name : test_laguerre.py """Tests for laguerre module. """ from functools import reduce import numpy as np import numpy.polynomial.laguerre as lag from numpy.polynomial.polynomial import polyval from numpy.testing import ( assert_almost_equal, assert_raises, assert_equal, assert_, ) L0 = np.array([1])/1 L1 = np.array([1, -1])/1 L2 = np.array([2, -4, 1])/2 L3 = np.array([6, -18, 9, -1])/6 L4 = np.array([24, -96, 72, -16, 1])/24 L5 = np.array([120, -600, 600, -200, 25, -1])/120 L6 = np.array([720, -4320, 5400, -2400, 450, -36, 1])/720 Llist = [L0, L1, L2, L3, L4, L5, L6] def trim(x): return lag.lagtrim(x, tol=1e-6) class TestConstants: def test_lagdomain(self): assert_equal(lag.lagdomain, [0, 1]) def test_lagzero(self): assert_equal(lag.lagzero, [0]) def test_lagone(self): assert_equal(lag.lagone, [1]) def test_lagx(self): assert_equal(lag.lagx, [1, -1]) class TestArithmetic: x = np.linspace(-3, 3, 100) def test_lagadd(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 = lag.lagadd([0]*i + [1], [0]*j + [1]) assert_equal(trim(res), trim(tgt), err_msg=msg) def test_lagsub(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 = lag.lagsub([0]*i + [1], [0]*j + [1]) assert_equal(trim(res), trim(tgt), err_msg=msg) def test_lagmulx(self): assert_equal(lag.lagmulx([0]), [0]) assert_equal(lag.lagmulx([1]), [1, -1]) for i in range(1, 5): ser = [0]*i + [1] tgt = [0]*(i - 1) + [-i, 2*i + 1, -(i + 1)] assert_almost_equal(lag.lagmulx(ser), tgt) def test_lagmul(self): # check values of result for i in range(5): pol1 = [0]*i + [1] val1 = lag.lagval(self.x, pol1) for j in range(5): msg = f"At i={i}, j={j}" pol2 = [0]*j + [1] val2 = lag.lagval(self.x, pol2) pol3 = lag.lagmul(pol1, pol2) val3 = lag.lagval(self.x, pol3) assert_(len(pol3) == i + j + 1, msg) assert_almost_equal(val3, val1*val2, err_msg=msg) def test_lagdiv(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 = lag.lagadd(ci, cj) quo, rem = lag.lagdiv(tgt, ci) res = lag.lagadd(lag.lagmul(quo, ci), rem) assert_almost_equal(trim(res), trim(tgt), err_msg=msg) def test_lagpow(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(lag.lagmul, [c]*j, np.array([1])) res = lag.lagpow(c, j) assert_equal(trim(res), trim(tgt), err_msg=msg) class TestEvaluation: # coefficients of 1 + 2*x + 3*x**2 c1d = np.array([9., -14., 6.]) 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_lagval(self): #check empty input assert_equal(lag.lagval([], [1]).size, 0) #check normal input) x = np.linspace(-1, 1) y = [polyval(x, c) for c in Llist] for i in range(7): msg = f"At i={i}" tgt = y[i] res = lag.lagval(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(lag.lagval(x, [1]).shape, dims) assert_equal(lag.lagval(x, [1, 0]).shape, dims) assert_equal(lag.lagval(x, [1, 0, 0]).shape, dims) def test_lagval2d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test exceptions assert_raises(ValueError, lag.lagval2d, x1, x2[:2], self.c2d) #test values tgt = y1*y2 res = lag.lagval2d(x1, x2, self.c2d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = lag.lagval2d(z, z, self.c2d) assert_(res.shape == (2, 3)) def test_lagval3d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test exceptions assert_raises(ValueError, lag.lagval3d, x1, x2, x3[:2], self.c3d) #test values tgt = y1*y2*y3 res = lag.lagval3d(x1, x2, x3, self.c3d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = lag.lagval3d(z, z, z, self.c3d) assert_(res.shape == (2, 3)) def test_laggrid2d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test values tgt = np.einsum('i,j->ij', y1, y2) res = lag.laggrid2d(x1, x2, self.c2d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = lag.laggrid2d(z, z, self.c2d) assert_(res.shape == (2, 3)*2) def test_laggrid3d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test