def test_jtheta_identities():
"""
Tests the some of the jacobi identidies found in Abramowitz,
Sec. 16.28, Pg. 576. The identies are tested to 1 part in 10^98.
"""
mp.dps = 110
eps1 = ldexp(eps, 30)
for i in range(10):
qstring = str(random.random())
q = mpf(qstring)
zstring = str(10 * random.random())
z = mpf(zstring)
# Abramowitz 16.28.1
# v_1(z, q)**2 * v_4(0, q)**2 = v_3(z, q)**2 * v_2(0, q)**2
# - v_2(z, q)**2 * v_3(0, q)**2
term1 = (jtheta(1, z, q)**2) * (jtheta(4, zero, q)**2)
term2 = (jtheta(3, z, q)**2) * (jtheta(2, zero, q)**2)
term3 = (jtheta(2, z, q)**2) * (jtheta(3, zero, q)**2)
equality = term1 - term2 + term3
assert (equality.ae(0, eps1))
zstring = str(100 * random.random())
z = mpf(zstring)
# Abramowitz 16.28.2
# v_2(z, q)**2 * v_4(0, q)**2 = v_4(z, q)**2 * v_2(0, q)**2
# - v_1(z, q)**2 * v_3(0, q)**2
term1 = (jtheta(2, z, q)**2) * (jtheta(4, zero, q)**2)
term2 = (jtheta(4, z, q)**2) * (jtheta(2, zero, q)**2)
term3 = (jtheta(1, z, q)**2) * (jtheta(3, zero, q)**2)
equality = term1 - term2 + term3
assert (equality.ae(0, eps1))
# Abramowitz 16.28.3
# v_3(z, q)**2 * v_4(0, q)**2 = v_4(z, q)**2 * v_3(0, q)**2
# - v_1(z, q)**2 * v_2(0, q)**2
term1 = (jtheta(3, z, q)**2) * (jtheta(4, zero, q)**2)
term2 = (jtheta(4, z, q)**2) * (jtheta(3, zero, q)**2)
term3 = (jtheta(1, z, q)**2) * (jtheta(2, zero, q)**2)
equality = term1 - term2 + term3
assert (equality.ae(0, eps1))
# Abramowitz 16.28.4
# v_4(z, q)**2 * v_4(0, q)**2 = v_3(z, q)**2 * v_3(0, q)**2
# - v_2(z, q)**2 * v_2(0, q)**2
term1 = (jtheta(4, z, q)**2) * (jtheta(4, zero, q)**2)
term2 = (jtheta(3, z, q)**2) * (jtheta(3, zero, q)**2)
term3 = (jtheta(2, z, q)**2) * (jtheta(2, zero, q)**2)
equality = term1 - term2 + term3
assert (equality.ae(0, eps1))
# Abramowitz 16.28.5
# v_2(0, q)**4 + v_4(0, q)**4 == v_3(0, q)**4
term1 = (jtheta(2, zero, q))**4
term2 = (jtheta(4, zero, q))**4
term3 = (jtheta(3, zero, q))**4
equality = term1 + term2 - term3
assert (equality.ae(0, eps1))
mp.dps = 15
def test_jtheta_identities():
"""
Tests the some of the jacobi identidies found in Abramowitz,
Sec. 16.28, Pg. 576. The identies are tested to 1 part in 10^98.
"""
mp.dps = 110
eps1 = ldexp(eps, 30)
for i in range(10):
qstring = str(random.random())
q = mpf(qstring)
zstring = str(10*random.random())
z = mpf(zstring)
# Abramowitz 16.28.1
# v_1(z, q)**2 * v_4(0, q)**2 = v_3(z, q)**2 * v_2(0, q)**2
# - v_2(z, q)**2 * v_3(0, q)**2
term1 = (jtheta(1, z, q)**2) * (jtheta(4, zero, q)**2)
term2 = (jtheta(3, z, q)**2) * (jtheta(2, zero, q)**2)
term3 = (jtheta(2, z, q)**2) * (jtheta(3, zero, q)**2)
equality = term1 - term2 + term3
assert(equality.ae(0, eps1))
zstring = str(100*random.random())
z = mpf(zstring)
# Abramowitz 16.28.2
# v_2(z, q)**2 * v_4(0, q)**2 = v_4(z, q)**2 * v_2(0, q)**2
# - v_1(z, q)**2 * v_3(0, q)**2
term1 = (jtheta(2, z, q)**2) * (jtheta(4, zero, q)**2)
term2 = (jtheta(4, z, q)**2) * (jtheta(2, zero, q)**2)
term3 = (jtheta(1, z, q)**2) * (jtheta(3, zero, q)**2)
equality = term1 - term2 + term3
assert(equality.ae(0, eps1))
# Abramowitz 16.28.3
# v_3(z, q)**2 * v_4(0, q)**2 = v_4(z, q)**2 * v_3(0, q)**2
# - v_1(z, q)**2 * v_2(0, q)**2
term1 = (jtheta(3, z, q)**2) * (jtheta(4, zero, q)**2)
term2 = (jtheta(4, z, q)**2) * (jtheta(3, zero, q)**2)
term3 = (jtheta(1, z, q)**2) * (jtheta(2, zero, q)**2)
equality = term1 - term2 + term3
assert(equality.ae(0, eps1))
# Abramowitz 16.28.4
# v_4(z, q)**2 * v_4(0, q)**2 = v_3(z, q)**2 * v_3(0, q)**2
# - v_2(z, q)**2 * v_2(0, q)**2
term1 = (jtheta(4, z, q)**2) * (jtheta(4, zero, q)**2)
term2 = (jtheta(3, z, q)**2) * (jtheta(3, zero, q)**2)
term3 = (jtheta(2, z, q)**2) * (jtheta(2, zero, q)**2)
equality = term1 - term2 + term3
assert(equality.ae(0, eps1))
# Abramowitz 16.28.5
# v_2(0, q)**4 + v_4(0, q)**4 == v_3(0, q)**4
term1 = (jtheta(2, zero, q))**4
term2 = (jtheta(4, zero, q))**4
term3 = (jtheta(3, zero, q))**4
equality = term1 + term2 - term3
assert(equality.ae(0, eps1))
mp.dps = 15
def test_jsn():
"""
Test some special cases of the sn(z, q) function.
