"""Tests for polynomial module.
"""
from functools import reduce
from fractions import Fraction
import numpy as np
import numpy.polynomial.polynomial as poly
import numpy.polynomial.polyutils as pu
import pickle
from copy import deepcopy
from numpy.testing import (
assert_almost_equal, assert_raises, assert_equal, assert_,
assert_array_equal, assert_raises_regex, assert_warns)
def trim(x):
return poly.polytrim(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 TestConstants:
def test_polydomain(self):
assert_equal(poly.polydomain, [-1, 1])
def test_polyzero(self):
assert_equal(poly.polyzero, [0])
def test_polyone(self):
assert_equal(poly.polyone, [1])
def test_polyx(self):
assert_equal(poly.polyx, [0, 1])
def test_copy(self):
x = poly.Polynomial([1, 2, 3])
y = deepcopy(x)
assert_equal(x, y)
def test_pickle(self):
x = poly.Polynomial([1, 2, 3])
y = pickle.loads(pickle.dumps(x))
assert_equal(x, y)
class TestArithmetic:
def test_polyadd(self):
for i in range(5):
for j in range(5):
msg = f"At i={i}, j={j}"
tgt = np.zeros(max(i, j) + 1)
tgt[i] += 1
tgt[j] += 1
res = poly.polyadd([0]*i + [1], [0]*j + [1])
assert_equal(trim(res), trim(tgt), err_msg=msg)
def test_polysub(self):
for i in range(5):
for j in range(5):
msg = f"At i={i}, j={j}"
tgt = np.zeros(max(i, j) + 1)
tgt[i] += 1
tgt[j] -= 1
res = poly.polysub([0]*i + [1], [0]*j + [1])
assert_equal(trim(res), trim(tgt), err_msg=msg)
def test_polymulx(self):
assert_equal(poly.polymulx([0]), [0])
assert_equal(poly.polymulx([1]), [0, 1])
for i in range(1, 5):
ser = [0]*i + [1]
tgt = [0]*(i + 1) + [1]
assert_equal(poly.polymulx(ser), tgt)
def test_polymul(self):
for i in range(5):
for j in range(5):
msg = f"At i={i}, j={j}"
tgt = np.zeros(i + j + 1)
tgt[i + j] += 1
res = poly.polymul([0]*i + [1], [0]*j + [1])
assert_equal(trim(res), trim(tgt), err_msg=msg)
def test_polydiv(self):
# check zero division
assert_raises(ZeroDivisionError, poly.polydiv, [1], [0])
# check scalar division
quo, rem = poly.polydiv([2], [2])
assert_equal((quo, rem), (1, 0))
quo, rem = poly.polydiv([2, 2], [2])
assert_equal((quo, rem), ((1, 1), 0))
# check rest.
for i in range(5):
for j in range(5):
msg = f"At i={i}, j={j}"
ci = [0]*i + [1, 2]
cj = [0]*j + [1, 2]
tgt = poly.polyadd(ci, cj)
quo, rem = poly.polydiv(tgt, ci)
res = poly.polyadd(poly.polymul(quo, ci), rem)
assert_equal(res, tgt, err_msg=msg)
def test_polypow(self):
for i in range(5):
for j in range(5):
msg = f"At i={i}, j={j}"
c = np.arange(i + 1)
tgt = reduce(poly.polymul, [c]*j, np.array([1]))
res = poly.polypow(c, j)
assert_equal(trim(res), trim(tgt), err_msg=msg)
class TestFraction:
def test_Fraction(self):
# assert we can use Polynomials with coefficients of object dtype
f = Fraction(2, 3)
one = Fraction(1, 1)
zero = Fraction(0, 1)
p = poly.Polynomial([f, f], domain=[zero, one], window=[zero, one])
x = 2 * p + p ** 2
assert_equal(x.coef, np.array([Fraction(16, 9), Fraction(20, 9),
Fraction(4, 9)], dtype=object))
assert_equal(p.domain, [zero, one])
assert_equal(p.coef.dtype, np.dtypes.ObjectDType())
assert_(isinstance(p(f), Fraction))
assert_equal(p(f), Fraction(10, 9))
p_deriv = poly.Polynomial([Fraction(2, 3)], domain=[zero, one],
window=[zero, one])
assert_equal(p.deriv(), p_deriv)
class TestEvaluation:
# coefficients of 1 + 2*x + 3*x**2
c1d = np.array([1., 2., 3.])
