def test_odefun_harmonic():
mp.dps = 15
# Harmonic oscillator
f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0])
for x in [0, 1, 2.5, 8, 3.7]: # we go back to 3.7 to check caching
c, s = f(x)
assert c.ae(cos(x))
assert s.ae(sin(x))
def clenshaw_curtis_nodes(n):
"""Return Clenshaw-Curtis nodes (actually returns n+1 nodes)."""
r = set()
for k in range(n+1):
r.add(mpmath.cos(k*mpmath.pi/mpmath.mpf(n)))
return sorted(r)
def test_odefun_harmonic():
mp.dps = 15
# Harmonic oscillator
f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0])
for x in [0, 1, 2.5, 8, 3.7]: # we go back to 3.7 to check caching
c, s = f(x)
assert c.ae(cos(x))
assert s.ae(sin(x))
def clenshaw_curtis_nodes(n):
"""Return Clenshaw-Curtis nodes."""
r = set()
for i in range(n):
r.add(mpmath.cos(i*mpmath.pi/mpmath.mpf(n-1)))
return map_to_zero_one(r)
def test_odefun_sinc_large():
mp.dps = 15
# Sinc function; test for large x
f = sinc
g = odefun(lambda x, y: [(cos(x) - y[0]) / x],
1, [f(1)],
tol=0.01,
degree=5)
assert abs(f(100) - g(100)[0]) / f(100) < 0.01
def clenshaw_curtis_weights(n):
"""Return Clenshaw-Curtis weights (actually the n+1 weights)."""
w = []
for k in range(n+1):
c_k = 1.0 if k % n == 0 else 2.0
t_k = k*mpmath.pi/mpmath.mpf(n)
acc = 1.0
for j in range(1, n/2+1):
b_j = 2.0 if j < n/2 else 1.0
acc -= b_j / (4.0*j**2 - 1.0) * mpmath.cos(2*j*t_k)
w.append(c_k / n * acc)
return w
p = np.poly1d([1, 0, 0, 0, 1])
# use mayavi if possible.
# otherwise, fall back to matplotlib.
if use_mayavi:
plot_both_mayavi(p)
nroot_real_mayavi(5)
nroot_imag_mayavi(5)
else:
plot_both_matplotlib(p)
nroot_real_matplotlib(5)
nroot_imag_matplotlib(5)
# All the different integration examples.
f1 = lambda z: z.conjugate()
f2 = lambda z: mp.exp(z)
c1 = lambda t: mp.cos(t) + 1.0j * mp.sin(t)
c2 = lambda t: t + 1.0j * t
c3 = lambda t: t
c4 = lambda t: 1 + 1.0j * t
c5 = lambda t: mp.cos(t) + 1.0j + 1.0j * mp.sin(t)
print "z conjugate counterclockwise along the unit ball starting and ending at 1."
print contour_int(f1, c1, 0, 2 * np.pi)
print "z conjugate along a straight line from 0 to 1+1j."
print contour_int(f1, c2, 0, 1)
print "z conjugate along the real axis from 0 to 1, then along the line from 1 to 1+1j."
print contour_int(f1, c3, 0, 1) + contour_int(f1, c4, 0, 1)
print "z conjugate along the unit ball centered at 1j from 0 to 1+1j."
print contour_int(f1, c5, -np.pi / 2, 0.)
print "e^z counterclockwise along the unit ball starting and ending at 1."
print contour_int(f2, c1, 0, 2 * np.pi)
print "e^z along a straight line from 0 to 1+1j."
def test_odefun_sinc_large():
mp.dps = 15
# Sinc function; test for large x
f = sinc
g = odefun(lambda x, y: [(cos(x)-y[0])/x], 1, [f(1)], tol=0.01, degree=5)
assert abs(f(100) - g(100)[0])/f(100) < 0.01
def pi_inner_integral(y):
vardict = {'y': y}
out = maple_link.eval_stack(pi_stack, vardict)
return out * cos(int(j) * y)
def pi_inner_integral(y):
vardict = {'y':y}
out = maple_link.eval_stack(pi_stack, vardict)
return out*cos(int(j)*y)
def f(x):
return d_polynomial(x, list_a) - sm.cos(x)
p = np.poly1d([1, 0, 0, 0, 1])
# use mayavi if possible.
# otherwise, fall back to matplotlib.
if use_mayavi:
plot_both_mayavi(p)
nroot_real_mayavi(5)
nroot_imag_mayavi(5)
else:
plot_both_matplotlib(p)
nroot_real_matplotlib(5)
nroot_imag_matplotlib(5)
# All the different integration examples.
f1 = lambda z: z.conjugate()
f2 = lambda z: mp.exp(z)
c1 = lambda t: mp.cos(t) + 1.0j * mp.sin(t)
c2 = lambda t: t + 1.0j * t
c3 = lambda t: t
c4 = lambda t: 1 + 1.0j * t
c5 = lambda t: mp.cos(t) + 1.0j + 1.0j * mp.sin(t)
print "z conjugate counterclockwise along the unit ball starting and ending at 1."
print contour_int(f1, c1, 0, 2 * np.pi)
print "z conjugate along a straight line from 0 to 1+1j."
print contour_int(f1, c2, 0, 1)
print "z conjugate along the real axis from 0 to 1, then along the line from 1 to 1+1j."
print contour_int(f1, c3, 0, 1) + contour_int(f1, c4, 0, 1)
print "z conjugate along the unit ball centered at 1j from 0 to 1+1j."
print contour_int(f1, c5, -np.pi / 2, 0.)
print "e^z counterclockwise along the unit ball starting and ending at 1."
print contour_int(f2, c1, 0, 2 * np.pi)
print "e^z along a straight line from 0 to 1+1j."
def f(x):
return d_polynomial(x, list_a) - sm.cos(x);