"""Tests for chebyshev module.
"""
from __future__ import division, absolute_import, print_function
import numpy as np
import numpy.polynomial.chebyshev as cheb
from numpy.polynomial.polynomial import polyval
from numpy.testing import (
assert_almost_equal, assert_raises, assert_equal, assert_,
run_module_suite
)
def trim(x):
return cheb.chebtrim(x, tol=1e-6)
T0 = [1]
T1 = [0, 1]
T2 = [-1, 0, 2]
T3 = [0, -3, 0, 4]
T4 = [1, 0, -8, 0, 8]
T5 = [0, 5, 0, -20, 0, 16]
T6 = [-1, 0, 18, 0, -48, 0, 32]
T7 = [0, -7, 0, 56, 0, -112, 0, 64]
T8 = [1, 0, -32, 0, 160, 0, -256, 0, 128]
T9 = [0, 9, 0, -120, 0, 432, 0, -576, 0, 256]
Tlist = [T0, T1, T2, T3, T4, T5, T6, T7, T8, T9]
class TestPrivate(object):
def test__cseries_to_zseries(self):
for i in range(5):
inp = np.array([2] + [1]*i, np.double)
tgt = np.array([.5]*i + [2] + [.5]*i, np.double)
res = cheb._cseries_to_zseries(inp)
assert_equal(res, tgt)
def test__zseries_to_cseries(self):
for i in range(5):
inp = np.array([.5]*i + [2] + [.5]*i, np.double)
tgt = np.array([2] + [1]*i, np.double)
res = cheb._zseries_to_cseries(inp)
assert_equal(res, tgt)
class TestConstants(object):
def test_chebdomain(self):
assert_equal(cheb.chebdomain, [-1, 1])
def test_chebzero(self):
assert_equal(cheb.chebzero, [0])
def test_chebone(self):
assert_equal(cheb.chebone, [1])
def test_chebx(self):
assert_equal(cheb.chebx, [0, 1])
class TestArithmetic(object):
def test_chebadd(self):
for i in range(5):
for j in range(5):
msg = "At i=%d, j=%d" % (i, j)
tgt = np.zeros(max(i, j) + 1)
tgt[i] += 1
tgt[j] += 1
res = cheb.chebadd([0]*i + [1], [0]*j + [1])
assert_equal(trim(res), trim(tgt), err_msg=msg)
def test_chebsub(self):
for i in range(5):
for j in range(5):
msg = "At i=%d, j=%d" % (i, j)
tgt = np.zeros(max(i, j) + 1)
tgt[i] += 1
tgt[j] -= 1
res = cheb.chebsub([0]*i + [1], [0]*j + [1])
assert_equal(trim(res), trim(tgt), err_msg=msg)
def test_chebmulx(self):
assert_equal(cheb.chebmulx([0]), [0])
assert_equal(cheb.chebmulx([1]), [0, 1])
for i in range(1, 5):
ser = [0]*i + [1]
tgt = [0]*(i - 1) + [.5, 0, .5]
assert_equal(cheb.chebmulx(ser), tgt)
def test_chebmul(self):
for i in range(5):
for j in range(5):
msg = "At i=%d, j=%d" % (i, j)
tgt = np.zeros(i + j + 1)
tgt[i + j] += .5
tgt[abs(i - j)] += .5
res = cheb.chebmul([0]*i + [1], [0]*j + [1])
assert_equal(trim(res), trim(tgt), err_msg=msg)
def test_chebdiv(self):
for i in range(5):
for j in range(5):
msg = "At i=%d, j=%d" % (i, j)
ci = [0]*i + [1]
cj = [0]*j + [1]
tgt = cheb.chebadd(ci, cj)
quo, rem = cheb.chebdiv(tgt, ci)
res = cheb.chebadd(cheb.chebmul(quo, ci), rem)
assert_equal(trim(res), trim(tgt), err_msg=msg)
class TestEvaluation(object):
# coefficients of 1 + 2*x + 3*x**2
c1d = np.array([2.5, 2., 1.5])
c2d = np.einsum('i,j->ij', c1d, c1d)
c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d)
# some random values in [-1, 1)
x = np.random.random((3, 5))*2 - 1
y = polyval(x, [1., 2., 3.])
