic.jacobi_elliptic_cn'
print >> sys.stderr, '\tu: %1.12f' % u
print >> sys.stderr, '\tm: %1.12f' % m
zero = mpf('0')
onehalf = mpf('0.5')
one = mpf('1')
two = mpf('2')
if m == zero: # cn collapses to cos(u)
if verbose:
print >> sys.stderr, 'cn: special case, m == 0'
return cos(u)
elif m == one: # cn collapses to sech(u)
if verbose:
print >> sys.stderr, 'cn: special case, m == 1'
return sech(u)
else:
k = sqrt(m) # convert m to k
q = calculate_nome(k)
kprimesquared = one - k**2
kprime = sqrt(kprimesquared)
v = (pi * u) / (two * ellipk(k**2))
sum = zero
term = zero # series starts at zero
if verbose:
print >> sys.stderr, 'elliptic.jacobi_elliptic_cn: calculating'
while True:
factor1 = (q**(term + onehalf)) / (one + q**(two * term + one))
factor2 = cos((two * term + one) * v)
term_n = factor1 * factor2
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if not factor2 == zero:
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + one
answer = (two * pi) / (sqrt(m) * ellipk(k**2)) * sum
return answer
def chebcoeff(f,a,b,j,N):
s = mpf(0)
h = mpf(0.5)
for k in range(1, N+1):
t = cos(pi*(k-h)/N)
s += f(t*(b-a)*h + (b+a)*h) * cos(pi*j*(k-h)/N)
return 2*s/N
def jacobi_theta_3(z, m):
"""
Implements the jacobi theta function 2, using the series expansion
found in Abramowitz & Stegun [4].
z is any complex number, but only reals here?
m is the parameter, which must be converted to the nome
"""
m = convert_lossless(m)
z = convert_lossless(z)
k = sqrt(m)
q = calculate_nome(k)
if abs(q) >= mpf('1'):
raise ValueError
delta_sum = 1
term = 1 # series starts at 1
zero = mpf('0')
sum = zero
if z == zero:
factor2 = mpf('1')
while True:
factor1 = q**(term*term)
term_n = factor1 # suboptimal, kept for readability
sum = sum + term_n
if factor1 == mpf('0'): # all further terms will be zero
break
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
else:
while True:
factor1 = q**(term*term)
if z == zero:
factor2 = 1
else:
factor2 = cos(2*term*z)
term_n = factor1 * factor2
sum = sum + term_n
if factor1 == mpf('0'): # all further terms will be zero
break
if factor2 != mpf('0'): # check precision iff cos != 0
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
return 1 + 2*sum
def jacobi_theta_3(z, m):
"""
Implements the jacobi theta function 2, using the series expansion
found in Abramowitz & Stegun [4].
z is any complex number, but only reals here?
m is the parameter, which must be converted to the nome
"""
m = convert_lossless(m)
z = convert_lossless(z)
k = sqrt(m)
q = calculate_nome(k)
if abs(q) >= mpf('1'):
raise ValueError
delta_sum = 1
term = 1 # series starts at 1
zero = mpf('0')
sum = zero
if z == zero:
factor2 = mpf('1')
while True:
factor1 = q**(term * term)
term_n = factor1 # suboptimal, kept for readability
sum = sum + term_n
if factor1 == mpf('0'): # all further terms will be zero
break
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
else:
while True:
factor1 = q**(term * term)
if z == zero:
factor2 = 1
else:
factor2 = cos(2 * term * z)
term_n = factor1 * factor2
sum = sum + term_n
if factor1 == mpf('0'): # all further terms will be zero
break
if factor2 != mpf('0'): # check precision iff cos != 0
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
return 1 + 2 * sum
def chebyfit(f, interval, N, error=False):
"""
Chebyshev approximation: returns coefficients of a degree N-1
polynomial that approximates f on the interval [a, b]. With error=True,
also returns an estimate of the maximum error.
