def test_split(self):
# If this ever breaks, we should solve the problem by writing a
# new kind of Function that turns a lambda expression into a
# Function. That way, it will never be simplified.
# f = x^2 + x
f = Function.sum(Function.power(Function.identity(),
Function.constant(2)),
Function.identity())
# f(-1) = 0, f(0) = 0, f(-.5) = -.25
# The range of f on [-1,0] is [-.25,0]
# f([-1,0]) = [-1,1]
# f([-1,-.5]) = [-.75,.5]
# f([-.5,0]) = [-.5,.25]
# This will never finish, since it asks for the exact bounds
self.assertTrue(is_bounded(f, Interval(-1,0), Interval(-1/4,0))
is None)
# g = 1/2*x^3-3/2*x
g = Function.sum(Function.product(
Function.constant(.5),
Function.power(Function.identity(),
Function.constant(3))),
Function.product(Function.constant(-1.5),
Function.identity()))
self.assertEqual(is_bounded(g, Interval(-1.5,1.5), Interval(-.9,1)),
False)
# This encounters a ValueError on the first split so should
# return None
h = Function.quotient(Function.constant(1), Function.identity())
self.assertTrue(is_bounded(h, Interval(-1,1), Interval(-5,5)) is None)
def test_simple(self):
self.assertTrue(is_bounded(Function.identity(), Interval(0,1),
Interval(0,1)))
self.assertFalse(is_bounded(Function.identity(), Interval(0,2),
Interval(0,1)))
self.assertFalse(is_bounded(Function.identity(), Interval(-1,1),
Interval(0,1)))
self.assertFalse(is_bounded(Function.identity(), Interval(-1,2),
Interval(0,1)))
self.assertTrue(is_bounded(Function.constant(0.), Interval(3,4),
Interval(0,1)))
self.assertFalse(is_bounded(Function.constant(2.), Interval(3,4),
Interval(0,1)))
def test_cubic_derivative(self):
a = cubic_derivative_approximation(
Function.power(Function.identity(), Function.constant(3)),
4, 10)
self.assertAlmostEqual(a(0), 0)
self.assertAlmostEqual(a(2), 8)
self.assertAlmostEqual(a(5), 125)
self.assertAlmostEqual(a(12), 1728)
self.assertAlmostEqual(a(-3), -27)
def test_approx(self):
f = Function.identity()
for a in approximate(f, 0, 1):
self.assertTrue(isinstance(a, Function))
self.assertTrue(isinstance(a, CubicSpline))
# We may someday generate approximations that don't go through
# the endpoints, and remove these tests. Until then, they
# help verify that the approximation formulas are correct.
self.assertEqual(a(0.0), 0.0)
self.assertEqual(a(1.0), 1.0)
# (-x^2 - 1) ^ .5
bad = Function.power(Function.sum(Function.product(
Function.constant(-1),
Function.power(Function.identity(), Function.constant(2))),
Function.constant(-1)),
Function.constant(.5))
self.assertEqual(list(approximate(bad, 0, 1)), [])
def recursively_generate_cubics(f, left, right, depth = 5):
for a in approximate(f, left, right):
if is_bounded(Function.sum(f, Function.product(
Function.constant(-1), a)), Interval(left, right),
Interval(-.008, .008)):
return [a]
if depth == 0:
return []
middle = (left + right) / 2
return recursively_generate_cubics(f, left, middle, depth-1) + \
recursively_generate_cubics(f, middle, right, depth-1)
def __init__(self, t0, t1, f0, c0, c1, f1):
self.__t0 = t0
self.__t1 = t1
self.__f0 = f0
self.__c0 = c0
self.__c1 = c1
self.__f1 = f1
# The specific functional form will have consequences for the
# efficiency of interval arithmetic (and thus, slicing).
# t' = (t-t0)/(t1-t0)
t = Function.product(Function.sum(Function.identity(), Function.constant(-t0)),
Function.constant(1.0/(t1-t0)))
omt = Function.sum(Function.constant(1), Function.product(Function.constant(-1), t))
def term(const, f):
return Function.sum(Function.constant(const), Function.product(t, f))
a = -f0 + 3*c0 - 3*c1 + f1
b = 3*f0 - 6*c0 + 3*c1
c = -3*f0 + 3*c0
d = f0
WrappedFunction.__init__(self, term(d, term(c, term(b, Function.constant(a)))).weak_simplify())
def test_depth(self):
f = Function.sum(Function.power(Function.identity(),
Function.constant(2)),
Function.identity())
# This requires splitting in half
self.assertTrue(is_bounded(f, Interval(-1,0), Interval(-3/4,1/2), 1))
self.assertEqual(is_bounded(f, Interval(-1,0), Interval(-3/4,1/2), 0),
None)
# This requires 3 levels of recursion
self.assertTrue(is_bounded(f, Interval(-1,0), Interval(-3/8,1/8), 3))
self.assertEqual(is_bounded(f, Interval(-1,0), Interval(-3/8,1/8), 2),
None)
# This requires 8 levels of recursion
self.assertTrue(is_bounded(f, Interval(-1,0), Interval(-65/256,1/256),
8))
self.assertEqual(is_bounded(f, Interval(-1,0), Interval(-65/256,1/256),
7), None)
def term(const, f):
return Function.sum(Function.constant(const), Function.product(t, f))