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 = poly.polyval(x, [1., 2., 3.])
def test_polyval(self):
#check empty input
assert_equal(poly.polyval([], [1]).size, 0)
#check normal input)
x = np.linspace(-1, 1)
y = [x**i for i in range(5)]
for i in range(5):
tgt = y[i]
res = poly.polyval(x, [0]*i + [1])
assert_almost_equal(res, tgt)
tgt = x*(x**2 - 1)
res = poly.polyval(x, [0, -1, 0, 1])
assert_almost_equal(res, tgt)
#check that shape is preserved
for i in range(3):
dims = [2]*i
x = np.zeros(dims)
assert_equal(poly.polyval(x, [1]).shape, dims)
assert_equal(poly.polyval(x, [1, 0]).shape, dims)
assert_equal(poly.polyval(x, [1, 0, 0]).shape, dims)
#check masked arrays are processed correctly
mask = [False, True, False]
mx = np.ma.array([1, 2, 3], mask=mask)
res = np.polyval([7, 5, 3], mx)
assert_array_equal(res.mask, mask)
#check subtypes of ndarray are preserved
class C(np.ndarray):
pass
cx = np.array([1, 2, 3]).view(C)
assert_equal(type(np.polyval([2, 3, 4], cx)), C)
def test_polyvalfromroots(self):
# check exception for broadcasting x values over root array with
# too few dimensions
assert_raises(ValueError, poly.polyvalfromroots,
[1], [1], tensor=False)
# check empty input
assert_equal(poly.polyvalfromroots([], [1]).size, 0)
assert_(poly.polyvalfromroots([], [1]).shape == (0,))
# check empty input + multidimensional roots
assert_equal(poly.polyvalfromroots([], [[1] * 5]).size, 0)
assert_(poly.polyvalfromroots([], [[1] * 5]).shape == (5, 0))
# check scalar input
assert_equal(poly.polyvalfromroots(1, 1), 0)
assert_(poly.polyvalfromroots(1, np.ones((3, 3))).shape == (3,))
# check normal input)
x = np.linspace(-1, 1)
y = [x**i for i in range(5)]
for i in range(1, 5):
tgt = y[i]
res = poly.polyvalfromroots(x, [0]*i)
assert_almost_equal(res, tgt)
tgt = x*(x - 1)*(x + 1)
res = poly.polyvalfromroots(x, [-1, 0, 1])
assert_almost_equal(res, tgt)
# check that shape is preserved
for i in range(3):
dims = [2]*i
x = np.zeros(dims)
assert_equal(poly.polyvalfromroots(x, [1]).shape, dims)
assert_equal(poly.polyvalfromroots(x, [1, 0]).shape, dims)
assert_equal(poly.polyvalfromroots(x, [1, 0, 0]).shape, dims)
# check compatibility with factorization
ptest = [15, 2, -16, -2, 1]
r = poly.polyroots(ptest)
x = np.linspace(-1, 1)
assert_almost_equal(poly.polyval(x, ptest),
poly.polyvalfromroots(x, r))