values tgt = np.einsum('i,j,k->ijk', y1, y2, y3) res = lag.laggrid3d(x1, x2, x3, self.c3d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = lag.laggrid3d(z, z, z, self.c3d) assert_(res.shape == (2, 3)*3) class TestIntegral: def test_lagint(self): # check exceptions assert_raises(TypeError, lag.lagint, [0], .5) assert_raises(ValueError, lag.lagint, [0], -1) assert_raises(ValueError, lag.lagint, [0], 1, [0, 0]) assert_raises(ValueError, lag.lagint, [0], lbnd=[0]) assert_raises(ValueError, lag.lagint, [0], scl=[0]) assert_raises(TypeError, lag.lagint, [0], axis=.5) # test integration of zero polynomial for i in range(2, 5): k = [0]*(i - 2) + [1] res = lag.lagint([0], m=i, k=k) assert_almost_equal(res, [1, -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] lagpol = lag.poly2lag(pol) lagint = lag.lagint(lagpol, m=1, k=[i]) res = lag.lag2poly(lagint) 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] lagpol = lag.poly2lag(pol) lagint = lag.lagint(lagpol, m=1, k=[i], lbnd=-1) assert_almost_equal(lag.lagval(-1, lagint), 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] lagpol = lag.poly2lag(pol) lagint = lag.lagint(lagpol, m=1, k=[i], scl=2) res = lag.lag2poly(lagint) 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 = lag.lagint(tgt, m=1) res = lag.lagint(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 = lag.lagint(tgt, m=1, k=[k]) res = lag.lagint(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 = lag.lagint(tgt, m=1, k=[k], lbnd=-1) res = lag.lagint(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 = lag.lagint(tgt, m=1, k=[k], scl=2) res = lag.lagint(pol, m=j, k=list(range(j)), scl=2) assert_almost_equal(trim(res), trim(tgt)) def test_lagint_axis(self): # check that axis keyword works c2d = np.random.random((3, 4)) tgt = np.vstack([lag.lagint(c) for c in c2d.T]).T res = lag.lagint(c2d, axis=0) assert_almost_equal(res, tgt) tgt = np.vstack([lag.lagint(c) for c in c2d]) res = lag.lagint(c2d, axis=1) assert_almost_equal(res, tgt) tgt = np.vstack([lag.lagint(c, k=3) for c in c2d]) res = lag.lagint(c2d, k=3, axis=1) assert_almost_equal(res, tgt) class TestDerivative: def test_lagder(self): # check exceptions assert_raises(TypeError, lag.lagder, [0], .5) assert_raises(ValueError, lag.lagder, [0], -1) # check that zeroth derivative does nothing for i in range(5): tgt = [0]*i + [1] res = lag.lagder(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 = lag.lagder(lag.lagint(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 = lag.lagder(lag.lagint(tgt, m=j, scl=2), m=j, scl=.5) assert_almost_equal(trim(res), trim(tgt)) def test_lagder_axis(self): # check that axis keyword works c2d = np.random.random((3, 4)) tgt = np.vstack([lag.lagder(c) for c in c2d.T]).T res = lag.lagder(c2d, axis=0) assert_almost_equal(res, tgt) tgt = np.vstack([lag.lagder(c) for c in c2d]) res = lag.lagder(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_lagvander(self): # check for 1d x x = np.arange(3) v = lag.lagvander(x, 3) assert_(v.shape == (3, 4)) for i in range(4): coef = [0]*i + [1] assert_almost_equal(v[..., i], lag.lagval(x, coef)) # check for 2d x x = np.array([[1, 2], [3, 4], [5, 6]]) v = lag.lagvander(x, 3) assert_(v.shape == (3, 2, 4)) for i in range(4): coef = [0]*i + [1] assert_almost_equal(v[..., i], lag.lagval(x, coef)) def test_lagvander2d(self): # also tests lagval2d for non-square coefficient array x1, x2, x3 = self.x c = np.random.random((2, 3)) van = lag.lagvander2d(x1, x2, [1, 2]) tgt = lag.lagval2d(x1, x2, c) res = np.dot(van, c.flat) assert_almost_equal(res, tgt) # check shape van = lag.lagvander2d([x1], [x2], [1, 2]) assert_(van.shape == (1, 5, 6)) def test_lagvander3d(self): # also tests lagval3d for non-square coefficient array x1, x2, x3 = self.x c = np.random.random((2, 3, 4)) van = lag.lagvander3d(x1, x2, x3, [1, 2, 3]) tgt = lag.lagval3d(x1, x2, x3, c) res = np.dot(van, c.flat) assert_almost_equal(res, tgt) # check shape van = lag.lagvander3d([x1], [x2], [x3], [1, 2, 3]) assert_(van.shape == (1, 5, 24)) class TestFitting: def test_lagfit(self): def f(x): return x*(x - 1)*(x - 2) # Test exceptions assert_raises(ValueError, lag.lagfit, [1], [1], -1) assert_raises(TypeError, lag.lagfit, [[1]], [1], 0) assert_raises(TypeError, lag.lagfit, [], [1], 0) assert_raises(TypeError, lag.lagfit, [1], [[[1]]], 0) assert_raises(TypeError, lag.lagfit, [1, 2], [1], 0) assert_raises(TypeError, lag.lagfit, [1], [1, 2], 0) assert_raises(TypeError, lag.lagfit, [1], [1], 0, w=[[1]]) assert_raises(TypeError, lag.lagfit, [1], [1], 0, w=[1, 1]) assert_raises(ValueError, lag.lagfit, [1], [1], [-1,]) assert_raises(ValueError, lag.lagfit, [1], [1], [2, -1, 6]) assert_raises(TypeError, lag.lagfit, [1], [1], []) # Test fit x = np.linspace(0, 2) y = f(x) # coef3 = lag.lagfit(x, y, 3) assert_equal(len(coef3), 4) assert_almost_equal(lag.lagval(x, coef3), y) coef3 = lag.lagfit(x, y, [0, 1, 2, 3]) assert_equal(len(coef3), 4) assert_almost_equal(lag.lagval(x, coef3), y) # coef4 = lag.lagfit(x, y, 4) assert_equal(len(coef4), 5) assert_almost_equal(lag.lagval(x, coef4), y) coef4 = lag.lagfit(x, y, [0, 1, 2, 3, 4]) assert_equal(len(coef4), 5) assert_almost_equal(lag.lagval(x, coef4), y) # coef2d = lag.lagfit(x, np.array([y, y]).T, 3) assert_almost_equal(coef2d, np.array([coef3, coef3]).T) coef2d = lag.lagfit(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 = lag.lagfit(x, yw, 3, w=w) assert_almost_equal(wcoef3, coef3) wcoef3 = lag.lagfit(x, yw, [0, 1, 2, 3], w=w) assert_almost_equal(wcoef3, coef3) # wcoef2d = lag.lagfit(x, np.array([yw, yw]).T, 3, w=w) assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) wcoef2d = lag.lagfit(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(lag.lagfit(x, x, 1), [1, -1]) assert_almost_equal(lag.lagfit(x, x, [0, 1]), [1, -1]) class TestCompanion: def test_raises(self): assert_raises(ValueError, lag.lagcompanion, []) assert_raises(ValueError, lag.lagcompanion, [1]) def test_dimensions(self): for i in range(1, 5): coef = [0]*i + [1] assert_(lag.lagcompanion(coef).shape == (i, i)) def test_linear_root(self): assert_(lag.lagcompanion([1, 2])[0, 0] == 1.5) class TestGauss: def test_100(self): x, w = lag.laggauss(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 = lag.lagvander(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 = 1.0 assert_almost_equal(w.sum(), tgt) class TestMisc: def test_lagfromroots(self): res = lag.lagfromroots([]) 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 = lag.lagfromroots(roots) res = lag.lagval(roots, pol) tgt = 0 assert_(len(pol) == i + 1) assert_almost_equal(lag.lag2poly(pol)[-1], 1) assert_almost_equal(res, tgt) def test_lagroots(self): assert_almost_equal(lag.lagroots([1]), []) assert_almost_equal(lag.lagroots([0, 1]), [1]) for i in range(2, 5): tgt = np.linspace(0, 3, i) res = lag.lagroots(lag.lagfromroots(tgt)) assert_almost_equal(trim(res), trim(tgt)) def test_lagtrim(self): coef = [2, -1, 1, 0] # Test exceptions assert_raises(ValueError, lag.lagtrim, coef, -1) # Test results assert_equal(lag.lagtrim(coef), coef[:-1]) assert_equal(lag.lagtrim(coef, 1), coef[:-3]) assert_equal(lag.lagtrim(coef, 2), [0]) def test_lagline(self): assert_equal(lag.lagline(3, 4), [7, -4]) def test_lag2poly(self): for i in range(7): assert_almost_equal(lag.lag2poly([0]*i + [1]), Llist[i]) def test_poly2lag(self): for i in range(7): assert_almost_equal(lag.poly2lag(Llist[i]), [0]*i + [1]) def test_weight(self): x = np.linspace(0, 10, 11) tgt = np.exp(-x) res = lag.lagweight(x) assert_almost_equal(res, tgt)