name : test_polynomial.py """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])