manyi/venv/Lib/site-packages/numpy/polynomial/tests/test_classes.py

"""Test inter-conversion of different polynomial classes.

This tests the convert and cast methods of all the polynomial classes.

"""
from __future__ import division, absolute_import, print_function

import operator as op
from numbers import Number

import pytest
import numpy as np
from numpy.polynomial import (
    Polynomial, Legendre, Chebyshev, Laguerre, Hermite, HermiteE)
from numpy.testing import (
    assert_almost_equal, assert_raises, assert_equal, assert_,
    )
from numpy.compat import long


#
# fixtures
#

classes = (
    Polynomial, Legendre, Chebyshev, Laguerre,
    Hermite, HermiteE
    )
classids = tuple(cls.__name__ for cls in classes)

@pytest.fixture(params=classes, ids=classids)
def Poly(request):
    return request.param

#
# helper functions
#
random = np.random.random


def assert_poly_almost_equal(p1, p2, msg=""):
    try:
        assert_(np.all(p1.domain == p2.domain))
        assert_(np.all(p1.window == p2.window))
        assert_almost_equal(p1.coef, p2.coef)
    except AssertionError:
        msg = "Result: %s\nTarget: %s", (p1, p2)
        raise AssertionError(msg)


#
# Test conversion methods that depend on combinations of two classes.
#

Poly1 = Poly
Poly2 = Poly


def test_conversion(Poly1, Poly2):
    x = np.linspace(0, 1, 10)
    coef = random((3,))

    d1 = Poly1.domain + random((2,))*.25
    w1 = Poly1.window + random((2,))*.25
    p1 = Poly1(coef, domain=d1, window=w1)

    d2 = Poly2.domain + random((2,))*.25
    w2 = Poly2.window + random((2,))*.25
    p2 = p1.convert(kind=Poly2, domain=d2, window=w2)

    assert_almost_equal(p2.domain, d2)
    assert_almost_equal(p2.window, w2)
    assert_almost_equal(p2(x), p1(x))


def test_cast(Poly1, Poly2):
    x = np.linspace(0, 1, 10)
    coef = random((3,))

    d1 = Poly1.domain + random((2,))*.25
    w1 = Poly1.window + random((2,))*.25
    p1 = Poly1(coef, domain=d1, window=w1)

    d2 = Poly2.domain + random((2,))*.25
    w2 = Poly2.window + random((2,))*.25
    p2 = Poly2.cast(p1, domain=d2, window=w2)

    assert_almost_equal(p2.domain, d2)
    assert_almost_equal(p2.window, w2)
    assert_almost_equal(p2(x), p1(x))


#
# test methods that depend on one class
#


def test_identity(Poly):
    d = Poly.domain + random((2,))*.25
    w = Poly.window + random((2,))*.25
    x = np.linspace(d[0], d[1], 11)
    p = Poly.identity(domain=d, window=w)
    assert_equal(p.domain, d)
    assert_equal(p.window, w)
    assert_almost_equal(p(x), x)


def test_basis(Poly):
    d = Poly.domain + random((2,))*.25
    w = Poly.window + random((2,))*.25
    p = Poly.basis(5, domain=d, window=w)
    assert_equal(p.domain, d)
    assert_equal(p.window, w)
    assert_equal(p.coef, [0]*5 + [1])


def test_fromroots(Poly):
    # check that requested roots are zeros of a polynomial
    # of correct degree, domain, and window.
    d = Poly.domain + random((2,))*.25
    w = Poly.window + random((2,))*.25
    r = random((5,))
    p1 = Poly.fromroots(r, domain=d, window=w)
    assert_equal(p1.degree(), len(r))
    assert_equal(p1.domain, d)
    assert_equal(p1.window, w)
    assert_almost_equal(p1(r), 0)

    # check that polynomial is monic
    pdom = Polynomial.domain
    pwin = Polynomial.window
    p2 = Polynomial.cast(p1, domain=pdom, window=pwin)
    assert_almost_equal(p2.coef[-1], 1)


def test_fit(Poly):

    def f(x):
        return x*(x - 1)*(x - 2)
    x = np.linspace(0, 3)
    y = f(x)

    # check default value of domain and window
    p = Poly.fit(x, y, 3)
    assert_almost_equal(p.domain, [0, 3])
    assert_almost_equal(p(x), y)
    assert_equal(p.degree(), 3)