"""
a, b = AS_POINTS(interval)
orig = mp.prec
try:
mp.prec = orig + int(N**0.5) + 20
c = [chebcoeff(f,a,b,k,N) for k in range(N)]
d = [mpf(0)] * N
d[0] = -c[0]/2
h = mpf(0.5)
T = chebT(mpf(2)/(b-a), mpf(-1)*(b+a)/(b-a))
for k in range(N):
Tk = T.next()
for i in range(len(Tk)):
d[i] += c[k]*Tk[i]
d = d[::-1]
# Estimate maximum error
err = mpf(0)
for k in range(N):
x = cos(pi*k/N) * (b-a)*h + (b+a)*h
err = max(err, abs(f(x) - polyval(d, x)))
finally:
mp.prec = orig
if error:
return d, +err
else:
return d
def jacobi_elliptic_dn(u, m, verbose=False):
"""
Implements the jacobi elliptic cn function, using the expansion in
terms of q, from Abramowitz 16.23.3.
"""
u = convert_lossless(u)
m = convert_lossless(m)
if verbose:
print >> sys.stderr, '\nelliptic.jacobi_elliptic_dn'
print >> sys.stderr, '\tu: %1.12f' % u
print >> sys.stderr, '\tm: %1.12f' % m
zero = mpf('0')
onehalf = mpf('0.5')
one = mpf('1')
two = mpf('2')
if m == zero: # dn collapes to 1
return one
elif m == one: # dn collapses to sech(u)
return sech(u)
else:
k = sqrt(m) # convert m to k
q = calculate_nome(k)
v = (pi * u) / (two*ellipk(k**2))
sum = zero
term = one # series starts at one
if verbose:
print >> sys.stderr, 'elliptic.jacobi_elliptic_dn: calculating'
while True:
factor1 = (q**term) / (one + q**(two*term))
factor2 = cos(two*term*v)
term_n = factor1*factor2
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if not factor2 == zero:
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + one
K = ellipk(k**2)
answer = (pi / (two*K)) + (two*pi*sum)/(ellipk(k**2))
return answer
def jacobi_elliptic_dn(u, m, verbose=False):
"""
Implements the jacobi elliptic cn function, using the expansion in
terms of q, from Abramowitz 16.23.3.
"""
u = convert_lossless(u)
m = convert_lossless(m)
if verbose:
print >> sys.stderr, '\nelliptic.jacobi_elliptic_dn'
print >> sys.stderr, '\tu: %1.12f' % u
print >> sys.stderr, '\tm: %1.12f' % m
zero = mpf('0')
onehalf = mpf('0.5')
one = mpf('1')
two = mpf('2')
if m == zero: # dn collapes to 1
return one
elif m == one: # dn collapses to sech(u)
return sech(u)
else:
k = sqrt(m) # convert m to k
q = calculate_nome(k)
v = (pi * u) / (two * ellipk(k**2))
sum = zero
term = one # series starts at one
if verbose:
print >> sys.stderr, 'elliptic.jacobi_elliptic_dn: calculating'
while True:
factor1 = (q**term) / (one + q**(two * term))
factor2 = cos(two * term * v)
term_n = factor1 * factor2
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if not factor2 == zero:
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + one
K = ellipk(k**2)
answer = (pi / (two * K)) + (two * pi * sum) / (ellipk(k**2))
return answer
def main():
"""
main program
"""
#Collect user input
F1 = float(
input("What is the magnitude of the first force (in Newtons)? "))
theta1 = float(input("What is the angle of the first force (in deg)? "))
F2 = float(
input("What is the magnitude of the second force (in Newtons)? "))
theta2 = float(input("What is the angle of the second force (in deg)?"))
#Convert to radians
theta1_rad = functions.deg2rad(theta1)
theta2_rad = functions.deg2rad(theta2)
#x and y components of Fsum
F3x = F1 * functions.cos(theta1_rad) + F2 * functions.cos(theta2_rad)
F3y = F1 * functions.sin(theta1_rad) + F2 * functions.sin(theta2_rad)
print("Net force is: ", F3x, F3y)
def jcn(u, m):
"""
Computes of the Jacobi elliptic cn function in terms
of Jacobi theta functions.
