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| """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) | |