    # check with given domains and window
    d = Poly.domain + random((2,))*.25
    w = Poly.window + random((2,))*.25
    p = Poly.fit(x, y, 3, domain=d, window=w)
    assert_almost_equal(p(x), y)
    assert_almost_equal(p.domain, d)
    assert_almost_equal(p.window, w)
    p = Poly.fit(x, y, [0, 1, 2, 3], domain=d, window=w)
    assert_almost_equal(p(x), y)
    assert_almost_equal(p.domain, d)
    assert_almost_equal(p.window, w)

    # check with class domain default
    p = Poly.fit(x, y, 3, [])
    assert_equal(p.domain, Poly.domain)
    assert_equal(p.window, Poly.window)
    p = Poly.fit(x, y, [0, 1, 2, 3], [])
    assert_equal(p.domain, Poly.domain)
    assert_equal(p.window, Poly.window)

    # check that fit accepts weights.
    w = np.zeros_like(x)
    z = y + random(y.shape)*.25
    w[::2] = 1
    p1 = Poly.fit(x[::2], z[::2], 3)
    p2 = Poly.fit(x, z, 3, w=w)
    p3 = Poly.fit(x, z, [0, 1, 2, 3], w=w)
    assert_almost_equal(p1(x), p2(x))
    assert_almost_equal(p2(x), p3(x))


def test_equal(Poly):
    p1 = Poly([1, 2, 3], domain=[0, 1], window=[2, 3])
    p2 = Poly([1, 1, 1], domain=[0, 1], window=[2, 3])
    p3 = Poly([1, 2, 3], domain=[1, 2], window=[2, 3])
    p4 = Poly([1, 2, 3], domain=[0, 1], window=[1, 2])
    assert_(p1 == p1)
    assert_(not p1 == p2)
    assert_(not p1 == p3)
    assert_(not p1 == p4)


def test_not_equal(Poly):
    p1 = Poly([1, 2, 3], domain=[0, 1], window=[2, 3])
    p2 = Poly([1, 1, 1], domain=[0, 1], window=[2, 3])
    p3 = Poly([1, 2, 3], domain=[1, 2], window=[2, 3])
    p4 = Poly([1, 2, 3], domain=[0, 1], window=[1, 2])
    assert_(not p1 != p1)
    assert_(p1 != p2)
    assert_(p1 != p3)
    assert_(p1 != p4)


def test_add(Poly):
    # This checks commutation, not numerical correctness
    c1 = list(random((4,)) + .5)
    c2 = list(random((3,)) + .5)
    p1 = Poly(c1)
    p2 = Poly(c2)
    p3 = p1 + p2
    assert_poly_almost_equal(p2 + p1, p3)
    assert_poly_almost_equal(p1 + c2, p3)
    assert_poly_almost_equal(c2 + p1, p3)
    assert_poly_almost_equal(p1 + tuple(c2), p3)
    assert_poly_almost_equal(tuple(c2) + p1, p3)
    assert_poly_almost_equal(p1 + np.array(c2), p3)
    assert_poly_almost_equal(np.array(c2) + p1, p3)
    assert_raises(TypeError, op.add, p1, Poly([0], domain=Poly.domain + 1))
    assert_raises(TypeError, op.add, p1, Poly([0], window=Poly.window + 1))
    if Poly is Polynomial:
        assert_raises(TypeError, op.add, p1, Chebyshev([0]))
    else:
        assert_raises(TypeError, op.add, p1, Polynomial([0]))


def test_sub(Poly):
    # This checks commutation, not numerical correctness
    c1 = list(random((4,)) + .5)
    c2 = list(random((3,)) + .5)
    p1 = Poly(c1)
    p2 = Poly(c2)
    p3 = p1 - p2
    assert_poly_almost_equal(p2 - p1, -p3)
    assert_poly_almost_equal(p1 - c2, p3)
    assert_poly_almost_equal(c2 - p1, -p3)
    assert_poly_almost_equal(p1 - tuple(c2), p3)
    assert_poly_almost_equal(tuple(c2) - p1, -p3)
    assert_poly_almost_equal(p1 - np.array(c2), p3)
    assert_poly_almost_equal(np.array(c2) - p1, -p3)
    assert_raises(TypeError, op.sub, p1, Poly([0], domain=Poly.domain + 1))
    assert_raises(TypeError, op.sub, p1, Poly([0], window=Poly.window + 1))
    if Poly is Polynomial:
        assert_raises(TypeError, op.sub, p1, Chebyshev([0]))
    else:
        assert_raises(TypeError, op.sub, p1, Polynomial([0]))


