def test_solve_ex_2_10(self):
# Attempt to solve sqrt(2)
def g(x):
return x ** 2 + x - 2
# cannot be used for solving sqrt(2) as it does not converge
s = iterate.solve(g, estimate=1.0, iterations=5, logger=logger)
self.assertNotAlmostEqual(s, 1.414, 3)
def test_solve_example_2_2(self):
def g(x):
return 0.25 * math.exp(x)
solution = iterate.solve(g,
estimate=0.35,
iterations=4,
logger=test_logger)
self.assertAlmostEqual(solution, 0.357, 3)
def test_solve_ex_2_11(self):
"""
Two rearrangements of x^3 + 4x^2 - 10 = 0
"""
def f_a(x):
return x - x ** 3 - 4 * x ** 2 + 10
def f_b(x):
inner = 10 / x - 4 * x
logger.info("Performing sqrt({})".format(inner))
return math.sqrt(inner)
logger.info('-' * 40)
# f_a(x) cannot be used to solve x^3 + 4x^2 - 10 = 0 as it diverges and oscillates.
iterate.solve(f_a, estimate=1.5, iterations=5, logger=logger)
logger.info('-' * 40)
with self.assertRaises(ValueError):
# f_b(x) cannot be used to solve x^3 + 4x^2 - 10 = 0 as the 3rd iteration attempts to root a -ve number.
iterate.solve(f_b, estimate=1.5, iterations=5, logger=logger)
logger.info('-' * 40)
def test_solve_root_2(self):
# From section 1.1
def g(x):
return (x + 2 / x) / 2
# TODO: could parameterise to accept a stopping criterion (the examples don't use fixed # iterations)
solution = iterate.solve(g,
estimate=1.0,
iterations=6,
logger=test_logger)
self.assertAlmostEqual(solution, 1.414, 3)
def solve(f, *, estimate, iterations=5, logger=None, counter=0):
def df(argx):
# Use sympy to do the differentiation
x = sym.symbols('x')
return sym.diff(f(x), x).evalf(subs={x: argx})
def g(x):
return x - f(x) / df(x)
return iterate.solve(g,
estimate=estimate,
iterations=iterations,
logger=logger,
counter=counter)
def test_solve_ex_2_9_ii(self):
def f_ii(x):
return math.sqrt(5 * x - 2)
logger.info("-" * 10, "ii with x0 = 0.5", "-" * 20)
# Might be converging to root ~ 4.56
iterate.solve(f_ii, estimate=0.5, iterations=5, logger=logger)
logger.info("-" * 10, "ii with x0 = 4.5", "-" * 20)
# Appears to be slowly converging to root ~ 4.56
iterate.solve(f_ii, estimate=4.5, iterations=5, logger=logger)
logger.info("-" * 10, "ii with x0 = 10", "-" * 20)
# Appears to be converging towards root ~ 4.56
iterate.solve(f_ii, estimate=10, iterations=5, logger=logger)
logger.info("-" * 40)
def test_solve_ex_2_9_i(self):
def f_i(x):
return 0.2 * x ** 2 + 0.4
logger.info("-" * 10, "i with x0 = 0.5", "-" * 20)
# Seems to be converging
s = iterate.solve(f_i, estimate=0.5, iterations=5, logger=logger)
self.assertAlmostEqual(s, 0.438, 3)
logger.info("-" * 10, "i with x0 = 4.5", "-" * 20)
# Unclear if converging
s = iterate.solve(f_i, estimate=4.5, iterations=5, logger=logger)
self.assertAlmostEqual(s, 3.493, 3)
logger.info("-" * 10, "i with x0 = 10", "-" * 20)
# Diverges
iterate.solve(f_i, estimate=10, iterations=5, logger=logger)
logger.info("-" * 10, "i with x0 = 4.5", "-" * 20)
# Seems to be slowly converging to 0.438
iterate.solve(f_i, estimate=4.5, iterations=15, logger=logger)
""" b) ii)
f = x^3 - 5x + 4
f' = 3x^2 - 5
"""
def f(x):
return x**3 - 5 * x + 4
def df(x):
return 3 * x**2 - 5
return x - f(x) / df(x)
return g, h
iterate.solve(f, estimate=1.0, iterations=5, logger=logger)
print('-' * 40)
iterate.solve(g, estimate=0.35, iterations=5, logger=logger)
print('-' * 40)
iterate.solve(h, estimate=1.5, iterations=5, logger=logger)
print('-' * 40)
print("Exercise 2.14")
funcs = make_functions()
iterate.solve(funcs[0], estimate=1.3, iterations=5, logger=logger)
print('-' * 40)
iterate.solve(funcs[1], estimate=1.3, iterations=5, logger=logger)
# Ex 2.14 c)
ans = bisect.solve(lambda x: x**3 - 5 * x + 4,
interval=(1.1, 1.9),
