Chris@87: """Tests for legendre module. Chris@87: Chris@87: """ Chris@87: from __future__ import division, absolute_import, print_function Chris@87: Chris@87: import numpy as np Chris@87: import numpy.polynomial.legendre as leg Chris@87: from numpy.polynomial.polynomial import polyval Chris@87: from numpy.testing import ( Chris@87: TestCase, assert_almost_equal, assert_raises, Chris@87: assert_equal, assert_, run_module_suite) Chris@87: Chris@87: L0 = np.array([1]) Chris@87: L1 = np.array([0, 1]) Chris@87: L2 = np.array([-1, 0, 3])/2 Chris@87: L3 = np.array([0, -3, 0, 5])/2 Chris@87: L4 = np.array([3, 0, -30, 0, 35])/8 Chris@87: L5 = np.array([0, 15, 0, -70, 0, 63])/8 Chris@87: L6 = np.array([-5, 0, 105, 0, -315, 0, 231])/16 Chris@87: L7 = np.array([0, -35, 0, 315, 0, -693, 0, 429])/16 Chris@87: L8 = np.array([35, 0, -1260, 0, 6930, 0, -12012, 0, 6435])/128 Chris@87: L9 = np.array([0, 315, 0, -4620, 0, 18018, 0, -25740, 0, 12155])/128 Chris@87: Chris@87: Llist = [L0, L1, L2, L3, L4, L5, L6, L7, L8, L9] Chris@87: Chris@87: Chris@87: def trim(x): Chris@87: return leg.legtrim(x, tol=1e-6) Chris@87: Chris@87: Chris@87: class TestConstants(TestCase): Chris@87: Chris@87: def test_legdomain(self): Chris@87: assert_equal(leg.legdomain, [-1, 1]) Chris@87: Chris@87: def test_legzero(self): Chris@87: assert_equal(leg.legzero, [0]) Chris@87: Chris@87: def test_legone(self): Chris@87: assert_equal(leg.legone, [1]) Chris@87: Chris@87: def test_legx(self): Chris@87: assert_equal(leg.legx, [0, 1]) Chris@87: Chris@87: Chris@87: class TestArithmetic(TestCase): Chris@87: x = np.linspace(-1, 1, 100) Chris@87: Chris@87: def test_legadd(self): Chris@87: for i in range(5): Chris@87: for j in range(5): Chris@87: msg = "At i=%d, j=%d" % (i, j) Chris@87: tgt = np.zeros(max(i, j) + 1) Chris@87: tgt[i] += 1 Chris@87: tgt[j] += 1 Chris@87: res = leg.legadd([0]*i + [1], [0]*j + [1]) Chris@87: assert_equal(trim(res), trim(tgt), err_msg=msg) Chris@87: Chris@87: def test_legsub(self): Chris@87: for i in range(5): Chris@87: for j in range(5): Chris@87: msg = "At i=%d, j=%d" % (i, j) Chris@87: tgt = np.zeros(max(i, j) + 1) Chris@87: tgt[i] += 1 Chris@87: tgt[j] -= 1 Chris@87: res = leg.legsub([0]*i + [1], [0]*j + [1]) Chris@87: assert_equal(trim(res), trim(tgt), err_msg=msg) Chris@87: Chris@87: def test_legmulx(self): Chris@87: assert_equal(leg.legmulx([0]), [0]) Chris@87: assert_equal(leg.legmulx([1]), [0, 1]) Chris@87: for i in range(1, 5): Chris@87: tmp = 2*i + 1 Chris@87: ser = [0]*i + [1] Chris@87: tgt = [0]*(i - 1) + [i/tmp, 0, (i + 1)/tmp] Chris@87: assert_equal(leg.legmulx(ser), tgt) Chris@87: Chris@87: def test_legmul(self): Chris@87: # check values of result Chris@87: for i in range(5): Chris@87: pol1 = [0]*i + [1] Chris@87: val1 = leg.legval(self.x, pol1) Chris@87: for j in range(5): Chris@87: msg = "At i=%d, j=%d" % (i, j) Chris@87: pol2 = [0]*j + [1] Chris@87: val2 = leg.legval(self.x, pol2) Chris@87: pol3 = leg.legmul(pol1, pol2) Chris@87: val3 = leg.legval(self.x, pol3) Chris@87: assert_(len(pol3) == i + j + 1, msg) Chris@87: assert_almost_equal(val3, val1*val2, err_msg=msg) Chris@87: Chris@87: def test_legdiv(self): Chris@87: for i in range(5): Chris@87: for j in range(5): Chris@87: msg = "At i=%d, j=%d" % (i, j) Chris@87: ci = [0]*i + [1] Chris@87: cj = [0]*j + [1] Chris@87: tgt = leg.legadd(ci, cj) Chris@87: quo, rem = leg.legdiv(tgt, ci) Chris@87: res = leg.legadd(leg.legmul(quo, ci), rem) Chris@87: assert_equal(trim(res), trim(tgt), err_msg=msg) Chris@87: Chris@87: Chris@87: class TestEvaluation(TestCase): Chris@87: # coefficients of 1 + 2*x + 3*x**2 Chris@87: c1d = np.array([2., 2., 2.]) Chris@87: c2d = np.einsum('i,j->ij', c1d, c1d) Chris@87: c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d) Chris@87: Chris@87: # some random values in [-1, 1) Chris@87: x = np.random.random((3, 5))*2 - 1 Chris@87: y = polyval(x, [1., 2., 3.]) Chris@87: Chris@87: def test_legval(self): Chris@87: #check empty input Chris@87: assert_equal(leg.legval([], [1]).size, 0) Chris@87: Chris@87: #check normal input) Chris@87: x = np.linspace(-1, 1) Chris@87: y = [polyval(x, c) for c in Llist] Chris@87: for i in range(10): Chris@87: msg = "At i=%d" % i Chris@87: tgt = y[i] Chris@87: res = leg.legval(x, [0]*i + [1]) Chris@87: assert_almost_equal(res, tgt, err_msg=msg) Chris@87: Chris@87: #check that shape is preserved Chris@87: for i in range(3): Chris@87: dims = [2]*i Chris@87: x = np.zeros(dims) Chris@87: assert_equal(leg.legval(x, [1]).shape, dims) Chris@87: assert_equal(leg.legval(x, [1, 0]).shape, dims) Chris@87: assert_equal(leg.legval(x, [1, 0, 0]).shape, dims) Chris@87: Chris@87: def test_legval2d(self): Chris@87: x1, x2, x3 = self.x Chris@87: y1, y2, y3 = self.y Chris@87: Chris@87: #test exceptions Chris@87: assert_raises(ValueError, leg.legval2d, x1, x2[:2], self.c2d) Chris@87: Chris@87: #test values Chris@87: tgt = y1*y2 Chris@87: res = leg.legval2d(x1, x2, self.c2d) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: #test shape Chris@87: z = np.ones((2, 3)) Chris@87: res = leg.legval2d(z, z, self.c2d) Chris@87: assert_(res.shape == (2, 3)) Chris@87: Chris@87: def test_legval3d(self): Chris@87: x1, x2, x3 = self.x Chris@87: y1, y2, y3 = self.y Chris@87: Chris@87: #test exceptions Chris@87: assert_raises(ValueError, leg.legval3d, x1, x2, x3[:2], self.c3d) Chris@87: Chris@87: #test values Chris@87: tgt = y1*y2*y3 Chris@87: res = leg.legval3d(x1, x2, x3, self.c3d) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: #test shape Chris@87: z = np.ones((2, 3)) Chris@87: res = leg.legval3d(z, z, z, self.c3d) Chris@87: assert_(res.shape == (2, 3)) Chris@87: Chris@87: def test_leggrid2d(self): Chris@87: x1, x2, x3 = self.x Chris@87: y1, y2, y3 = self.y Chris@87: Chris@87: #test values Chris@87: tgt = np.einsum('i,j->ij', y1, y2) Chris@87: res = leg.leggrid2d(x1, x2, self.c2d) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: #test shape Chris@87: z = np.ones((2, 3)) Chris@87: res = leg.leggrid2d(z, z, self.c2d) Chris@87: assert_(res.shape == (2, 3)*2) Chris@87: Chris@87: def test_leggrid3d(self): Chris@87: x1, x2, x3 = self.x Chris@87: y1, y2, y3 = self.y Chris@87: Chris@87: #test values Chris@87: tgt = np.einsum('i,j,k->ijk', y1, y2, y3) Chris@87: res = leg.leggrid3d(x1, x2, x3, self.c3d) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: #test shape Chris@87: z = np.ones((2, 3)) Chris@87: res = leg.leggrid3d(z, z, z, self.c3d) Chris@87: assert_(res.shape == (2, 3)*3) Chris@87: Chris@87: Chris@87: class TestIntegral(TestCase): Chris@87: Chris@87: def test_legint(self): Chris@87: # check exceptions Chris@87: assert_raises(ValueError, leg.legint, [0], .5) Chris@87: assert_raises(ValueError, leg.legint, [0], -1) Chris@87: assert_raises(ValueError, leg.legint, [0], 1, [0, 0]) Chris@87: Chris@87: # test integration of zero polynomial Chris@87: for i in range(2, 5): Chris@87: k = [0]*(i - 2) + [1] Chris@87: res = leg.legint([0], m=i, k=k) Chris@87: assert_almost_equal(res, [0, 1]) Chris@87: Chris@87: # check single integration with integration constant Chris@87: for i in range(5): Chris@87: scl = i + 1 Chris@87: pol = [0]*i + [1] Chris@87: tgt = [i] + [0]*i + [1/scl] Chris@87: legpol = leg.poly2leg(pol) Chris@87: legint = leg.legint(legpol, m=1, k=[i]) Chris@87: res = leg.leg2poly(legint) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: # check single integration with integration constant and lbnd Chris@87: for i in range(5): Chris@87: scl = i + 1 Chris@87: pol = [0]*i + [1] Chris@87: legpol = leg.poly2leg(pol) Chris@87: legint = leg.legint(legpol, m=1, k=[i], lbnd=-1) Chris@87: assert_almost_equal(leg.legval(-1, legint), i) Chris@87: Chris@87: # check single integration with integration constant and scaling Chris@87: for i in range(5): Chris@87: scl = i + 1 Chris@87: pol = [0]*i + [1] Chris@87: tgt = [i] + [0]*i + [2/scl] Chris@87: legpol = leg.poly2leg(pol) Chris@87: legint = leg.legint(legpol, m=1, k=[i], scl=2) Chris@87: res = leg.leg2poly(legint) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: # check multiple integrations with default k Chris@87: for i in range(5): Chris@87: for j in range(2, 5): Chris@87: pol = [0]*i + [1] Chris@87: tgt = pol[:] Chris@87: for k in range(j): Chris@87: tgt = leg.legint(tgt, m=1) Chris@87: res = leg.legint(pol, m=j) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: # check multiple integrations with defined k Chris@87: for i in range(5): Chris@87: for j in range(2, 5): Chris@87: pol = [0]*i + [1] Chris@87: tgt = pol[:] Chris@87: for k in range(j): Chris@87: tgt = leg.legint(tgt, m=1, k=[k]) Chris@87: res = leg.legint(pol, m=j, k=list(range(j))) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: # check multiple integrations with lbnd Chris@87: for i in range(5): Chris@87: for j in range(2, 5): Chris@87: pol = [0]*i + [1] Chris@87: tgt = pol[:] Chris@87: for k in range(j): Chris@87: tgt = leg.legint(tgt, m=1, k=[k], lbnd=-1) Chris@87: res = leg.legint(pol, m=j, k=list(range(j)), lbnd=-1) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: # check multiple integrations with scaling Chris@87: for i in range(5): Chris@87: for j in range(2, 5): Chris@87: pol = [0]*i + [1] Chris@87: tgt = pol[:] Chris@87: for k in range(j): Chris@87: tgt = leg.legint(tgt, m=1, k=[k], scl=2) Chris@87: res = leg.legint(pol, m=j, k=list(range(j)), scl=2) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: def test_legint_axis(self): Chris@87: # check that axis keyword works Chris@87: c2d = np.random.random((3, 4)) Chris@87: Chris@87: tgt = np.vstack([leg.legint(c) for c in c2d.T]).T Chris@87: res = leg.legint(c2d, axis=0) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: tgt = np.vstack([leg.legint(c) for c in c2d]) Chris@87: res = leg.legint(c2d, axis=1) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: tgt = np.vstack([leg.legint(c, k=3) for c in c2d]) Chris@87: res = leg.legint(c2d, k=3, axis=1) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: Chris@87: class TestDerivative(TestCase): Chris@87: Chris@87: def test_legder(self): Chris@87: # check exceptions Chris@87: assert_raises(ValueError, leg.legder, [0], .5) Chris@87: assert_raises(ValueError, leg.legder, [0], -1) Chris@87: Chris@87: # check that zeroth deriviative does nothing Chris@87: for i in range(5): Chris@87: tgt = [0]*i + [1] Chris@87: res = leg.legder(tgt, m=0) Chris@87: assert_equal(trim(res), trim(tgt)) Chris@87: Chris@87: # check that derivation is the inverse of integration Chris@87: for i in range(5): Chris@87: for j in range(2, 5): Chris@87: tgt = [0]*i + [1] Chris@87: res = leg.legder(leg.legint(tgt, m=j), m=j) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: # check derivation with scaling Chris@87: for i in range(5): Chris@87: for j in range(2, 5): Chris@87: tgt = [0]*i + [1] Chris@87: res = leg.legder(leg.legint(tgt, m=j, scl=2), m=j, scl=.5) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: def test_legder_axis(self): Chris@87: # check that axis keyword works Chris@87: c2d = np.random.random((3, 4)) Chris@87: Chris@87: tgt = np.vstack([leg.legder(c) for c in c2d.T]).T Chris@87: res = leg.legder(c2d, axis=0) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: tgt = np.vstack([leg.legder(c) for c in c2d]) Chris@87: res = leg.legder(c2d, axis=1) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: Chris@87: class TestVander(TestCase): Chris@87: # some random values in [-1, 1) Chris@87: x = np.random.random((3, 5))*2 - 1 Chris@87: Chris@87: def test_legvander(self): Chris@87: # check for 1d x Chris@87: x = np.arange(3) Chris@87: v = leg.legvander(x, 3) Chris@87: assert_(v.shape == (3, 4)) Chris@87: for i in range(4): Chris@87: coef = [0]*i + [1] Chris@87: assert_almost_equal(v[..., i], leg.legval(x, coef)) Chris@87: Chris@87: # check for 2d x Chris@87: x = np.array([[1, 2], [3, 4], [5, 6]]) Chris@87: v = leg.legvander(x, 3) Chris@87: assert_(v.shape == (3, 2, 4)) Chris@87: for i in range(4): Chris@87: coef = [0]*i + [1] Chris@87: assert_almost_equal(v[..., i], leg.legval(x, coef)) Chris@87: Chris@87: def test_legvander2d(self): Chris@87: # also tests polyval2d for non-square coefficient array Chris@87: x1, x2, x3 = self.x Chris@87: c = np.random.random((2, 3)) Chris@87: van = leg.legvander2d(x1, x2, [1, 2]) Chris@87: tgt = leg.legval2d(x1, x2, c) Chris@87: res = np.dot(van, c.flat) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: # check shape Chris@87: van = leg.legvander2d([x1], [x2], [1, 2]) Chris@87: assert_(van.shape == (1, 5, 6)) Chris@87: Chris@87: def test_legvander3d(self): Chris@87: # also tests polyval3d for non-square coefficient array Chris@87: x1, x2, x3 = self.x Chris@87: c = np.random.random((2, 3, 4)) Chris@87: van = leg.legvander3d(x1, x2, x3, [1, 2, 3]) Chris@87: tgt = leg.legval3d(x1, x2, x3, c) Chris@87: res = np.dot(van, c.flat) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: # check shape Chris@87: van = leg.legvander3d([x1], [x2], [x3], [1, 2, 3]) Chris@87: assert_(van.shape == (1, 5, 24)) Chris@87: Chris@87: Chris@87: class TestFitting(TestCase): Chris@87: Chris@87: def test_legfit(self): Chris@87: def f(x): Chris@87: return x*(x - 1)*(x - 2) Chris@87: Chris@87: # Test exceptions Chris@87: assert_raises(ValueError, leg.legfit, [1], [1], -1) Chris@87: assert_raises(TypeError, leg.legfit, [[1]], [1], 0) Chris@87: assert_raises(TypeError, leg.legfit, [], [1], 0) Chris@87: assert_raises(TypeError, leg.legfit, [1], [[[1]]], 0) Chris@87: assert_raises(TypeError, leg.legfit, [1, 2], [1], 0) Chris@87: assert_raises(TypeError, leg.legfit, [1], [1, 2], 0) Chris@87: assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[[1]]) Chris@87: assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[1, 1]) Chris@87: Chris@87: # Test fit Chris@87: x = np.linspace(0, 2) Chris@87: y = f(x) Chris@87: # Chris@87: coef3 = leg.legfit(x, y, 3) Chris@87: assert_equal(len(coef3), 4) Chris@87: assert_almost_equal(leg.legval(x, coef3), y) Chris@87: # Chris@87: coef4 = leg.legfit(x, y, 4) Chris@87: assert_equal(len(coef4), 5) Chris@87: assert_almost_equal(leg.legval(x, coef4), y) Chris@87: # Chris@87: coef2d = leg.legfit(x, np.array([y, y]).T, 3) Chris@87: assert_almost_equal(coef2d, np.array([coef3, coef3]).T) Chris@87: # test weighting Chris@87: w = np.zeros_like(x) Chris@87: yw = y.copy() Chris@87: w[1::2] = 1 Chris@87: y[0::2] = 0 Chris@87: wcoef3 = leg.legfit(x, yw, 3, w=w) Chris@87: assert_almost_equal(wcoef3, coef3) Chris@87: # Chris@87: wcoef2d = leg.legfit(x, np.array([yw, yw]).T, 3, w=w) Chris@87: assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) Chris@87: # test scaling with complex values x points whose square Chris@87: # is zero when summed. Chris@87: x = [1, 1j, -1, -1j] Chris@87: assert_almost_equal(leg.legfit(x, x, 1), [0, 1]) Chris@87: Chris@87: Chris@87: class TestCompanion(TestCase): Chris@87: Chris@87: def test_raises(self): Chris@87: assert_raises(ValueError, leg.legcompanion, []) Chris@87: assert_raises(ValueError, leg.legcompanion, [1]) Chris@87: Chris@87: def test_dimensions(self): Chris@87: for i in range(1, 5): Chris@87: coef = [0]*i + [1] Chris@87: assert_(leg.legcompanion(coef).shape == (i, i)) Chris@87: Chris@87: def test_linear_root(self): Chris@87: assert_(leg.legcompanion([1, 2])[0, 0] == -.5) Chris@87: Chris@87: Chris@87: class TestGauss(TestCase): Chris@87: Chris@87: def test_100(self): Chris@87: x, w = leg.leggauss(100) Chris@87: Chris@87: # test orthogonality. Note that the results need to be normalized, Chris@87: # otherwise the huge values that can arise from fast growing Chris@87: # functions like Laguerre can be very confusing. Chris@87: v = leg.legvander(x, 99) Chris@87: vv = np.dot(v.T * w, v) Chris@87: vd = 1/np.sqrt(vv.diagonal()) Chris@87: vv = vd[:, None] * vv * vd Chris@87: assert_almost_equal(vv, np.eye(100)) Chris@87: Chris@87: # check that the integral of 1 is correct Chris@87: tgt = 2.0 Chris@87: assert_almost_equal(w.sum(), tgt) Chris@87: Chris@87: Chris@87: class TestMisc(TestCase): Chris@87: Chris@87: def test_legfromroots(self): Chris@87: res = leg.legfromroots([]) Chris@87: assert_almost_equal(trim(res), [1]) Chris@87: for i in range(1, 5): Chris@87: roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2]) Chris@87: pol = leg.legfromroots(roots) Chris@87: res = leg.legval(roots, pol) Chris@87: tgt = 0 Chris@87: assert_(len(pol) == i + 1) Chris@87: assert_almost_equal(leg.leg2poly(pol)[-1], 1) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: def test_legroots(self): Chris@87: assert_almost_equal(leg.legroots([1]), []) Chris@87: assert_almost_equal(leg.legroots([1, 2]), [-.5]) Chris@87: for i in range(2, 5): Chris@87: tgt = np.linspace(-1, 1, i) Chris@87: res = leg.legroots(leg.legfromroots(tgt)) Chris@87: assert_almost_equal(trim(res), trim(tgt)) Chris@87: Chris@87: def test_legtrim(self): Chris@87: coef = [2, -1, 1, 0] Chris@87: Chris@87: # Test exceptions Chris@87: assert_raises(ValueError, leg.legtrim, coef, -1) Chris@87: Chris@87: # Test results Chris@87: assert_equal(leg.legtrim(coef), coef[:-1]) Chris@87: assert_equal(leg.legtrim(coef, 1), coef[:-3]) Chris@87: assert_equal(leg.legtrim(coef, 2), [0]) Chris@87: Chris@87: def test_legline(self): Chris@87: assert_equal(leg.legline(3, 4), [3, 4]) Chris@87: Chris@87: def test_leg2poly(self): Chris@87: for i in range(10): Chris@87: assert_almost_equal(leg.leg2poly([0]*i + [1]), Llist[i]) Chris@87: Chris@87: def test_poly2leg(self): Chris@87: for i in range(10): Chris@87: assert_almost_equal(leg.poly2leg(Llist[i]), [0]*i + [1]) Chris@87: Chris@87: def test_weight(self): Chris@87: x = np.linspace(-1, 1, 11) Chris@87: tgt = 1. Chris@87: res = leg.legweight(x) Chris@87: assert_almost_equal(res, tgt) Chris@87: Chris@87: Chris@87: if __name__ == "__main__": Chris@87: run_module_suite()