`u` is any complex number, `m` must be in the unit disk
The cn-function is doubly periodic in the complex
plane with periods `4 K(m)` and `4 i K(1-m)`
(see :func:`ellipk`)::
>>> from mpmath import *
>>> mp.dps = 25
>>> print jcn(2, 0.25)
-0.2698649654510865792581416
>>> print jcn(2+4*ellipk(0.25), 0.25)
-0.2698649654510865792581416
>>> print chop(jcn(2+4*j*ellipk(1-0.25), 0.25))
-0.2698649654510865792581416
"""
if abs(m) < eps:
return cos(u)
elif m == one:
return sech(u)
else:
extra = 10
try:
mp.prec += extra
q = calculate_nome(sqrt(m))
v3 = jtheta(3, zero, q)
v2 = jtheta(2, zero, q)
v04 = jtheta(4, zero, q)
arg1 = u / (v3*v3)
v1 = jtheta(2, arg1, q)
v4 = jtheta(4, arg1, q)
cn = (v04/v2)*(v1/v4)
finally:
mp.prec -= extra
return cn
def _djacobi_theta2(z, q, nd):
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
s = s1 + ((a * sn * 3**nd) >> wp)
else:
s = c1 + ((a * cn * 3**nd) >> wp)
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if nd&1:
s += (a * sn * (2*n+1)**nd) >> wp
else:
s += (a * cn * (2*n+1)**nd) >> wp
n += 1
s = -(s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
sre = s1 + ((are * sn * 3**nd) >> wp)
sim = ((aim * sn * 3**nd) >> wp)
else:
sre = c1 + ((are * cn * 3**nd) >> wp)
sim = ((aim * cn * 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
sre += ((are * sn * n**nd) >> wp)
sim += ((aim * sn * n**nd) >> wp)
else:
sre += ((are * cn * n**nd) >> wp)
sim += ((aim * cn * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
prec0 = mp.prec
mp.prec = wp
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre = s1re + ((a * snre * 3**nd) >> wp)
sim = s1im + ((a * snim * 3**nd) >> wp)
else:
sre = c1re + ((a * cnre * 3**nd) >> wp)
sim = c1im + ((a * cnim * 3**nd) >> wp)
n = 5
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += ((a * snre * n**nd) >> wp)
sim += ((a * snim * n**nd) >> wp)
else:
sre += ((a * cnre * n**nd) >> wp)
sim += ((a * cnim * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = mp.prec
mp.prec = wp
# cos(2*z), siz(2*z) with z complex
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre = s1re + (((are * snre - aim * snim) * 3**nd) >> wp)
sim = s1im + (((are * snim + aim * snre)* 3**nd) >> wp)
else:
sre = c1re + (((are * cnre - aim * cnim) * 3**nd) >> wp)
sim = c1im + (((are * cnim + aim * cnre)* 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (((are * snre - aim * snim) * n**nd) >> wp)
sim += (((aim * snre + are * snim) * n**nd) >> wp)
else:
sre += (((are * cnre - aim * cnim) * n**nd) >> wp)
sim += (((aim * cnre + are * cnim) * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
s *= nthroot(q, 4)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
def jacobi_theta_2(z, m, verbose=False):
"""
Implements the jacobi theta function 2, using the series expansion
found in Abramowitz & Stegun [4].
z is any complex number, but only reals here?
