name : test_chebyshev.py """Tests for chebyshev module. """ from functools import reduce 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_, ) 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: 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: 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: def test_chebadd(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 = 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 = f"At i={i}, j={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 = f"At i={i}, j={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 = f"At i={i}, j={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) def test_chebpow(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(cheb.chebmul, [c]*j, np.array([1])) res = cheb.chebpow(c, j) assert_equal(trim(res), trim(tgt), err_msg=msg) class TestEvaluation: # 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 = f"At i={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: def test_chebint(self): # check exceptions assert_raises(TypeError, 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(TypeError, 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: def test_chebder(self): # check exceptions assert_raises(TypeError, 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: # 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: 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: 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: 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: 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: 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)