# check multidimensional arrays of roots and values
# check tensor=False
rshape = (3, 5)
x = np.arange(-3, 2)
r = np.random.randint(-5, 5, size=rshape)
res = poly.polyvalfromroots(x, r, tensor=False)
tgt = np.empty(r.shape[1:])
for ii in range(tgt.size):
tgt[ii] = poly.polyvalfromroots(x[ii], r[:, ii])
assert_equal(res, tgt)
# check tensor=True
x = np.vstack([x, 2*x])
res = poly.polyvalfromroots(x, r, tensor=True)
tgt = np.empty(r.shape[1:] + x.shape)
for ii in range(r.shape[1]):
for jj in range(x.shape[0]):
tgt[ii, jj, :] = poly.polyvalfromroots(x[jj], r[:, ii])
assert_equal(res, tgt)
def test_polyval2d(self):
x1, x2, x3 = self.x
y1, y2, y3 = self.y
#test exceptions
assert_raises_regex(ValueError, 'incompatible',
poly.polyval2d, x1, x2[:2], self.c2d)
#test values
tgt = y1*y2
res = poly.polyval2d(x1, x2, self.c2d)
assert_almost_equal(res, tgt)
#test shape
z = np.ones((2, 3))
res = poly.polyval2d(z, z, self.c2d)
assert_(res.shape == (2, 3))
def test_polyval3d(self):
x1, x2, x3 = self.x
y1, y2, y3 = self.y
#test exceptions
assert_raises_regex(ValueError, 'incompatible',
poly.polyval3d, x1, x2, x3[:2], self.c3d)
#test values
tgt = y1*y2*y3
res = poly.polyval3d(x1, x2, x3, self.c3d)
assert_almost_equal(res, tgt)
#test shape
z = np.ones((2, 3))
res = poly.polyval3d(z, z, z, self.c3d)
assert_(res.shape == (2, 3))
def test_polygrid2d(self):
x1, x2, x3 = self.x
y1, y2, y3 = self.y
#test values
tgt = np.einsum('i,j->ij', y1, y2)
res = poly.polygrid2d(x1, x2, self.c2d)
assert_almost_equal(res, tgt)
#test shape
z = np.ones((2, 3))
res = poly.polygrid2d(z, z, self.c2d)
assert_(res.shape == (2, 3)*2)
def test_polygrid3d(self):
x1, x2, x3 = self.x
y1, y2, y3 = self.y
#test values
tgt = np.einsum('i,j,k->ijk', y1, y2, y3)
res = poly.polygrid3d(x1, x2, x3, self.c3d)
assert_almost_equal(res, tgt)
#test shape
z = np.ones((2, 3))
res = poly.polygrid3d(z, z, z, self.c3d)
assert_(res.shape == (2, 3)*3)
class TestIntegral:
def test_polyint(self):
# check exceptions
assert_raises(TypeError, poly.polyint, [0], .5)
assert_raises(ValueError, poly.polyint, [0], -1)
assert_raises(ValueError, poly.polyint, [0], 1, [0, 0])
assert_raises(ValueError, poly.polyint, [0], lbnd=[0])
assert_raises(ValueError, poly.polyint, [0], scl=[0])
assert_raises(TypeError, poly.polyint, [0], axis=.5)
assert_raises(TypeError, poly.polyint, [1, 1], 1.)