m is the parameter, which must be converted to the nome
"""
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_2'
m = convert_lossless(m)
z = convert_lossless(z)
k = sqrt(m)
q = calculate_nome(k)
if verbose:
print >> sys.stderr, '\tk: %f ' % k
print >> sys.stderr, '\tq: %f ' % q
if abs(q) >= mpf('1'):
raise ValueError
sum = 0
term = 0 # series starts at zero
zero = mpf('0')
if q == zero:
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_2: q == 0, return 0'
return zero
else:
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_2: calculating'
if z == zero:
factor2 = mpf('1')
while True:
factor1 = q**(term*(term + 1))
term_n = factor1 # suboptimal, kept for readability
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if factor1 == zero: # all further terms will be zero
break
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
else:
while True:
factor1 = q**(term*(term + 1))
if z == zero:
factor2 = 1
else:
factor2 = cos((2*term + 1)*z)
term_n = factor1 * factor2
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if factor1 == zero: # all further terms will be zero
break
if factor2 != zero: # check precision iff cos != 0
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
return (2*q**(0.25))*sum
# can't get here
print >> sys.stderr, 'elliptic.jacobi_theta_2 in impossible state'
sys.exit(2)
def jacobi_theta_4(z, m):
"""
Implements the series expansion of the jacobi theta function
1, where z == 0.
z is any complex number, but only reals here?
m is the parameter, which must be converted to the nome
"""
m = convert_lossless(m)
z = convert_lossless(z)
k = sqrt(m)
q = calculate_nome(k)
if abs(q) >= mpf('1'):
raise ValueError
sum = 0
delta_sum = 1
term = 1 # series starts at 1
zero = mpf('0')
if z == zero:
factor2 = mpf('1')
while True:
if (term % 2) == 0:
factor0 = 1
else:
factor0 = -1
factor1 = q**(term * term)
term_n = factor0 * factor1 * factor2
sum = sum + term_n
if factor1 == mpf('0'): # all further terms will be zero
break
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
else:
while True:
if (term % 2) == 0:
factor0 = 1
else:
factor0 = -1
factor1 = q**(term * term)
if z == zero:
factor2 = 1
else:
factor2 = cos(2 * term * z)
term_n = factor0 * factor1 * factor2
sum = sum + term_n
if factor1 == mpf('0'): # all further terms will be zero
break
if factor2 != mpf('0'): # check precision iff cos != 0
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
return 1 + 2 * sum
def jacobi_theta_2(z, m, verbose=False):
"""
Implements the jacobi theta function 2, using the series expansion
found in Abramowitz & Stegun [4].
z is any complex number, but only reals here?
m is the parameter, which must be converted to the nome
"""
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_2'
m = convert_lossless(m)
z = convert_lossless(z)
k = sqrt(m)
q = calculate_nome(k)
if verbose:
print >> sys.stderr, '\tk: %f ' % k
print >> sys.stderr, '\tq: %f ' % q
if abs(q) >= mpf('1'):
raise ValueError
sum = 0
term = 0 # series starts at zero
zero = mpf('0')
if q == zero:
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_2: q == 0, return 0'
return zero
else:
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_2: calculating'
if z == zero:
factor2 = mpf('1')
while True:
factor1 = q**(term * (term + 1))
term_n = factor1 # suboptimal, kept for readability
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if factor1 == zero: # all further terms will be zero
break
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
else:
while True:
factor1 = q**(term * (term + 1))
if z == zero:
factor2 = 1
else:
factor2 = cos((2 * term + 1) * z)
term_n = factor1 * factor2
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if factor1 == zero: # all further terms will be zero
break
if factor2 != zero: # check precision iff cos != 0
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
return (2 * q**(0.25)) * sum
# can't get here
print >> sys.stderr, 'elliptic.jacobi_theta_2 in impossible state'
sys.exit(2)
def _jacobi_theta2(z, q):