def test_chebval(self):
#check empty input
assert_equal(cheb.chebval([], [1]).size, 0)
#check normal input)
x = np.linspace(-1, 1)
y = [polyval(x, c) for c in Tlist]
for i in range(10):
msg = "At i=%d" % i
tgt = y[i]
res = cheb.chebval(x, [0]*i + [1])
assert_almost_equal(res, tgt, err_msg=msg)
#check that shape is preserved
for i in range(3):
dims = [2]*i
x = np.zeros(dims)
assert_equal(cheb.chebval(x, [1]).shape, dims)
assert_equal(cheb.chebval(x, [1, 0]).shape, dims)
assert_equal(cheb.chebval(x, [1, 0, 0]).shape, dims)
def test_chebval2d(self):
x1, x2, x3 = self.x
y1, y2, y3 = self.y
#test exceptions
assert_raises(ValueError, cheb.chebval2d, x1, x2[:2], self.c2d)
#test values
tgt = y1*y2
res = cheb.chebval2d(x1, x2, self.c2d)
assert_almost_equal(res, tgt)
#test shape
z = np.ones((2, 3))
res = cheb.chebval2d(z, z, self.c2d)
assert_(res.shape == (2, 3))
def test_chebval3d(self):
x1, x2, x3 = self.x
y1, y2, y3 = self.y
#test exceptions
assert_raises(ValueError, cheb.chebval3d, x1, x2, x3[:2], self.c3d)
#test values
tgt = y1*y2*y3
res = cheb.chebval3d(x1, x2, x3, self.c3d)
assert_almost_equal(res, tgt)
#test shape
z = np.ones((2, 3))
res = cheb.chebval3d(z, z, z, self.c3d)
assert_(res.shape == (2, 3))
def test_chebgrid2d(self):
x1, x2, x3 = self.x
y1, y2, y3 = self.y
#test values
tgt = np.einsum('i,j->ij', y1, y2)
res = cheb.chebgrid2d(x1, x2, self.c2d)
assert_almost_equal(res, tgt)
#test shape
z = np.ones((2, 3))
res = cheb.chebgrid2d(z, z, self.c2d)
assert_(res.shape == (2, 3)*2)
def test_chebgrid3d(self):
x1, x2, x3 = self.x
y1, y2, y3 = self.y
#test values
tgt = np.einsum('i,j,k->ijk', y1, y2, y3)
res = cheb.chebgrid3d(x1, x2, x3, self.c3d)
assert_almost_equal(res, tgt)
#test shape
z = np.ones((2, 3))
res = cheb.chebgrid3d(z, z, z, self.c3d)
assert_(res.shape == (2, 3)*3)
class TestIntegral(object):
def test_chebint(self):
# check exceptions
assert_raises(ValueError, cheb.chebint, [0], .5)
assert_raises(ValueError, cheb.chebint, [0], -1)
assert_raises(ValueError, cheb.chebint, [0], 1, [0, 0])
assert_raises(ValueError, cheb.chebint, [0], lbnd=[0])
assert_raises(ValueError, cheb.chebint, [0], scl=[0])
assert_raises(ValueError, cheb.chebint, [0], axis=.5)
# test integration of zero polynomial
for i in range(2, 5):
k = [0]*(i - 2) + [1]
res = cheb.chebint([0], m=i, k=k)
assert_almost_equal(res, [0, 1])
# check single integration with integration constant
for i in range(5):
scl = i + 1
pol = [0]*i + [1]
tgt = [i] + [0]*i + [1/scl]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i])
res = cheb.cheb2poly(chebint)
assert_almost_equal(trim(res), trim(tgt))
# check single integration with integration constant and lbnd
for i in range(5):
scl = i + 1
pol = [0]*i + [1]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i], lbnd=-1)
assert_almost_equal(cheb.chebval(-1, chebint), i)
# check single integration with integration constant and scaling
for i in range(5):
scl = i + 1
pol = [0]*i + [1]
tgt = [i] + [0]*i + [2/scl]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i], scl=2)
res = cheb.cheb2poly(chebint)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with default k
for i in range(5):