def test_mul(Poly):
    c1 = list(random((4,)) + .5)
    c2 = list(random((3,)) + .5)
    p1 = Poly(c1)
    p2 = Poly(c2)
    p3 = p1 * p2
    assert_poly_almost_equal(p2 * p1, p3)
    assert_poly_almost_equal(p1 * c2, p3)
    assert_poly_almost_equal(c2 * p1, p3)
    assert_poly_almost_equal(p1 * tuple(c2), p3)
    assert_poly_almost_equal(tuple(c2) * p1, p3)
    assert_poly_almost_equal(p1 * np.array(c2), p3)
    assert_poly_almost_equal(np.array(c2) * p1, p3)
    assert_poly_almost_equal(p1 * 2, p1 * Poly([2]))
    assert_poly_almost_equal(2 * p1, p1 * Poly([2]))
    assert_raises(TypeError, op.mul, p1, Poly([0], domain=Poly.domain + 1))
    assert_raises(TypeError, op.mul, p1, Poly([0], window=Poly.window + 1))
    if Poly is Polynomial:
        assert_raises(TypeError, op.mul, p1, Chebyshev([0]))
    else:
        assert_raises(TypeError, op.mul, p1, Polynomial([0]))


def test_floordiv(Poly):
    c1 = list(random((4,)) + .5)
    c2 = list(random((3,)) + .5)
    c3 = list(random((2,)) + .5)
    p1 = Poly(c1)
    p2 = Poly(c2)
    p3 = Poly(c3)
    p4 = p1 * p2 + p3
    c4 = list(p4.coef)
    assert_poly_almost_equal(p4 // p2, p1)
    assert_poly_almost_equal(p4 // c2, p1)
    assert_poly_almost_equal(c4 // p2, p1)
    assert_poly_almost_equal(p4 // tuple(c2), p1)
    assert_poly_almost_equal(tuple(c4) // p2, p1)
    assert_poly_almost_equal(p4 // np.array(c2), p1)
    assert_poly_almost_equal(np.array(c4) // p2, p1)
    assert_poly_almost_equal(2 // p2, Poly([0]))
    assert_poly_almost_equal(p2 // 2, 0.5*p2)
    assert_raises(
        TypeError, op.floordiv, p1, Poly([0], domain=Poly.domain + 1))
    assert_raises(
        TypeError, op.floordiv, p1, Poly([0], window=Poly.window + 1))
    if Poly is Polynomial:
        assert_raises(TypeError, op.floordiv, p1, Chebyshev([0]))
    else:
        assert_raises(TypeError, op.floordiv, p1, Polynomial([0]))


def test_truediv(Poly):
    # true division is valid only if the denominator is a Number and
    # not a python bool.
    p1 = Poly([1,2,3])
    p2 = p1 * 5

    for stype in np.ScalarType:
        if not issubclass(stype, Number) or issubclass(stype, bool):
            continue
        s = stype(5)
        assert_poly_almost_equal(op.truediv(p2, s), p1)
        assert_raises(TypeError, op.truediv, s, p2)
    for stype in (int, long, float):
        s = stype(5)
        assert_poly_almost_equal(op.truediv(p2, s), p1)
        assert_raises(TypeError, op.truediv, s, p2)
    for stype in [complex]:
        s = stype(5, 0)
        assert_poly_almost_equal(op.truediv(p2, s), p1)
        assert_raises(TypeError, op.truediv, s, p2)
    for s in [tuple(), list(), dict(), bool(), np.array([1])]:
        assert_raises(TypeError, op.truediv, p2, s)
        assert_raises(TypeError, op.truediv, s, p2)
    for ptype in classes:
        assert_raises(TypeError, op.truediv, p2, ptype(1))


