name : test_legendre.py """Tests for legendre module. """ from functools import reduce import numpy as np import numpy.polynomial.legendre as leg from numpy.polynomial.polynomial import polyval from numpy.testing import ( assert_almost_equal, assert_raises, assert_equal, assert_, ) L0 = np.array([1]) L1 = np.array([0, 1]) L2 = np.array([-1, 0, 3])/2 L3 = np.array([0, -3, 0, 5])/2 L4 = np.array([3, 0, -30, 0, 35])/8 L5 = np.array([0, 15, 0, -70, 0, 63])/8 L6 = np.array([-5, 0, 105, 0, -315, 0, 231])/16 L7 = np.array([0, -35, 0, 315, 0, -693, 0, 429])/16 L8 = np.array([35, 0, -1260, 0, 6930, 0, -12012, 0, 6435])/128 L9 = np.array([0, 315, 0, -4620, 0, 18018, 0, -25740, 0, 12155])/128 Llist = [L0, L1, L2, L3, L4, L5, L6, L7, L8, L9] def trim(x): return leg.legtrim(x, tol=1e-6) class TestConstants: def test_legdomain(self): assert_equal(leg.legdomain, [-1, 1]) def test_legzero(self): assert_equal(leg.legzero, [0]) def test_legone(self): assert_equal(leg.legone, [1]) def test_legx(self): assert_equal(leg.legx, [0, 1]) class TestArithmetic: x = np.linspace(-1, 1, 100) def test_legadd(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 = leg.legadd([0]*i + [1], [0]*j + [1]) assert_equal(trim(res), trim(tgt), err_msg=msg) def test_legsub(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 = leg.legsub([0]*i + [1], [0]*j + [1]) assert_equal(trim(res), trim(tgt), err_msg=msg) def test_legmulx(self): assert_equal(leg.legmulx([0]), [0]) assert_equal(leg.legmulx([1]), [0, 1]) for i in range(1, 5): tmp = 2*i + 1 ser = [0]*i + [1] tgt = [0]*(i - 1) + [i/tmp, 0, (i + 1)/tmp] assert_equal(leg.legmulx(ser), tgt) def test_legmul(self): # check values of result for i in range(5): pol1 = [0]*i + [1] val1 = leg.legval(self.x, pol1) for j in range(5): msg = f"At i={i}, j={j}" pol2 = [0]*j + [1] val2 = leg.legval(self.x, pol2) pol3 = leg.legmul(pol1, pol2) val3 = leg.legval(self.x, pol3) assert_(len(pol3) == i + j + 1, msg) assert_almost_equal(val3, val1*val2, err_msg=msg) def test_legdiv(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 = leg.legadd(ci, cj) quo, rem = leg.legdiv(tgt, ci) res = leg.legadd(leg.legmul(quo, ci), rem) assert_equal(trim(res), trim(tgt), err_msg=msg) def test_legpow(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(leg.legmul, [c]*j, np.array([1])) res = leg.legpow(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., 2., 2.]) 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_legval(self): #check empty input assert_equal(leg.legval([], [1]).size, 0) #check normal input) x = np.linspace(-1, 1) y = [polyval(x, c) for c in Llist] for i in range(10): msg = f"At i={i}" tgt = y[i] res = leg.legval(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(leg.legval(x, [1]).shape, dims) assert_equal(leg.legval(x, [1, 0]).shape, dims) assert_equal(leg.legval(x, [1, 0, 0]).shape, dims) def test_legval2d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test exceptions assert_raises(ValueError, leg.legval2d, x1, x2[:2], self.c2d) #test values tgt = y1*y2 res = leg.legval2d(x1, x2, self.c2d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = leg.legval2d(z, z, self.c2d) assert_(res.shape == (2, 3)) def test_legval3d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test exceptions assert_raises(ValueError, leg.legval3d, x1, x2, x3[:2], self.c3d) #test values tgt = y1*y2*y3 res = leg.legval3d(x1, x2, x3, self.c3d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = leg.legval3d(z, z, z, self.c3d) assert_(res.shape == (2, 3)) def test_leggrid2d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test values tgt = np.einsum('i,j->ij', y1, y2) res = leg.leggrid2d(x1, x2, self.c2d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = leg.leggrid2d(z, z, self.c2d) assert_(res.shape == (2, 3)*2) def test_leggrid3d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test values tgt = np.einsum('i,j,k->ijk', y1, y2, y3) res = leg.leggrid3d(x1, x2, x3, self.c3d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = leg.leggrid3d(z, z, z, self.c3d) assert_(res.shape == (2, 3)*3) class TestIntegral: def test_legint(self): # check exceptions assert_raises(TypeError, leg.legint, [0], .5) assert_raises(ValueError, leg.legint, [0], -1) assert_raises(ValueError, leg.legint, [0], 1, [0, 0]) assert_raises(ValueError, leg.legint, [0], lbnd=[0]) assert_raises(ValueError, leg.legint, [0], scl=[0]) assert_raises(TypeError, leg.legint, [0], axis=.5) # test integration of zero polynomial for i in range(2, 5): k = [0]*(i - 2) + [1] res = leg.legint([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] legpol = leg.poly2leg(pol) legint = leg.legint(legpol, m=1, k=[i]) res = leg.leg2poly(legint) 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] legpol = leg.poly2leg(pol) legint = leg.legint(legpol, m=1, k=[i], lbnd=-1) assert_almost_equal(leg.legval(-1, legint), 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] legpol = leg.poly2leg(pol) legint = leg.legint(legpol, m=1, k=[i], scl=2) res = leg.leg2poly(legint) 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 = leg.legint(tgt, m=1) res = leg.legint(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 = leg.legint(tgt, m=1, k=[k]) res = leg.legint(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 = leg.legint(tgt, m=1, k=[k], lbnd=-1) res = leg.legint(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 = leg.legint(tgt, m=1, k=[k], scl=2) res = leg.legint(pol, m=j, k=list(range(j)), scl=2) assert_almost_equal(trim(res), trim(tgt)) def test_legint_axis(self): # check that axis keyword works c2d = np.random.random((3, 4)) tgt = np.vstack([leg.legint(c) for c in c2d.T]).T res = leg.legint(c2d, axis=0) assert_almost_equal(res, tgt) tgt = np.vstack([leg.legint(c) for c in c2d]) res = leg.legint(c2d, axis=1) assert_almost_equal(res, tgt) tgt = np.vstack([leg.legint(c, k=3) for c in c2d]) res = leg.legint(c2d, k=3, axis=1) assert_almost_equal(res, tgt) def test_legint_zerointord(self): assert_equal(leg.legint((1, 2, 3), 0), (1, 2, 3)) class TestDerivative: def test_legder(self): # check exceptions assert_raises(TypeError, leg.legder, [0], .5) assert_raises(ValueError, leg.legder, [0], -1) # check that zeroth derivative does nothing for i in range(5): tgt = [0]*i + [1] res = leg.legder(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 = leg.legder(leg.legint(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 = leg.legder(leg.legint(tgt, m=j, scl=2), m=j, scl=.5) assert_almost_equal(trim(res), trim(tgt)) def test_legder_axis(self): # check that axis keyword works c2d = np.random.random((3, 4)) tgt = np.vstack([leg.legder(c) for c in c2d.T]).T res = leg.legder(c2d, axis=0) assert_almost_equal(res, tgt) tgt = np.vstack([leg.legder(c) for c in c2d]) res = leg.legder(c2d, axis=1) assert_almost_equal(res, tgt) def test_legder_orderhigherthancoeff(self): c = (1, 2, 3, 4) assert_equal(leg.legder(c, 4), [0]) class TestVander: # some random values in [-1, 1) x = np.random.random((3, 5))*2 - 1 def test_legvander(self): # check for 1d x x = np.arange(3) v = leg.legvander(x, 3) assert_(v.shape == (3, 4)) for i in range(4): coef = [0]*i + [1] assert_almost_equal(v[..., i], leg.legval(x, coef)) # check for 2d x x = np.array([[1, 2], [3, 4], [5, 6]]) v = leg.legvander(x, 3) assert_(v.shape == (3, 2, 4)) for i in range(4): coef = [0]*i + [1] assert_almost_equal(v[..., i], leg.legval(x, coef)) def test_legvander2d(self): # also tests polyval2d for non-square coefficient array x1, x2, x3 = self.x c = np.random.random((2, 3)) van = leg.legvander2d(x1, x2, [1, 2]) tgt = leg.legval2d(x1, x2, c) res = np.dot(van, c.flat) assert_almost_equal(res, tgt) # check shape van = leg.legvander2d([x1], [x2], [1, 2]) assert_(van.shape == (1, 5, 6)) def test_legvander3d(self): # also tests polyval3d for non-square coefficient array x1, x2, x3 = self.x c = np.random.random((2, 3, 4)) van = leg.legvander3d(x1, x2, x3, [1, 2, 3]) tgt = leg.legval3d(x1, x2, x3, c) res = np.dot(van, c.flat) assert_almost_equal(res, tgt) # check shape van = leg.legvander3d([x1], [x2], [x3], [1, 2, 3]) assert_(van.shape == (1, 5, 24)) def test_legvander_negdeg(self): assert_raises(ValueError, leg.legvander, (1, 