for j in range(2, 5):
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j):
tgt = cheb.chebint(tgt, m=1)
res = cheb.chebint(pol, m=j)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with defined k
for i in range(5):
for j in range(2, 5):
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j):
tgt = cheb.chebint(tgt, m=1, k=[k])
res = cheb.chebint(pol, m=j, k=list(range(j)))
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with lbnd
for i in range(5):
for j in range(2, 5):
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j):
tgt = cheb.chebint(tgt, m=1, k=[k], lbnd=-1)
res = cheb.chebint(pol, m=j, k=list(range(j)), lbnd=-1)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with scaling
for i in range(5):
for j in range(2, 5):
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j):
tgt = cheb.chebint(tgt, m=1, k=[k], scl=2)
res = cheb.chebint(pol, m=j, k=list(range(j)), scl=2)
assert_almost_equal(trim(res), trim(tgt))
def test_chebint_axis(self):
# check that axis keyword works
c2d = np.random.random((3, 4))
tgt = np.vstack([cheb.chebint(c) for c in c2d.T]).T
res = cheb.chebint(c2d, axis=0)
assert_almost_equal(res, tgt)
tgt = np.vstack([cheb.chebint(c) for c in c2d])
res = cheb.chebint(c2d, axis=1)
assert_almost_equal(res, tgt)
tgt = np.vstack([cheb.chebint(c, k=3) for c in c2d])
res = cheb.chebint(c2d, k=3, axis=1)
assert_almost_equal(res, tgt)
class TestDerivative(object):
def test_chebder(self):
# check exceptions
assert_raises(ValueError, cheb.chebder, [0], .5)
assert_raises(ValueError, cheb.chebder, [0], -1)
# check that zeroth derivative does nothing
for i in range(5):
tgt = [0]*i + [1]
res = cheb.chebder(tgt, m=0)
assert_equal(trim(res), trim(tgt))
# check that derivation is the inverse of integration
for i in range(5):
for j in range(2, 5):
tgt = [0]*i + [1]
res = cheb.chebder(cheb.chebint(tgt, m=j), m=j)
assert_almost_equal(trim(res), trim(tgt))
# check derivation with scaling
for i in range(5):
for j in range(2, 5):
tgt = [0]*i + [1]
res = cheb.chebder(cheb.chebint(tgt, m=j, scl=2), m=j, scl=.5)
assert_almost_equal(trim(res), trim(tgt))
def test_chebder_axis(self):
# check that axis keyword works
c2d = np.random.random((3, 4))
tgt = np.vstack([cheb.chebder(c) for c in c2d.T]).T
res = cheb.chebder(c2d, axis=0)
assert_almost_equal(res, tgt)
tgt = np.vstack([cheb.chebder(c) for c in c2d])
res = cheb.chebder(c2d, axis=1)
assert_almost_equal(res, tgt)
class TestVander(object):
# some random values in [-1, 1)
x = np.random.random((3, 5))*2 - 1
def test_chebvander(self):
# check for 1d x
x = np.arange(3)
v = cheb.chebvander(x, 3)
assert_(v.shape == (3, 4))
for i in range(4):
coef = [0]*i + [1]
assert_almost_equal(v[..., i], cheb.chebval(x, coef))
# check for 2d x
x = np.array([[1, 2], [3, 4], [5, 6]])
v = cheb.chebvander(x, 3)
assert_(v.shape == (3, 2, 4))
for i in range(4):
coef = [0]*i + [1]
assert_almost_equal(v[..., i], cheb.chebval(x, coef))
def test_chebvander2d(self):
# also tests chebval2d for non-square coefficient array
x1, x2, x3 = self.x
c = np.random.random((2, 3))
van = cheb.chebvander2d(x1, x2, [1, 2])
tgt = cheb.chebval2d(x1, x2, c)
res = np.dot(van, c.flat)
