def plot_convergence():
"""Compare convergence of various methods.
Methods: Newton's method, Broyden's method, and function iteration.
References
----------
.. [RA20] Randall Romero-Aguilar. A Python version of Miranda and Fackler’s
CompEcon toolbox. 2020. URL: https://github.com/randall-romero/CompEcon.
"""
def f(x):
fval = np.exp(x) - 1
x_values.append(x)
return fval
def f_2(x):
fval = np.exp(x) - 1
x_values.append(x)
return fval, np.exp(x)
get_log_error = lambda z: np.log10(np.abs(z)).flatten() # noqa
# Newton
x_values = []
_ = newton_method(f_2, 2)
error_newton = get_log_error(x_values)
# Broyden
x_values = []
_ = root(f,
2.0,
method="broyden1",
options={"jac_options": {
"alpha": -1 / np.exp(2)
}})
error_broyden = get_log_error(x_values)
# Function iteration
x_values = []
_ = funcit(f, x0=2)
error_funcit = get_log_error(x_values)
# Plot results.
plt.figure(figsize=[10, 6])
plt.plot(error_newton, label="Newton's Method")
plt.plot(error_broyden, label="Broyden's Method")
plt.plot(error_funcit, label="Function Iteration")
plt.title(r"Convergence rates for $f(x)= exp(x)-1$ with $x_0=2$")
plt.xlabel("Iteration")
plt.ylabel("Log10 Error")
plt.xlim(0, 10)
plt.ylim(-5, 2)
plt.legend()
def test_3():
"""Newton method is working."""
def _jacobian(x):
return 3 * x**2
def _value(x):
return x**3 - 2
def f(x):
return _value(x), _jacobian(x)
x = newton_method(f, 0.4)
np.testing.assert_almost_equal(f(x)[0], 0)
def test_exercise_4():
"""Test for exercise 4."""
for x0 in [-0.01, 0.01]:
x = newton_method(newton_pathological_example, 0.45)
print(f"candidate for root {x[0][0]:+5.3f}")