extra1 = 10
extra2 = 20
# the loops below break when the fixed precision quantities
# a and b go to zero;
# right shifting small negative numbers by wp one obtains -1, not zero,
# so the condition a**2 + b**2 > MIN is used to break the loops.
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
s = x2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << (wp+1)) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
sre = (1<<wp) + are
sim = aim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
else:
if isinstance(q, mpf) and isinstance(z, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
s = c1 + ((a * cn) >> wp)
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
s += (a * cn) >> wp
s = (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
s *= nthroot(q, 4)
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
sre = c1 + ((are * cn) >> wp)
sim = ((aim * cn) >> wp)
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
sre += ((are * cn) >> wp)
sim += ((aim * cn) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
prec0 = mp.prec
mp.prec = wp
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre = c1re + ((a * cnre) >> wp)
sim = c1im + ((a * cnim) >> wp)
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += ((a * cnre) >> wp)
sim += ((a * cnim) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = mp.prec
mp.prec = wp
# cos(z), siz(z) with z complex
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
n = 1
termre = c1re
termim = c1im
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 3
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((are * cnim + aim * cnre) >> wp)
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((aim * cnre + are * cnim) >> wp)
sre += ((are * cnre - aim * cnim) >> wp)
sim += ((aim * cnre + are * cnim) >> wp)
n += 2
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
s *= nthroot(q, 4)
return s
def _djacobi_theta3(z, q, nd):
"""nd=1,2,3 order of the derivative with respect to z"""
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
s += (a * sn) >> wp
else:
s += (a * cn) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
s += (a * sn * n**nd) >> wp
else:
s += (a * cn * n**nd) >> wp
n += 1
s = -(s << (nd+1))
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
sre = (are * sn) >> wp
sim = (aim * sn) >> wp
else:
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
sre += (are * sn * n**nd) >> wp
sim += (aim * sn * n**nd) >> wp
else:
sre += (are * cn * n**nd) >> wp
sim += (aim * cn * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd&1):
sre = (a * snre) >> wp
sim = (a * snim) >> wp
else:
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (a * snre * n**nd) >> wp
sim += (a * snim * n**nd) >> wp
else:
sre += (a * cnre * n**nd) >> wp
sim += (a * cnim * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd&1):
sre = (are * snre - aim * snim) >> wp
sim = (aim * snre + are * snim) >> wp
else:
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if(nd&1):
sre += ((are * snre - aim * snim) * n**nd) >> wp
sim += ((aim * snre + are * snim) * n**nd) >> wp
else:
sre += ((are * cnre - aim * cnim) * n**nd) >> wp
sim += ((aim * cnre + are * cnim) * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
def _jacobi_theta3(z, q):
extra1 = 10
extra2 = 20
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
s = x
a = b = x
x2 = (x*x) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
sre = are = bre = xre
sim = aim = bim = xim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
else:
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
s += (a * cn) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
s += (a * cn) >> wp
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
sre += (are * cn) >> wp
sim += (aim * cn) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (a * cnre) >> wp
sim += (a * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (are * cnre - aim * cnim) >> wp
sim += (aim * cnre + are * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
def jacobi_theta_4(z, m):
"""
Implements the series expansion of the jacobi theta function
1, where z == 0.
z is any complex number, but only reals here?
m is the parameter, which must be converted to the nome
"""
m = convert_lossless(m)
z = convert_lossless(z)
k = sqrt(m)
q = calculate_nome(k)
if abs(q) >= mpf('1'):
raise ValueError
sum = 0
delta_sum = 1
term = 1 # series starts at 1
zero = mpf('0')
if z == zero:
factor2 = mpf('1')
while True:
if (term % 2) == 0:
factor0 = 1
else:
factor0 = -1
factor1 = q**(term*term)
term_n = factor0 * factor1 * factor2
sum = sum + term_n
if factor1 == mpf('0'): # all further terms will be zero
break
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
else:
while True:
if (term % 2) == 0:
factor0 = 1
else:
factor0 = -1
factor1 = q**(term*term)
if z == zero:
factor2 = 1
else:
factor2 = cos(2*term*z)
term_n = factor0 * factor1 * factor2
sum = sum + term_n
if factor1 == mpf('0'): # all further terms will be zero
break
if factor2 != mpf('0'): # check precision iff cos != 0
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
return 1 + 2*sum
import numpy as np import matplotlib.pyplot as plt from functions import sin, cos x = np.linspace(0, 10, 10000) plt.plot(x, sin(x)) plt.plot(x, cos(x)) plt.show()
def cos(self):
return F.cos(self)