assert_almost_equal(res, tgt)
# check shape
van = cheb.chebvander2d([x1], [x2], [1, 2])
assert_(van.shape == (1, 5, 6))
def test_chebvander3d(self):
# also tests chebval3d for non-square coefficient array
x1, x2, x3 = self.x
c = np.random.random((2, 3, 4))
van = cheb.chebvander3d(x1, x2, x3, [1, 2, 3])
tgt = cheb.chebval3d(x1, x2, x3, c)
res = np.dot(van, c.flat)
assert_almost_equal(res, tgt)
# check shape
van = cheb.chebvander3d([x1], [x2], [x3], [1, 2, 3])
assert_(van.shape == (1, 5, 24))
class TestFitting(object):
def test_chebfit(self):
def f(x):
return x*(x - 1)*(x - 2)
def f2(x):
return x**4 + x**2 + 1
# Test exceptions
assert_raises(ValueError, cheb.chebfit, [1], [1], -1)
assert_raises(TypeError, cheb.chebfit, [[1]], [1], 0)
assert_raises(TypeError, cheb.chebfit, [], [1], 0)
assert_raises(TypeError, cheb.chebfit, [1], [[[1]]], 0)
assert_raises(TypeError, cheb.chebfit, [1, 2], [1], 0)
assert_raises(TypeError, cheb.chebfit, [1], [1, 2], 0)
assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[[1]])
assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[1, 1])
assert_raises(ValueError, cheb.chebfit, [1], [1], [-1,])
assert_raises(ValueError, cheb.chebfit, [1], [1], [2, -1, 6])
assert_raises(TypeError, cheb.chebfit, [1], [1], [])
# Test fit
x = np.linspace(0, 2)
y = f(x)
#
coef3 = cheb.chebfit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(cheb.chebval(x, coef3), y)
coef3 = cheb.chebfit(x, y, [0, 1, 2, 3])
assert_equal(len(coef3), 4)
assert_almost_equal(cheb.chebval(x, coef3), y)
#
coef4 = cheb.chebfit(x, y, 4)
assert_equal(len(coef4), 5)
assert_almost_equal(cheb.chebval(x, coef4), y)
coef4 = cheb.chebfit(x, y, [0, 1, 2, 3, 4])
assert_equal(len(coef4), 5)
assert_almost_equal(cheb.chebval(x, coef4), y)
# check things still work if deg is not in strict increasing
coef4 = cheb.chebfit(x, y, [2, 3, 4, 1, 0])
assert_equal(len(coef4), 5)
assert_almost_equal(cheb.chebval(x, coef4), y)
#
coef2d = cheb.chebfit(x, np.array([y, y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
coef2d = cheb.chebfit(x, np.array([y, y]).T, [0, 1, 2, 3])
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = cheb.chebfit(x, yw, 3, w=w)
assert_almost_equal(wcoef3, coef3)
wcoef3 = cheb.chebfit(x, yw, [0, 1, 2, 3], w=w)
assert_almost_equal(wcoef3, coef3)
#
wcoef2d = cheb.chebfit(x, np.array([yw, yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
wcoef2d = cheb.chebfit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
# test scaling with complex values x points whose square
# is zero when summed.
x = [1, 1j, -1, -1j]
assert_almost_equal(cheb.chebfit(x, x, 1), [0, 1])
assert_almost_equal(cheb.chebfit(x, x, [0, 1]), [0, 1])
# test fitting only even polynomials
x = np.linspace(-1, 1)
y = f2(x)
coef1 = cheb.chebfit(x, y, 4)
assert_almost_equal(cheb.chebval(x, coef1), y)
coef2 = cheb.chebfit(x, y, [0, 2, 4])
assert_almost_equal(cheb.chebval(x, coef2), y)
assert_almost_equal(coef1, coef2)
class TestInterpolate(object):
def f(self, x):
return x * (x - 1) * (x - 2)
def test_raises(self):
assert_raises(ValueError, cheb.chebinterpolate, self.f, -1)
assert_raises(TypeError, cheb.chebinterpolate, self.f, 10.)
