def test_complex_halley():
"""Test Halley's works with complex roots"""
def f(x, *a):
return a[0] * x**2 + a[1] * x + a[2]
def f_1(x, *a):
return 2 * a[0] * x + a[1]
def f_2(x, *a):
retval = 2 * a[0]
try:
size = len(x)
except TypeError:
return retval
else:
return [retval] * size
z = complex(1.0, 2.0)
coeffs = (2.0, 3.0, 4.0)
y = zeros.newton(f, z, args=coeffs, fprime=f_1, fprime2=f_2, tol=1e-6)
# (-0.75000000000000078+1.1989578808281789j)
assert_allclose(f(y, *coeffs), 0, atol=1e-6)
z = [z] * 10
coeffs = (2.0, 3.0, 4.0)
y = zeros.newton(f, z, args=coeffs, fprime=f_1, fprime2=f_2, tol=1e-6)
assert_allclose(f(y, *coeffs), 0, atol=1e-6)
def test_zero_der_nz_dp():
"""Test secant method with a non-zero dp, but an infinite newton step"""
# pick a symmetrical functions and choose a point on the side that with dx
# makes a secant that is a flat line with zero slope, EG: f = (x - 100)**2,
# which has a root at x = 100 and is symmetrical around the line x = 100
# we have to pick a really big number so that it is consistently true
# now find a point on each side so that the secant has a zero slope
dx = np.finfo(float).eps**0.33
# 100 - p0 = p1 - 100 = p0 * (1 + dx) + dx - 100
# -> 200 = p0 * (2 + dx) + dx
p0 = (200.0 - dx) / (2.0 + dx)
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "RMS of")
x = zeros.newton(lambda y: (y - 100.0)**2, x0=[p0] * 10)
assert_allclose(x, [100] * 10)
# test scalar cases too
p0 = (2.0 - 1e-4) / (2.0 + 1e-4)
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "Tolerance of")
x = zeros.newton(lambda y: (y - 1.0)**2, x0=p0, disp=False)
assert_allclose(x, 1)
with pytest.raises(RuntimeError, match="Tolerance of"):
x = zeros.newton(lambda y: (y - 1.0)**2, x0=p0, disp=True)
p0 = (-2.0 + 1e-4) / (2.0 + 1e-4)
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "Tolerance of")
x = zeros.newton(lambda y: (y + 1.0)**2, x0=p0, disp=False)
assert_allclose(x, -1)
with pytest.raises(RuntimeError, match="Tolerance of"):
x = zeros.newton(lambda y: (y + 1.0)**2, x0=p0, disp=True)
def test_array_newton_integers(self):
# test secant with float
x = zeros.newton(lambda y, z: z - y**2, [4.0] * 2,
args=([15.0, 17.0], ))
assert_allclose(x, (3.872983346207417, 4.123105625617661))
# test integer becomes float
x = zeros.newton(lambda y, z: z - y**2, [4] * 2, args=([15, 17], ))
assert_allclose(x, (3.872983346207417, 4.123105625617661))
def test_deriv_zero_warning(self):
func = lambda x: x**2 - 2.0
dfunc = lambda x: 2 * x
assert_warns(RuntimeWarning,
zeros.newton,
func,
0.0,
dfunc,
disp=False)
with pytest.raises(RuntimeError, match="Derivative was zero"):
zeros.newton(func, 0.0, dfunc)
def test_newton_full_output(self):
# Test the full_output capability, both when converging and not.
# Use simple polynomials, to avoid hitting platform dependencies
# (e.g., exp & trig) in number of iterations
x0 = 3
expected_counts = [(6, 7), (5, 10), (3, 9)]
for derivs in range(3):
kwargs = {
"tol": 1e-6,
"full_output": True,
}
for k, v in [["fprime", self.f1_1], ["fprime2",
self.f1_2]][:derivs]:
kwargs[k] = v
x, r = zeros.newton(self.f1, x0, disp=False, **kwargs)
assert_(r.converged)
assert_equal(x, r.root)
assert_equal((r.iterations, r.function_calls),
expected_counts[derivs])
if derivs == 0:
assert r.function_calls <= r.iterations + 1
else:
assert_equal(r.function_calls, (derivs + 1) * r.iterations)
# Now repeat, allowing one fewer iteration to force convergence failure
iters = r.iterations - 1
x, r = zeros.newton(self.f1,
x0,
maxiter=iters,
disp=False,
**kwargs)
assert_(not r.converged)
assert_equal(x, r.root)
assert_equal(r.iterations, iters)
if derivs == 1:
# Check that the correct Exception is raised and
# validate the start of the message.
with pytest.raises(
RuntimeError,
match=
"Failed to converge after %d iterations, value is .*" %
(iters),
):
x, r = zeros.newton(self.f1,
x0,
maxiter=iters,
disp=True,
**kwargs)
def test_newton(self):
for f, f_1, f_2 in [
(self.f1, self.f1_1, self.f1_2),
(self.f2, self.f2_1, self.f2_2),
]:
x = zeros.newton(f, 3, tol=1e-6)
assert_allclose(f(x), 0, atol=1e-6)
x = zeros.newton(f, 3, x1=5, tol=1e-6) # secant, x0 and x1
assert_allclose(f(x), 0, atol=1e-6)
x = zeros.newton(f, 3, fprime=f_1, tol=1e-6) # newton
assert_allclose(f(x), 0, atol=1e-6)
x = zeros.newton(f, 3, fprime=f_1, fprime2=f_2, tol=1e-6) # halley
assert_allclose(f(x), 0, atol=1e-6)
def test_array_newton(self):
"""test newton with array"""
def f1(x, *a):
b = a[0] + x * a[3]
return a[1] - a[2] * (np.exp(b / a[5]) - 1.0) - b / a[4] - x
def f1_1(x, *a):
b = a[3] / a[5]
return -a[2] * np.exp(a[0] / a[5] + x * b) * b - a[3] / a[4] - 1
def f1_2(x, *a):
b = a[3] / a[5]
return -a[2] * np.exp(a[0] / a[5] + x * b) * b**2
a0 = np.array([
5.32725221,
5.48673747,
5.49539973,
5.36387202,
4.80237316,
1.43764452,
5.23063958,
5.46094772,
5.50512718,
5.42046290,
])
a1 = (np.sin(range(10)) + 1.0) * 7.0
args = (a0, a1, 1e-09, 0.004, 10, 0.27456)
x0 = [7.0] * 10
x = zeros.newton(f1, x0, f1_1, args)
x_expected = (
6.17264965,
11.7702805,
12.2219954,
7.11017681,
1.18151293,
0.143707955,
4.31928228,
10.5419107,
12.7552490,
8.91225749,
)
assert_allclose(x, x_expected)
# test halley's
x = zeros.newton(f1, x0, f1_1, args, fprime2=f1_2)
assert_allclose(x, x_expected)
# test secant
x = zeros.newton(f1, x0, args=args)
assert_allclose(x, x_expected)
def test_gh9551_raise_error_if_disp_true():
"""Test that if disp is true then zero derivative raises RuntimeError"""
def f(x):
return x * x + 1
def f_p(x):
return 2 * x
assert_warns(RuntimeWarning, zeros.newton, f, 1.0, f_p, disp=False)
with pytest.raises(
RuntimeError,
match=
r"^Derivative was zero\. Failed to converge after \d+ iterations, value is [+-]?\d*\.\d+\.$",
):
zeros.newton(f, 1.0, f_p)
root = zeros.newton(f, complex(10.0, 10.0), f_p)
assert_allclose(root, complex(0.0, 1.0))
def test_array_newton_complex(self):
def f(x):
return x + 1 + 1j
def fprime(x):
return 1.0
t = np.full(4, 1j)
x = zeros.newton(f, t, fprime=fprime)
assert_allclose(f(x), 0.0)
# should work even if x0 is not complex
t = np.ones(4)
x = zeros.newton(f, t, fprime=fprime)
assert_allclose(f(x), 0.0)
x = zeros.newton(f, t)
assert_allclose(f(x), 0.0)
def test_gh_9608_preserve_array_shape():
"""
Test that shape is preserved for array inputs even if fprime or fprime2 is
scalar
"""
def f(x):
return x**2
def fp(x):
return 2 * x
def fpp(x):
return 2
x0 = np.array([-2], dtype=np.float32)
rt, r = newton(f, x0, fprime=fp, fprime2=fpp, full_output=True)
assert r.converged
x0_array = np.array([-2, -3], dtype=np.float32)
# This next invocation should fail
with pytest.raises(IndexError):
result = zeros.newton(f,
x0_array,
fprime=fp,
fprime2=fpp,
full_output=True)
def fpp_array(x):
return np.full(np.shape(x), 2, dtype=np.float32)
result = zeros.newton(f,
x0_array,
fprime=fp,
fprime2=fpp_array,
full_output=True)
assert result.converged.all()
def test_array_newton_failures():
"""Test that array newton fails as expected"""
