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+"""Tests for polynomial module.
+
+"""
+from functools import reduce
+
+import numpy as np
+import numpy.polynomial.polynomial as poly
+import pickle
+from copy import deepcopy
+from numpy.testing import (
+ assert_almost_equal, assert_raises, assert_equal, assert_,
+ assert_warns, assert_array_equal, assert_raises_regex)
+
+
+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 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)
+ with assert_warns(DeprecationWarning):
+ 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])
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