def test_dimensions(self):
for deg in range(1, 5):
assert_(cheb.chebinterpolate(self.f, deg).shape == (deg + 1,))
def test_approximation(self):
def powx(x, p):
return x**p
x = np.linspace(-1, 1, 10)
for deg in range(0, 10):
for p in range(0, deg + 1):
c = cheb.chebinterpolate(powx, deg, (p,))
assert_almost_equal(cheb.chebval(x, c), powx(x, p), decimal=12)
class TestCompanion(object):
def test_raises(self):
assert_raises(ValueError, cheb.chebcompanion, [])
assert_raises(ValueError, cheb.chebcompanion, [1])
def test_dimensions(self):
for i in range(1, 5):
coef = [0]*i + [1]
assert_(cheb.chebcompanion(coef).shape == (i, i))
def test_linear_root(self):
assert_(cheb.chebcompanion([1, 2])[0, 0] == -.5)
class TestGauss(object):
def test_100(self):
x, w = cheb.chebgauss(100)
# test orthogonality. Note that the results need to be normalized,
# otherwise the huge values that can arise from fast growing
# functions like Laguerre can be very confusing.
v = cheb.chebvander(x, 99)
vv = np.dot(v.T * w, v)
vd = 1/np.sqrt(vv.diagonal())
vv = vd[:, None] * vv * vd
assert_almost_equal(vv, np.eye(100))
# check that the integral of 1 is correct
tgt = np.pi
assert_almost_equal(w.sum(), tgt)
class TestMisc(object):
def test_chebfromroots(self):
res = cheb.chebfromroots([])
assert_almost_equal(trim(res), [1])
for i in range(1, 5):
roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2])
tgt = [0]*i + [1]
res = cheb.chebfromroots(roots)*2**(i-1)
assert_almost_equal(trim(res), trim(tgt))
def test_chebroots(self):
assert_almost_equal(cheb.chebroots([1]), [])
assert_almost_equal(cheb.chebroots([1, 2]), [-.5])
for i in range(2, 5):
tgt = np.linspace(-1, 1, i)
res = cheb.chebroots(cheb.chebfromroots(tgt))
assert_almost_equal(trim(res), trim(tgt))
def test_chebtrim(self):
coef = [2, -1, 1, 0]
# Test exceptions
assert_raises(ValueError, cheb.chebtrim, coef, -1)
# Test results
assert_equal(cheb.chebtrim(coef), coef[:-1])
assert_equal(cheb.chebtrim(coef, 1), coef[:-3])
assert_equal(cheb.chebtrim(coef, 2), [0])
def test_chebline(self):
assert_equal(cheb.chebline(3, 4), [3, 4])
def test_cheb2poly(self):
for i in range(10):
assert_almost_equal(cheb.cheb2poly([0]*i + [1]), Tlist[i])
def test_poly2cheb(self):
for i in range(10):
assert_almost_equal(cheb.poly2cheb(Tlist[i]), [0]*i + [1])
def test_weight(self):
x = np.linspace(-1, 1, 11)[1:-1]
tgt = 1./(np.sqrt(1 + x) * np.sqrt(1 - x))
res = cheb.chebweight(x)
assert_almost_equal(res, tgt)
def test_chebpts1(self):
#test exceptions
assert_raises(ValueError, cheb.chebpts1, 1.5)
assert_raises(ValueError, cheb.chebpts1, 0)
#test points
tgt = [0]
assert_almost_equal(cheb.chebpts1(1), tgt)
tgt = [-0.70710678118654746, 0.70710678118654746]
assert_almost_equal(cheb.chebpts1(2), tgt)
tgt = [-0.86602540378443871, 0, 0.86602540378443871]
assert_almost_equal(cheb.chebpts1(3), tgt)
tgt = [-0.9238795325, -0.3826834323, 0.3826834323, 0.9238795325]
assert_almost_equal(cheb.chebpts1(4), tgt)
def test_chebpts2(self):
#test exceptions
assert_raises(ValueError, cheb.chebpts2, 1.5)
assert_raises(ValueError, cheb.chebpts2, 1)
#test points
tgt = [-1, 1]
assert_almost_equal(cheb.chebpts2(2), tgt)
tgt = [-1, 0, 1]
assert_almost_equal(cheb.chebpts2(3), tgt)
tgt = [-1, -0.5, .5, 1]
assert_almost_equal(cheb.chebpts2(4), tgt)
tgt = [-1.0, -0.707106781187, 0, 0.707106781187, 1.0]
assert_almost_equal(cheb.chebpts2(5), tgt)
if __name__ == "__main__":
run_module_suite()