# p = 0.68 # [MPa]
# dp = -0.068 * 1e6 # [Pa]
# T = 323 # [K]
diameter = 0.10 # [m]
# L = 100 # [m]
roughness = 0.00015 # [m]
rho = 988.1 # [kg/m**3]
mu = 5.4790e-04 # [Pa*s]
u = 2.488 # [m/s]
reynolds_number = rho * u * diameter / mu # Reynolds number
def colebrook_eqn(darcy_friction, re, dia):
return 1 / np.sqrt(darcy_friction) + 2 * np.log10(
roughness / 3.7 / dia + 2.51 / re / np.sqrt(darcy_friction))
# only some failures
with pytest.warns(RuntimeWarning):
result = zeros.newton(
colebrook_eqn,
x0=[0.01, 0.2, 0.02223, 0.3],
maxiter=2,
args=[reynolds_number, diameter],
full_output=True,
)
assert not result.converged.all()
# they all fail
with pytest.raises(RuntimeError):
result = zeros.newton(
colebrook_eqn,
x0=[0.01] * 2,
maxiter=2,
args=[reynolds_number, diameter],
full_output=True,
)
def test_array_newton_zero_der_failures(self):
# test derivative zero warning
assert_warns(
RuntimeWarning,
zeros.newton,
lambda y: y**2 - 2,
[0.0, 0.0],
lambda y: 2 * y,
)
# test failures and zero_der
with pytest.warns(RuntimeWarning):
results = zeros.newton(lambda y: y**2 - 2, [0.0, 0.0],
lambda y: 2 * y,
full_output=True)
assert_allclose(results.root, 0)
assert results.zero_der.all()
assert not results.converged.any()
def test_gh8904_zeroder_at_root_fails():
"""Test that Newton or Halley don't warn if zero derivative at root"""
# a function that has a zero derivative at it's root
def f_zeroder_root(x):
return x**3 - x**2
# should work with secant
r = zeros.newton(f_zeroder_root, x0=0)
assert_allclose(r, 0, atol=sp_zeros._xtol, rtol=sp_zeros._rtol)
# test again with array
r = zeros.newton(f_zeroder_root, x0=[0] * 10)
assert_allclose(r, 0, atol=sp_zeros._xtol, rtol=sp_zeros._rtol)
# 1st derivative
def fder(x):
return 3 * x**2 - 2 * x
# 2nd derivative
def fder2(x):
return 6 * x - 2
# should work with newton and halley
r = zeros.newton(f_zeroder_root, x0=0, fprime=fder)
assert_allclose(r, 0, atol=sp_zeros._xtol, rtol=sp_zeros._rtol)
r = zeros.newton(f_zeroder_root, x0=0, fprime=fder, fprime2=fder2)
assert_allclose(r, 0, atol=sp_zeros._xtol, rtol=sp_zeros._rtol)
# test again with array
r = zeros.newton(f_zeroder_root, x0=[0] * 10, fprime=fder)
assert_allclose(r, 0, atol=sp_zeros._xtol, rtol=sp_zeros._rtol)
r = zeros.newton(f_zeroder_root, x0=[0] * 10, fprime=fder, fprime2=fder2)
assert_allclose(r, 0, atol=sp_zeros._xtol, rtol=sp_zeros._rtol)
# also test that if a root is found we do not raise RuntimeWarning even if
# the derivative is zero, EG: at x = 0.5, then fval = -0.125 and
# fder = -0.25 so the next guess is 0.5 - (-0.125/-0.5) = 0 which is the
# root, but if the solver continued with that guess, then it will calculate
# a zero derivative, so it should return the root w/o RuntimeWarning
r = zeros.newton(f_zeroder_root, x0=0.5, fprime=fder)
assert_allclose(r, 0, atol=sp_zeros._xtol, rtol=sp_zeros._rtol)
# test again with array
r = zeros.newton(f_zeroder_root, x0=[0.5] * 10, fprime=fder)
assert_allclose(r, 0, atol=sp_zeros._xtol, rtol=sp_zeros._rtol)
def test_jitted():
f1j = numba.njit()(f1)
assert_allclose(zeros.j_newton(f1j, 0), zeros.newton(f1, 0))
def test_array_secant_active_zero_der(self):
"""test secant doesn't continue to iterate zero derivatives"""
x = zeros.newton(lambda x, *a: x * x - a[0],
x0=[4.123, 5],
args=[np.array([17, 25])])
assert_allclose(x, (4.123105625617661, 5.0))