def test_mod(Poly):
    # This checks commutation, not numerical correctness
    c1 = list(random((4,)) + .5)
    c2 = list(random((3,)) + .5)
    c3 = list(random((2,)) + .5)
    p1 = Poly(c1)
    p2 = Poly(c2)
    p3 = Poly(c3)
    p4 = p1 * p2 + p3
    c4 = list(p4.coef)
    assert_poly_almost_equal(p4 % p2, p3)
    assert_poly_almost_equal(p4 % c2, p3)
    assert_poly_almost_equal(c4 % p2, p3)
    assert_poly_almost_equal(p4 % tuple(c2), p3)
    assert_poly_almost_equal(tuple(c4) % p2, p3)
    assert_poly_almost_equal(p4 % np.array(c2), p3)
    assert_poly_almost_equal(np.array(c4) % p2, p3)
    assert_poly_almost_equal(2 % p2, Poly([2]))
    assert_poly_almost_equal(p2 % 2, Poly([0]))
    assert_raises(TypeError, op.mod, p1, Poly([0], domain=Poly.domain + 1))
    assert_raises(TypeError, op.mod, p1, Poly([0], window=Poly.window + 1))
    if Poly is Polynomial:
        assert_raises(TypeError, op.mod, p1, Chebyshev([0]))
    else:
        assert_raises(TypeError, op.mod, p1, Polynomial([0]))


def test_divmod(Poly):
    # This checks commutation, not numerical correctness
    c1 = list(random((4,)) + .5)
    c2 = list(random((3,)) + .5)
    c3 = list(random((2,)) + .5)
    p1 = Poly(c1)
    p2 = Poly(c2)
    p3 = Poly(c3)
    p4 = p1 * p2 + p3
    c4 = list(p4.coef)
    quo, rem = divmod(p4, p2)
    assert_poly_almost_equal(quo, p1)
    assert_poly_almost_equal(rem, p3)
    quo, rem = divmod(p4, c2)
    assert_poly_almost_equal(quo, p1)
    assert_poly_almost_equal(rem, p3)
    quo, rem = divmod(c4, p2)
    assert_poly_almost_equal(quo, p1)
    assert_poly_almost_equal(rem, p3)
    quo, rem = divmod(p4, tuple(c2))
    assert_poly_almost_equal(quo, p1)
    assert_poly_almost_equal(rem, p3)
    quo, rem = divmod(tuple(c4), p2)
    assert_poly_almost_equal(quo, p1)
    assert_poly_almost_equal(rem, p3)
    quo, rem = divmod(p4, np.array(c2))
    assert_poly_almost_equal(quo, p1)
    assert_poly_almost_equal(rem, p3)
    quo, rem = divmod(np.array(c4), p2)
    assert_poly_almost_equal(quo, p1)
    assert_poly_almost_equal(rem, p3)
    quo, rem = divmod(p2, 2)
    assert_poly_almost_equal(quo, 0.5*p2)
    assert_poly_almost_equal(rem, Poly([0]))
    quo, rem = divmod(2, p2)
    assert_poly_almost_equal(quo, Poly([0]))
    assert_poly_almost_equal(rem, Poly([2]))
    assert_raises(TypeError, divmod, p1, Poly([0], domain=Poly.domain + 1))
    assert_raises(TypeError, divmod, p1, Poly([0], window=Poly.window + 1))
    if Poly is Polynomial:
        assert_raises(TypeError, divmod, p1, Chebyshev([0]))
    else:
        assert_raises(TypeError, divmod, p1, Polynomial([0]))


def test_roots(Poly):
    d = Poly.domain * 1.25 + .25
    w = Poly.window
    tgt = np.linspace(d[0], d[1], 5)
    res = np.sort(Poly.fromroots(tgt, domain=d, window=w).roots())
    assert_almost_equal(res, tgt)
    # default domain and window
    res = np.sort(Poly.fromroots(tgt).roots())
    assert_almost_equal(res, tgt)


def test_degree(Poly):
    p = Poly.basis(5)
    assert_equal(p.degree(), 5)


def test_copy(Poly):
    p1 = Poly.basis(5)
    p2 = p1.copy()
    assert_(p1 == p2)
    assert_(p1 is not p2)
    assert_(p1.coef is not p2.coef)
    assert_(p1.domain is not p2.domain)
    assert_(p1.window is not p2.window)