2, 3), -1) class TestFitting: def test_legfit(self): def f(x): return x*(x - 1)*(x - 2) def f2(x): return x**4 + x**2 + 1 # Test exceptions assert_raises(ValueError, leg.legfit, [1], [1], -1) assert_raises(TypeError, leg.legfit, [[1]], [1], 0) assert_raises(TypeError, leg.legfit, [], [1], 0) assert_raises(TypeError, leg.legfit, [1], [[[1]]], 0) assert_raises(TypeError, leg.legfit, [1, 2], [1], 0) assert_raises(TypeError, leg.legfit, [1], [1, 2], 0) assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[[1]]) assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[1, 1]) assert_raises(ValueError, leg.legfit, [1], [1], [-1,]) assert_raises(ValueError, leg.legfit, [1], [1], [2, -1, 6]) assert_raises(TypeError, leg.legfit, [1], [1], []) # Test fit x = np.linspace(0, 2) y = f(x) # coef3 = leg.legfit(x, y, 3) assert_equal(len(coef3), 4) assert_almost_equal(leg.legval(x, coef3), y) coef3 = leg.legfit(x, y, [0, 1, 2, 3]) assert_equal(len(coef3), 4) assert_almost_equal(leg.legval(x, coef3), y) # coef4 = leg.legfit(x, y, 4) assert_equal(len(coef4), 5) assert_almost_equal(leg.legval(x, coef4), y) coef4 = leg.legfit(x, y, [0, 1, 2, 3, 4]) assert_equal(len(coef4), 5) assert_almost_equal(leg.legval(x, coef4), y) # check things still work if deg is not in strict increasing coef4 = leg.legfit(x, y, [2, 3, 4, 1, 0]) assert_equal(len(coef4), 5) assert_almost_equal(leg.legval(x, coef4), y) # coef2d = leg.legfit(x, np.array([y, y]).T, 3) assert_almost_equal(coef2d, np.array([coef3, coef3]).T) coef2d = leg.legfit(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 = leg.legfit(x, yw, 3, w=w) assert_almost_equal(wcoef3, coef3) wcoef3 = leg.legfit(x, yw, [0, 1, 2, 3], w=w) assert_almost_equal(wcoef3, coef3) # wcoef2d = leg.legfit(x, np.array([yw, yw]).T, 3, w=w) assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) wcoef2d = leg.legfit(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(leg.legfit(x, x, 1), [0, 1]) assert_almost_equal(leg.legfit(x, x, [0, 1]), [0, 1]) # test fitting only even Legendre polynomials x = np.linspace(-1, 1) y = f2(x) coef1 = leg.legfit(x, y, 4) assert_almost_equal(leg.legval(x, coef1), y) coef2 = leg.legfit(x, y, [0, 2, 4]) assert_almost_equal(leg.legval(x, coef2), y) assert_almost_equal(coef1, coef2) class TestCompanion: def test_raises(self): assert_raises(ValueError, leg.legcompanion, []) assert_raises(ValueError, leg.legcompanion, [1]) def test_dimensions(self): for i in range(1, 5): coef = [0]*i + [1] assert_(leg.legcompanion(coef).shape == (i, i)) def test_linear_root(self): assert_(leg.legcompanion([1, 2])[0, 0] == -.5) class TestGauss: def test_100(self): x, w = leg.leggauss(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 = leg.legvander(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 = 2.0 assert_almost_equal(w.sum(), tgt) class TestMisc: def test_legfromroots(self): res = leg.legfromroots([]) 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]) pol = leg.legfromroots(roots) res = leg.legval(roots, pol) tgt = 0 assert_(len(pol) == i + 1) assert_almost_equal(leg.leg2poly(pol)[-1], 1) assert_almost_equal(res, tgt) def test_legroots(self): assert_almost_equal(leg.legroots([1]), []) assert_almost_equal(leg.legroots([1, 2]), [-.5]) for i in range(2, 5): tgt = np.linspace(-1, 1, i) res = leg.legroots(leg.legfromroots(tgt)) assert_almost_equal(trim(res), trim(tgt)) def test_legtrim(self): coef = [2, -1, 1, 0] # Test exceptions assert_raises(ValueError, leg.legtrim, coef, -1) # Test results assert_equal(leg.legtrim(coef), coef[:-1]) assert_equal(leg.legtrim(coef, 1), coef[:-3]) assert_equal(leg.legtrim(coef, 2), [0]) def test_legline(self): assert_equal(leg.legline(3, 4), [3, 4]) def test_legline_zeroscl(self): assert_equal(leg.legline(3, 0), [3]) def test_leg2poly(self): for i in range(10): assert_almost_equal(leg.leg2poly([0]*i + [1]), Llist[i]) def test_poly2leg(self): for i in range(10): assert_almost_equal(leg.poly2leg(Llist[i]), [0]*i + [1]) def test_weight(self): x = np.linspace(-1, 1, 11) tgt = 1. res = leg.legweight(x) assert_almost_equal(res, tgt)