# test integration of zero polynomial
for i in range(2, 5):
k = [0]*(i - 2) + [1]
res = poly.polyint([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]
res = poly.polyint(pol, m=1, k=[i])
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]
res = poly.polyint(pol, m=1, k=[i], lbnd=-1)
assert_almost_equal(poly.polyval(-1, res), 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]
res = poly.polyint(pol, m=1, k=[i], scl=2)
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 = poly.polyint(tgt, m=1)
res = poly.polyint(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 = poly.polyint(tgt, m=1, k=[k])
res = poly.polyint(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 = poly.polyint(tgt, m=1, k=[k], lbnd=-1)
res = poly.polyint(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 = poly.polyint(tgt, m=1, k=[k], scl=2)
res = poly.polyint(pol, m=j, k=list(range(j)), scl=2)
assert_almost_equal(trim(res), trim(tgt))
def test_polyint_axis(self):
# check that axis keyword works
c2d = np.random.random((3, 4))
tgt = np.vstack([poly.polyint(c) for c in c2d.T]).T
res = poly.polyint(c2d, axis=0)
assert_almost_equal(res, tgt)
tgt = np.vstack([poly.polyint(c) for c in c2d])
res = poly.polyint(c2d, axis=1)
assert_almost_equal(res, tgt)
tgt = np.vstack([poly.polyint(c, k=3) for c in c2d])
res = poly.polyint(c2d, k=3, axis=1)
assert_almost_equal(res, tgt)
class TestDerivative:
def test_polyder(self):
# check exceptions
assert_raises(TypeError, poly.polyder, [0], .5)
assert_raises(ValueError, poly.polyder, [0], -1)
# check that zeroth derivative does nothing
for i in range(5):
tgt = [0]*i + [1]
res = poly.polyder(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 = poly.polyder(poly.polyint(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 = poly.polyder(poly.polyint(tgt, m=j, scl=2), m=j, scl=.5)
assert_almost_equal(trim(res), trim(tgt))
def test_polyder_axis(self):
# check that axis keyword works
c2d = np.random.random((3, 4))
tgt = np.vstack([poly.polyder(c) for c in c2d.T]).T
res = poly.polyder(c2d, axis=0)
assert_almost_equal(res, tgt)
tgt = np.vstack([poly.polyder(c) for c in c2d])
res = poly.polyder(c2d, axis=1)
assert_almost_equal(res, tgt)
class TestVander:
# some random values in [-1, 1)
x = np.random.random((3, 5))*2 - 1
def test_polyvander(self):
# check for 1d x
x = np.arange(3)
v = poly.polyvander(x, 3)
assert_(v.shape == (3, 4))
for i in range(4):
coef = [0]*i + [1]
assert_almost_equal(v[..., i], poly.polyval(x, coef))
# check for 2d x
x = np.array([[1, 2], [3, 4], [5, 6]])
v = poly.polyvander(x, 3)
assert_(v.shape == (3, 2, 4))
for i in range(4):
coef = [0]*i + [1]
assert_almost_equal(v[..., i], poly.polyval(x, coef))
def test_polyvander2d(self):
# also tests polyval2d for non-square coefficient array
x1, x2, x3 = self.x
c = np.random.random((2, 3))
van = poly.polyvander2d(x1, x2, [1, 2])
tgt = poly.polyval2d(x1, x2, c)
res = np.dot(van, c.flat)
assert_almost_equal(res, tgt)
# check shape
van = poly.polyvander2d([x1], [x2], [1, 2])
assert_(van.shape == (1, 5, 6))
def test_polyvander3d(self):
# also tests polyval3d for non-square coefficient array
x1, x2, x3 = self.x
c = np.random.random((2, 3, 4))
van = poly.polyvander3d(x1, x2, x3, [1, 2, 3])
tgt = poly.polyval3d(x1, x2, x3, c)
res = np.dot(van, c.flat)
assert_almost_equal(res, tgt)
# check shape
van = poly.polyvander3d([x1], [x2], [x3], [1, 2, 3])
assert_(van.shape == (1, 5, 24))
def test_polyvandernegdeg(self):
x = np.arange(3)
assert_raises(ValueError, poly.polyvander, x, -1)
class TestCompanion:
def test_raises(self):
assert_raises(ValueError, poly.polycompanion, [])
assert_raises(ValueError, poly.polycompanion, [1])
def test_dimensions(self):