def test_integ(Poly):
    P = Polynomial
    # Check defaults
    p0 = Poly.cast(P([1*2, 2*3, 3*4]))
    p1 = P.cast(p0.integ())
    p2 = P.cast(p0.integ(2))
    assert_poly_almost_equal(p1, P([0, 2, 3, 4]))
    assert_poly_almost_equal(p2, P([0, 0, 1, 1, 1]))
    # Check with k
    p0 = Poly.cast(P([1*2, 2*3, 3*4]))
    p1 = P.cast(p0.integ(k=1))
    p2 = P.cast(p0.integ(2, k=[1, 1]))
    assert_poly_almost_equal(p1, P([1, 2, 3, 4]))
    assert_poly_almost_equal(p2, P([1, 1, 1, 1, 1]))
    # Check with lbnd
    p0 = Poly.cast(P([1*2, 2*3, 3*4]))
    p1 = P.cast(p0.integ(lbnd=1))
    p2 = P.cast(p0.integ(2, lbnd=1))
    assert_poly_almost_equal(p1, P([-9, 2, 3, 4]))
    assert_poly_almost_equal(p2, P([6, -9, 1, 1, 1]))
    # Check scaling
    d = 2*Poly.domain
    p0 = Poly.cast(P([1*2, 2*3, 3*4]), domain=d)
    p1 = P.cast(p0.integ())
    p2 = P.cast(p0.integ(2))
    assert_poly_almost_equal(p1, P([0, 2, 3, 4]))
    assert_poly_almost_equal(p2, P([0, 0, 1, 1, 1]))


def test_deriv(Poly):
    # Check that the derivative is the inverse of integration. It is
    # assumes that the integration has been checked elsewhere.
    d = Poly.domain + random((2,))*.25
    w = Poly.window + random((2,))*.25
    p1 = Poly([1, 2, 3], domain=d, window=w)
    p2 = p1.integ(2, k=[1, 2])
    p3 = p1.integ(1, k=[1])
    assert_almost_equal(p2.deriv(1).coef, p3.coef)
    assert_almost_equal(p2.deriv(2).coef, p1.coef)
    # default domain and window
    p1 = Poly([1, 2, 3])
    p2 = p1.integ(2, k=[1, 2])
    p3 = p1.integ(1, k=[1])
    assert_almost_equal(p2.deriv(1).coef, p3.coef)
    assert_almost_equal(p2.deriv(2).coef, p1.coef)


def test_linspace(Poly):
    d = Poly.domain + random((2,))*.25
    w = Poly.window + random((2,))*.25
    p = Poly([1, 2, 3], domain=d, window=w)
    # check default domain
    xtgt = np.linspace(d[0], d[1], 20)
    ytgt = p(xtgt)
    xres, yres = p.linspace(20)
    assert_almost_equal(xres, xtgt)
    assert_almost_equal(yres, ytgt)
    # check specified domain
    xtgt = np.linspace(0, 2, 20)
    ytgt = p(xtgt)
    xres, yres = p.linspace(20, domain=[0, 2])
    assert_almost_equal(xres, xtgt)
    assert_almost_equal(yres, ytgt)


def test_pow(Poly):
    d = Poly.domain + random((2,))*.25
    w = Poly.window + random((2,))*.25
    tgt = Poly([1], domain=d, window=w)
    tst = Poly([1, 2, 3], domain=d, window=w)
    for i in range(5):
        assert_poly_almost_equal(tst**i, tgt)
        tgt = tgt * tst
    # default domain and window
    tgt = Poly([1])
    tst = Poly([1, 2, 3])
    for i in range(5):
        assert_poly_almost_equal(tst**i, tgt)
        tgt = tgt * tst
    # check error for invalid powers
    assert_raises(ValueError, op.pow, tgt, 1.5)
    assert_raises(ValueError, op.pow, tgt, -1)


def test_call(Poly):
    P = Polynomial
    d = Poly.domain
    x = np.linspace(d[0], d[1], 11)

    # Check defaults
    p = Poly.cast(P([1, 2, 3]))
    tgt = 1 + x*(2 + 3*x)
    res = p(x)
    assert_almost_equal(res, tgt)


def test_cutdeg(Poly):
    p = Poly([1, 2, 3])
    assert_raises(ValueError, p.cutdeg, .5)
    assert_raises(ValueError, p.cutdeg, -1)
    assert_equal(len(p.cutdeg(3)), 3)
    assert_equal(len(p.cutdeg(2)), 3)
    assert_equal(len(p.cutdeg(1)), 2)
    assert_equal(len(p.cutdeg(0)), 1)


def test_truncate(Poly):
    p = Poly([1, 2, 3])
    assert_raises(ValueError, p.truncate, .5)
    assert_raises(ValueError, p.truncate, 0)
    assert_equal(len(p.truncate(4)), 3)
    assert_equal(len(p.truncate(3)), 3)
    assert_equal(len(p.truncate(2)), 2)
    assert_equal(len(p.truncate(1)), 1)