"""
mp.dps = 100
# trival case
result = jsn(zero, zero)
assert (result == zero)
# Abramowitz Table 16.5
#
# sn(0, m) = 0
for i in range(10):
qstring = str(random.random())
q = mpf(qstring)
equality = jsn(zero, q)
assert (equality.ae(0))
# Abramowitz Table 16.6.1
#
# sn(z, 0) = sin(z), m == 0
#
# sn(z, 1) = tanh(z), m == 1
#
# It would be nice to test these, but I find that they run
# in to numerical trouble. I'm currently treating as a boundary
# case for sn function.
mp.dps = 25
arg = one / 10
#N[JacobiSN[1/10, 2^-100], 25]
res = mpf('0.09983341664682815230681420')
m = ldexp(one, -100)
result = jsn(arg, m)
assert (result.ae(res))
# N[JacobiSN[1/10, 1/10], 25]
res = mpf('0.09981686718599080096451168')
result = jsn(arg, arg)
assert (result.ae(res))
mp.dps = 15
def test_jsn():
"""
Test some special cases of the sn(z, q) function.
"""
mp.dps = 100
# trival case
result = jsn(zero, zero)
assert(result == zero)
# Abramowitz Table 16.5
#
# sn(0, m) = 0
for i in range(10):
qstring = str(random.random())
q = mpf(qstring)
equality = jsn(zero, q)
assert(equality.ae(0))
# Abramowitz Table 16.6.1
#
# sn(z, 0) = sin(z), m == 0
#
# sn(z, 1) = tanh(z), m == 1
#
# It would be nice to test these, but I find that they run
# in to numerical trouble. I'm currently treating as a boundary
# case for sn function.
mp.dps = 25
arg = one/10
#N[JacobiSN[1/10, 2^-100], 25]
res = mpf('0.09983341664682815230681420')
m = ldexp(one, -100)
result = jsn(arg, m)
assert(result.ae(res))
# N[JacobiSN[1/10, 1/10], 25]
res = mpf('0.09981686718599080096451168')
result = jsn(arg, arg)
assert(result.ae(res))
mp.dps = 15
def test_jcn():
"""
Test some special cases of the cn(z, q) function.
"""
mp.dps = 100
# Abramowitz Table 16.5
# cn(0, q) = 1
qstring = str(random.random())
q = mpf(qstring)
cn = jcn(zero, q)
assert (cn.ae(one))
# Abramowitz Table 16.6.2
#
# cn(u, 0) = cos(u), m == 0
#
# cn(u, 1) = sech(z), m == 1
#
# It would be nice to test these, but I find that they run
# in to numerical trouble. I'm currently treating as a boundary
# case for cn function.
mp.dps = 25
arg = one / 10
m = ldexp(one, -100)
#N[JacobiCN[1/10, 2^-100], 25]
res = mpf('0.9950041652780257660955620')
result = jcn(arg, m)
assert (result.ae(res))
# N[JacobiCN[1/10, 1/10], 25]
res = mpf('0.9950058256237368748520459')
result = jcn(arg, arg)
assert (result.ae(res))
mp.dps = 15
def test_jcn():
"""
Test some special cases of the cn(z, q) function.
"""
mp.dps = 100
# Abramowitz Table 16.5
# cn(0, q) = 1
qstring = str(random.random())
q = mpf(qstring)
cn = jcn(zero, q)
assert(cn.ae(one))
# Abramowitz Table 16.6.2
#
# cn(u, 0) = cos(u), m == 0
#
# cn(u, 1) = sech(z), m == 1
#
# It would be nice to test these, but I find that they run
# in to numerical trouble. I'm currently treating as a boundary
# case for cn function.
mp.dps = 25
arg = one/10
m = ldexp(one, -100)
#N[JacobiCN[1/10, 2^-100], 25]
res = mpf('0.9950041652780257660955620')
result = jcn(arg, m)
assert(result.ae(res))
# N[JacobiCN[1/10, 1/10], 25]
res = mpf('0.9950058256237368748520459')
result = jcn(arg, arg)
assert(result.ae(res))
mp.dps = 15