for i in range(1, 5):
coef = [0]*i + [1]
assert_(poly.polycompanion(coef).shape == (i, i))
def test_linear_root(self):
assert_(poly.polycompanion([1, 2])[0, 0] == -.5)
class TestMisc:
def test_polyfromroots(self):
res = poly.polyfromroots([])
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 = Tlist[i]
res = poly.polyfromroots(roots)*2**(i-1)
assert_almost_equal(trim(res), trim(tgt))
def test_polyroots(self):
assert_almost_equal(poly.polyroots([1]), [])
assert_almost_equal(poly.polyroots([1, 2]), [-.5])
for i in range(2, 5):
tgt = np.linspace(-1, 1, i)
res = poly.polyroots(poly.polyfromroots(tgt))
assert_almost_equal(trim(res), trim(tgt))
def test_polyfit(self):
def f(x):
return x*(x - 1)*(x - 2)
def f2(x):
return x**4 + x**2 + 1
# Test exceptions
assert_raises(ValueError, poly.polyfit, [1], [1], -1)
assert_raises(TypeError, poly.polyfit, [[1]], [1], 0)
assert_raises(TypeError, poly.polyfit, [], [1], 0)
assert_raises(TypeError, poly.polyfit, [1], [[[1]]], 0)
assert_raises(TypeError, poly.polyfit, [1, 2], [1], 0)
assert_raises(TypeError, poly.polyfit, [1], [1, 2], 0)
assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[[1]])
assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[1, 1])
assert_raises(ValueError, poly.polyfit, [1], [1], [-1,])
assert_raises(ValueError, poly.polyfit, [1], [1], [2, -1, 6])
assert_raises(TypeError, poly.polyfit, [1], [1], [])
# Test fit
x = np.linspace(0, 2)
y = f(x)
#
coef3 = poly.polyfit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(poly.polyval(x, coef3), y)
coef3 = poly.polyfit(x, y, [0, 1, 2, 3])
assert_equal(len(coef3), 4)
assert_almost_equal(poly.polyval(x, coef3), y)
#
coef4 = poly.polyfit(x, y, 4)
assert_equal(len(coef4), 5)
assert_almost_equal(poly.polyval(x, coef4), y)
coef4 = poly.polyfit(x, y, [0, 1, 2, 3, 4])
assert_equal(len(coef4), 5)
assert_almost_equal(poly.polyval(x, coef4), y)
#
coef2d = poly.polyfit(x, np.array([y, y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
coef2d = poly.polyfit(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
yw[0::2] = 0
wcoef3 = poly.polyfit(x, yw, 3, w=w)
assert_almost_equal(wcoef3, coef3)
wcoef3 = poly.polyfit(x, yw, [0, 1, 2, 3], w=w)
assert_almost_equal(wcoef3, coef3)
#
wcoef2d = poly.polyfit(x, np.array([yw, yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
wcoef2d = poly.polyfit(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(poly.polyfit(x, x, 1), [0, 1])
assert_almost_equal(poly.polyfit(x, x, [0, 1]), [0, 1])
# test fitting only even Polyendre polynomials
x = np.linspace(-1, 1)
y = f2(x)
coef1 = poly.polyfit(x, y, 4)
assert_almost_equal(poly.polyval(x, coef1), y)
coef2 = poly.polyfit(x, y, [0, 2, 4])
assert_almost_equal(poly.polyval(x, coef2), y)
assert_almost_equal(coef1, coef2)
def test_polytrim(self):
coef = [2, -1, 1, 0]
# Test exceptions
assert_raises(ValueError, poly.polytrim, coef, -1)
# Test results
assert_equal(poly.polytrim(coef), coef[:-1])
assert_equal(poly.polytrim(coef, 1), coef[:-3])
assert_equal(poly.polytrim(coef, 2), [0])
def test_polyline(self):
assert_equal(poly.polyline(3, 4), [3, 4])
def test_polyline_zero(self):
assert_equal(poly.polyline(3, 0), [3])
def test_fit_degenerate_domain(self):
p = poly.Polynomial.fit([1], [2], deg=0)
assert_equal(p.coef, [2.])
p = poly.Polynomial.fit([1, 1], [2, 2.1], deg=0)
assert_almost_equal(p.coef, [2.05])
with assert_warns(pu.RankWarning):
p = poly.Polynomial.fit([1, 1], [2, 2.1], deg=1)
def test_result_type(self):
w = np.array([-1, 1], dtype=np.float32)
p = np.polynomial.Polynomial(w, domain=w, window=w)
v = p(2)
assert_equal(v.dtype, np.float32)
arr = np.polydiv(1, np.float32(1))
assert_equal(arr[0].dtype, np.float64)