def test_piecewise_convex():
def func(x):
if abs(x) > 2:
return x**2
if abs(x) > 1:
return abs(x)
return 1
(x, fx, niters) = brent(func, -10, +10)
assert_(-1 - 1e6 <= x <= 1 + 1e6)
assert_almost_equal(fx, 1)
assert_array_less(niters, 45)
(x, fx, niters) = brent(func, 1 / 2, +1)
assert_(-1 - 1e6 <= x <= 1 + 1e6)
assert_almost_equal(fx, 1)
assert_array_less(niters, 35)
(x, fx, niters) = brent(func, -1, -1 / 2)
assert_(-1 - 1e6 <= x <= 1 + 1e6)
assert_almost_equal(fx, 1)
assert_array_less(niters, 35)
(x, fx, niters) = brent(func, -1 / 2, +1 / 2)
assert_(-1 - 1e6 <= x <= 1 + 1e6)
assert_almost_equal(fx, 1)
assert_array_less(niters, 37)
def test_asymptotic():
def func(x):
return -3 + 1 / x
(x, _, niters) = brent(func, 1e-6, +10, rtol=1e-9)
assert_almost_equal(x, 10)
assert_array_less(niters, 42)
(x, _, niters) = brent(func, -10, -1e-6)
assert_almost_equal(x, -1e-06)
assert_array_less(niters, 42)
def test_convex():
def func(x, s):
return abs(x - s) + 3
(x, fx, niters) = brent(lambda x: func(x, 0), -10, +10)
assert_almost_equal(x, 0)
assert_almost_equal(fx, 3)
assert_array_less(niters, 37)
(x, fx, niters) = brent(lambda x: func(x, -9), -10, +10)
assert_almost_equal(x, -9)
assert_array_less(niters, 37)
def test_same_point():
def func(x):
return x**2
(x, _, niters) = brent(func, 1.2, 1.2)
assert_array_less(niters, 2)
assert_almost_equal(x, 1.2)
def _boxcox(lib, x):
from numpy_sugar import epsilon
from scipy.stats import boxcox_llf
from scipy.special import boxcox as bc
x = lib.asarray(x).astype(float)
m = x.min()
if m <= 0:
m = max(lib.abs(m), epsilon.small)
x = x + m + m / 2
lmb = brent(lambda lmb: -boxcox_llf(lmb, x), -5, +5)[0]
return bc(x, lmb)
def test_strictly_convex():
def func(x, s):
return (x - s)**2 - 0.8
(x, fx, niters) = brent(lambda x: func(x, 0), -10, 10)
assert_almost_equal(x, 0)
assert_almost_equal(fx, -0.8)
assert_array_less(niters, 7)
(x, fx, niters) = brent(lambda x: func(x, 0), 0, 10)
assert_almost_equal(x, 0)
assert_array_less(niters, 42)
(x, fx, niters) = brent(lambda x: func(x, 5), -1000, -100, rtol=1e-10)
assert_almost_equal(x, -100)
assert_almost_equal(fx, func(-100, 5), decimal=4)
assert_array_less(niters, 51)
(x, fx, niters) = brent(lambda x: func(x, 5), -100, -99, rtol=1e-10)
assert_almost_equal(x, -99)
assert_almost_equal(fx, func(-99, 5), decimal=4)
(x, fx, niters) = brent(lambda x: func(x, 5), 99, 100, rtol=1e-10)
assert_almost_equal(x, 99)
assert_almost_equal(fx, func(99, 5), decimal=4)
assert_array_less(niters, 36)
(x, fx, niters) = brent(lambda x: func(x, 5), -1, -1 / 2)
assert_almost_equal(x, -1 / 2)
assert_almost_equal(fx, func(-1 / 2, 5), decimal=4)
assert_array_less(niters, 35)
(x, fx, niters) = brent(lambda x: func(x, 5), 6.5, 6.5)
assert_almost_equal(x, 6.5)
assert_almost_equal(fx, func(6.5, 5), decimal=4)
assert_array_less(niters, 2)
(x, fx, niters) = brent(lambda x: func(x, 5), 6, 7, rtol=1e-9)
assert_almost_equal(x, 6)
assert_almost_equal(fx, func(6, 5), decimal=4)
assert_array_less(niters, 37)
def pcgc(y, kinship, K):
y = (y - np.mean(y)) / np.std(y)
P = np.sum(y > 0) / len(y)
t = st.norm.isf(K)
c = P * (1 - P) * st.norm.pdf(t)**2 / (K**2 * (1 - K)**2)
return brent(lambda h2: _pcgc_cost(y, kinship, c, h2), 1e-4, 1 - 1e-4)[0]