def test_trim(Poly):
    c = [1, 1e-6, 1e-12, 0]
    p = Poly(c)
    assert_equal(p.trim().coef, c[:3])
    assert_equal(p.trim(1e-10).coef, c[:2])
    assert_equal(p.trim(1e-5).coef, c[:1])


def test_mapparms(Poly):
    # check with defaults. Should be identity.
    d = Poly.domain
    w = Poly.window
    p = Poly([1], domain=d, window=w)
    assert_almost_equal([0, 1], p.mapparms())
    #
    w = 2*d + 1
    p = Poly([1], domain=d, window=w)
    assert_almost_equal([1, 2], p.mapparms())


def test_ufunc_override(Poly):
    p = Poly([1, 2, 3])
    x = np.ones(3)
    assert_raises(TypeError, np.add, p, x)
    assert_raises(TypeError, np.add, x, p)



class TestLatexRepr(object):
    """Test the latex repr used by ipython """

    def as_latex(self, obj):
        # right now we ignore the formatting of scalars in our tests, since
        # it makes them too verbose. Ideally, the formatting of scalars will
        # be fixed such that tests below continue to pass
        obj._repr_latex_scalar = lambda x: str(x)
        try:
            return obj._repr_latex_()
        finally:
            del obj._repr_latex_scalar

    def test_simple_polynomial(self):
        # default input
        p = Polynomial([1, 2, 3])
        assert_equal(self.as_latex(p),
            r'$x \mapsto 1.0 + 2.0\,x + 3.0\,x^{2}$')

        # translated input
        p = Polynomial([1, 2, 3], domain=[-2, 0])
        assert_equal(self.as_latex(p),
            r'$x \mapsto 1.0 + 2.0\,\left(1.0 + x\right) + 3.0\,\left(1.0 + x\right)^{2}$')

        # scaled input
        p = Polynomial([1, 2, 3], domain=[-0.5, 0.5])
        assert_equal(self.as_latex(p),
            r'$x \mapsto 1.0 + 2.0\,\left(2.0x\right) + 3.0\,\left(2.0x\right)^{2}$')

        # affine input
        p = Polynomial([1, 2, 3], domain=[-1, 0])
        assert_equal(self.as_latex(p),
            r'$x \mapsto 1.0 + 2.0\,\left(1.0 + 2.0x\right) + 3.0\,\left(1.0 + 2.0x\right)^{2}$')

    def test_basis_func(self):
        p = Chebyshev([1, 2, 3])
        assert_equal(self.as_latex(p),
            r'$x \mapsto 1.0\,{T}_{0}(x) + 2.0\,{T}_{1}(x) + 3.0\,{T}_{2}(x)$')
        # affine input - check no surplus parens are added
        p = Chebyshev([1, 2, 3], domain=[-1, 0])
        assert_equal(self.as_latex(p),
            r'$x \mapsto 1.0\,{T}_{0}(1.0 + 2.0x) + 2.0\,{T}_{1}(1.0 + 2.0x) + 3.0\,{T}_{2}(1.0 + 2.0x)$')

    def test_multichar_basis_func(self):
        p = HermiteE([1, 2, 3])
        assert_equal(self.as_latex(p),
            r'$x \mapsto 1.0\,{He}_{0}(x) + 2.0\,{He}_{1}(x) + 3.0\,{He}_{2}(x)$')


#
# Test class method that only exists for some classes
#


class TestInterpolate(object):

    def f(self, x):
        return x * (x - 1) * (x - 2)

    def test_raises(self):
        assert_raises(ValueError, Chebyshev.interpolate, self.f, -1)
        assert_raises(TypeError, Chebyshev.interpolate, self.f, 10.)

    def test_dimensions(self):
        for deg in range(1, 5):
            assert_(Chebyshev.interpolate(self.f, deg).degree() == deg)

    def test_approximation(self):

        def powx(x, p):
            return x**p

        x = np.linspace(0, 2, 10)
        for deg in range(0, 10):
            for t in range(0, deg + 1):
                p = Chebyshev.interpolate(powx, deg, domain=[0, 2], args=(t,))
                assert_almost_equal(p(x), powx(x, t), decimal=12)