def gr_roots(n):
"""Gauss Radau roots and weights for [-1, 1]
with -1 first
"""
leg = orthogonal.legendre(n - 1)
l = (leg + orthogonal.legendre(n))/np.poly1d((1, 1))
x = np.r_[-1, l[0].roots]
w = (1 - x)/(n * leg(x))**2
return x, w
def gr_roots(n):
"""Gauss Radau roots and weights for [-1, 1]
with -1 first
"""
leg = orthogonal.legendre(n - 1)
l = (leg + orthogonal.legendre(n)) / np.poly1d((1, 1))
x = np.r_[-1, l[0].roots]
w = (1 - x) / (n * leg(x))**2
return x, w
def test_p_roots():
weightf = orth.legendre(5).weight_func
verify_gauss_quad(orth.p_roots, orth.eval_legendre, weightf, -1., 1., 5)
verify_gauss_quad(orth.p_roots,
orth.eval_legendre,
weightf,
-1.,
1.,
25,
atol=1e-13)
verify_gauss_quad(orth.p_roots,
orth.eval_legendre,
weightf,
-1.,
1.,
100,
atol=1e-12)
x, w = orth.p_roots(5, False)
y, v, m = orth.p_roots(5, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
muI, muI_err = integrate.quad(weightf, -1, 1)
assert_allclose(m, muI, rtol=muI_err)
assert_raises(ValueError, orth.p_roots, 0)
assert_raises(ValueError, orth.p_roots, 3.3)
def get_model(self, x=None, param=None):
""" return the model for the given data.
The modelization is based on legendre polynomes that expect x to be between -1 and 1.
This will create a reshaped copy of x to scale it between -1 and 1 but
if x is already as such, save time by setting reshapex to False
Returns
-------
array (size of x)
"""
if param is not None:
self.setup(param)
if x is not None:
self.set_xsource(x)
if self.use_legendre:
model = np.asarray([
orthogonal.legendre(i)(self.xsource_scaled)
for i in range(self.DEGREE)
])
else:
model = np.asarray([self.xfit**i for i in range(self.DEGREE)])
return np.dot(model.T, self.parameters.T).T
def gl_roots(n):
"""Gauss Lobatto roots and weights for [-1, 1]
with -1 first and 1 last
"""
leg = orthogonal.legendre(n - 1)
x = np.r_[-1, leg.deriv().roots, 1]
w = 2 / (n * (n - 1) * leg(x)**2)
return x, w
def gl_roots(n):
"""Gauss Lobatto roots and weights for [-1, 1]
with -1 first and 1 last
"""
leg = orthogonal.legendre(n - 1)
x = np.r_[-1, leg.deriv().roots, 1]
w = 2/(n*(n - 1)*leg(x)**2)
return x, w
def test_p_roots():
weightf = orth.legendre(5).weight_func
verify_gauss_quad(orth.p_roots, orth.eval_legendre, weightf, -1.0, 1.0, 5)
verify_gauss_quad(orth.p_roots, orth.eval_legendre, weightf, -1.0, 1.0, 25, atol=1e-13)
verify_gauss_quad(orth.p_roots, orth.eval_legendre, weightf, -1.0, 1.0, 100, atol=1e-12)
x, w = orth.p_roots(5, False)
y, v, m = orth.p_roots(5, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
muI, muI_err = integrate.quad(weightf, -1, 1)
assert_allclose(m, muI, rtol=muI_err)
assert_raises(ValueError, orth.p_roots, 0)
assert_raises(ValueError, orth.p_roots, 3.3)
def fit_polynome(x, y, degree, variance=None):
""" """
from scipy.optimize import fmin
from scipy.special import orthogonal
xdata = (x - np.min(x)) / (np.max(x) - np.min(x)) * 2 - 1.
basemodel = np.asarray(
[orthogonal.legendre(i)(xdata) for i in range(degree)])
def get_model(parameters):
""" """
return np.dot(basemodel.T, parameters.T).T
def get_chi2(parameters):
res = (y - get_model(parameters))**2
if variance is not None:
res /= variance
return np.sum(res)
guess = np.zeros(degree)
guess[0] = np.median(y)
param = fmin(get_chi2, guess, disp=0)
return get_model(param)
def test_sh_legendre(self):
# P*_n(x) = P_n(2x-1)
psub = np.poly1d([2, -1])
Ps0 = orth.sh_legendre(0)
Ps1 = orth.sh_legendre(1)
Ps2 = orth.sh_legendre(2)
Ps3 = orth.sh_legendre(3)
Ps4 = orth.sh_legendre(4)
Ps5 = orth.sh_legendre(5)
pse0 = orth.legendre(0)(psub)
pse1 = orth.legendre(1)(psub)
pse2 = orth.legendre(2)(psub)
pse3 = orth.legendre(3)(psub)
pse4 = orth.legendre(4)(psub)
pse5 = orth.legendre(5)(psub)
assert_array_almost_equal(Ps0.c, pse0.c, 13)
assert_array_almost_equal(Ps1.c, pse1.c, 13)
assert_array_almost_equal(Ps2.c, pse2.c, 13)
assert_array_almost_equal(Ps3.c, pse3.c, 13)
assert_array_almost_equal(Ps4.c, pse4.c, 12)
assert_array_almost_equal(Ps5.c, pse5.c, 12)
def test_sh_legendre(self):
# P*_n(x) = P_n(2x-1)
psub = np.poly1d([2,-1])
Ps0 = orth.sh_legendre(0)
Ps1 = orth.sh_legendre(1)
Ps2 = orth.sh_legendre(2)
Ps3 = orth.sh_legendre(3)
Ps4 = orth.sh_legendre(4)
Ps5 = orth.sh_legendre(5)
pse0 = orth.legendre(0)(psub)
pse1 = orth.legendre(1)(psub)
pse2 = orth.legendre(2)(psub)
pse3 = orth.legendre(3)(psub)
pse4 = orth.legendre(4)(psub)
pse5 = orth.legendre(5)(psub)
assert_array_almost_equal(Ps0.c,pse0.c,13)
assert_array_almost_equal(Ps1.c,pse1.c,13)
assert_array_almost_equal(Ps2.c,pse2.c,13)
assert_array_almost_equal(Ps3.c,pse3.c,13)
assert_array_almost_equal(Ps4.c,pse4.c,12)
assert_array_almost_equal(Ps5.c,pse5.c,12)
