name : test_hermite_e.py """Tests for hermite_e module. """ from functools import reduce import numpy as np import numpy.polynomial.hermite_e as herme from numpy.polynomial.polynomial import polyval from numpy.testing import ( assert_almost_equal, assert_raises, assert_equal, assert_, ) He0 = np.array([1]) He1 = np.array([0, 1]) He2 = np.array([-1, 0, 1]) He3 = np.array([0, -3, 0, 1]) He4 = np.array([3, 0, -6, 0, 1]) He5 = np.array([0, 15, 0, -10, 0, 1]) He6 = np.array([-15, 0, 45, 0, -15, 0, 1]) He7 = np.array([0, -105, 0, 105, 0, -21, 0, 1]) He8 = np.array([105, 0, -420, 0, 210, 0, -28, 0, 1]) He9 = np.array([0, 945, 0, -1260, 0, 378, 0, -36, 0, 1]) Helist = [He0, He1, He2, He3, He4, He5, He6, He7, He8, He9] def trim(x): return herme.hermetrim(x, tol=1e-6) class TestConstants: def test_hermedomain(self): assert_equal(herme.hermedomain, [-1, 1]) def test_hermezero(self): assert_equal(herme.hermezero, [0]) def test_hermeone(self): assert_equal(herme.hermeone, [1]) def test_hermex(self): assert_equal(herme.hermex, [0, 1]) class TestArithmetic: x = np.linspace(-3, 3, 100) def test_hermeadd(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 = herme.hermeadd([0]*i + [1], [0]*j + [1]) assert_equal(trim(res), trim(tgt), err_msg=msg) def test_hermesub(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 = herme.hermesub([0]*i + [1], [0]*j + [1]) assert_equal(trim(res), trim(tgt), err_msg=msg) def test_hermemulx(self): assert_equal(herme.hermemulx([0]), [0]) assert_equal(herme.hermemulx([1]), [0, 1]) for i in range(1, 5): ser = [0]*i + [1] tgt = [0]*(i - 1) + [i, 0, 1] assert_equal(herme.hermemulx(ser), tgt) def test_hermemul(self): # check values of result for i in range(5): pol1 = [0]*i + [1] val1 = herme.hermeval(self.x, pol1) for j in range(5): msg = f"At i={i}, j={j}" pol2 = [0]*j + [1] val2 = herme.hermeval(self.x, pol2) pol3 = herme.hermemul(pol1, pol2) val3 = herme.hermeval(self.x, pol3) assert_(len(pol3) == i + j + 1, msg) assert_almost_equal(val3, val1*val2, err_msg=msg) def test_hermediv(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 = herme.hermeadd(ci, cj) quo, rem = herme.hermediv(tgt, ci) res = herme.hermeadd(herme.hermemul(quo, ci), rem) assert_equal(trim(res), trim(tgt), err_msg=msg) def test_hermepow(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(herme.hermemul, [c]*j, np.array([1])) res = herme.hermepow(c, j) assert_equal(trim(res), trim(tgt), err_msg=msg) class TestEvaluation: # coefficients of 1 + 2*x + 3*x**2 c1d = np.array([4., 2., 3.]) 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_hermeval(self): #check empty input assert_equal(herme.hermeval([], [1]).size, 0) #check normal input) x = np.linspace(-1, 1) y = [polyval(x, c) for c in Helist] for i in range(10): msg = f"At i={i}" tgt = y[i] res = herme.hermeval(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(herme.hermeval(x, [1]).shape, dims) assert_equal(herme.hermeval(x, [1, 0]).shape, dims) assert_equal(herme.hermeval(x, [1, 0, 0]).shape, dims) def test_hermeval2d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test exceptions assert_raises(ValueError, herme.hermeval2d, x1, x2[:2], self.c2d) #test values tgt = y1*y2 res = herme.hermeval2d(x1, x2, self.c2d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = herme.hermeval2d(z, z, self.c2d) assert_(res.shape == (2, 3)) def test_hermeval3d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test exceptions assert_raises(ValueError, herme.hermeval3d, x1, x2, x3[:2], self.c3d) #test values tgt = y1*y2*y3 res = herme.hermeval3d(x1, x2, x3, self.c3d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = herme.hermeval3d(z, z, z, self.c3d) assert_(res.shape == (2, 3)) def test_hermegrid2d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test values tgt = np.einsum('i,j->ij', y1, y2) res = herme.hermegrid2d(x1, x2, self.c2d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = herme.hermegrid2d(z, z, self.c2d) assert_(res.shape == (2, 3)*2) def test_hermegrid3d(self): x1, x2, x3 = self.x y1, y2, y3 = self.y #test values tgt = np.einsum('i,j,k->ijk', y1, y2, y3) res = herme.hermegrid3d(x1, x2, x3, self.c3d) assert_almost_equal(res, tgt) #test shape z = np.ones((2, 3)) res = herme.hermegrid3d(z, z, z, self.c3d) assert_(res.shape == (2, 3)*3) class TestIntegral: def test_hermeint(self): # check exceptions assert_raises(TypeError, herme.hermeint, [0], .5) assert_raises(ValueError, herme.hermeint, [0], -1) assert_raises(ValueError, herme.hermeint, [0], 1, [0, 0]) assert_raises(ValueError, herme.hermeint, [0], lbnd=[0]) assert_raises(ValueError, herme.hermeint, [0], scl=[0]) assert_raises(TypeError, herme.hermeint, [0], axis=.5) # test integration of zero polynomial for i in range(2, 5): k = [0]*(i - 2) + [1] res = herme.hermeint([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] hermepol = herme.poly2herme(pol) hermeint = herme.hermeint(hermepol, m=1, k=[i]) res = herme.herme2poly(hermeint) 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] hermepol = herme.poly2herme(pol) hermeint = herme.hermeint(hermepol, m=1, k=[i], lbnd=-1) assert_almost_equal(herme.hermeval(-1, hermeint), 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] hermepol = herme.poly2herme(pol) hermeint = herme.hermeint(hermepol, m=1, k=[i], scl=2) res = herme.herme2poly(hermeint) 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 = herme.hermeint(tgt, m=1) res = herme.hermeint(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 = herme.hermeint(tgt, m=1, k=[k]) res = herme.hermeint(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 = herme.hermeint(tgt, m=1, k=[k], lbnd=-1) res = herme.hermeint(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 = herme.hermeint(tgt, m=1, k=[k], scl=2) res = herme.hermeint(pol, m=j, k=list(range(j)), scl=2) assert_almost_equal(trim(res), trim(tgt)) def test_hermeint_axis(self): # check that axis keyword works c2d = np.random.random((3, 4)) tgt = np.vstack([herme.hermeint(c) for c in c2d.T]).T res = herme.hermeint(c2d, axis=0) assert_almost_equal(res, tgt) tgt = np.vstack([herme.hermeint(c) for c in c2d]) res = herme.hermeint(c2d, axis=1) assert_almost_equal(res, tgt) tgt = np.vstack([herme.hermeint(c, k=3) for c in c2d]) res = herme.hermeint(c2d, k=3, axis=1) assert_almost_equal(res, tgt) class TestDerivative: def test_hermeder(self): # check exceptions assert_raises(TypeError, herme.hermeder, [0], .5) assert_raises(ValueError, herme.hermeder, [0], -1) # check that zeroth derivative does nothing for i in range(5): tgt = [0]*i + [1] res = herme.hermeder(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 = herme.hermeder(herme.hermeint(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 = herme.hermeder( herme.hermeint(tgt, m=j, scl=2), m=j, scl=.5) assert_almost_equal(trim(res), trim(tgt)) def test_hermeder_axis(self): # check that axis keyword works c2d = np.random.random((3, 4)) tgt = np.vstack([herme.hermeder(c) for c in c2d.T]).T res = herme.hermeder(c2d, axis=0) assert_almost_equal(res, tgt) tgt = np.vstack([herme.hermeder(c) for c in c2d]) res = herme.hermeder(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_hermevander(self): # check for 1d x x = np.arange(3) v = herme.hermevander(x, 3) assert_(v.shape == (3, 4)) for i in range(4): coef = [0]*i + [1] assert_almost_equal(v[..., i], herme.hermeval(x, coef)) # check for 2d x x = np.array([[1, 2], [3, 4], [5, 6]]) v = herme.hermevander(x, 3) assert_(v.shape == (3, 2, 4)) for i in range(4): coef = [0]*i + [1] assert_almost_equal(v[..., i], herme.hermeval(x, coef)) def test_hermevander2d(self): # also tests hermeval2d for non-square coefficient array x1, x2, x3 = self.x c = np.random.random((2, 3)) van = herme.hermevander2d(x1, x2, [1, 2]) tgt = herme.hermeval2d(x1, x2, c) res = np.dot(van, c.flat) assert_almost_equal(res, tgt) # check shape van = herme.hermevander2d([x1], [x2], [1, 2]) assert_(van.shape == (1, 5, 6)) def test_hermevander3d(self): # also tests hermeval3d for non-square coefficient array x1, x2, x3 = self.x c = np.random.random((2, 3, 4)) van = herme.hermevander3d(x1, x2, x3, [1, 2, 3]) tgt = herme.hermeval3d(x1, x2, x3, c) res = np.dot(van, c.flat) assert_almost_equal(res, tgt) # check shape van = herme.hermevander3d([x1], [x2], [x3], [1, 2, 3]) assert_(van.shape == (1, 5, 24)) class TestFitting: def test_hermefit(self): def f(x): return x*(x - 1)*(x - 2) def f2(x): return x**4 + x**2 + 1 # Test exceptions assert_raises(ValueError, herme.hermefit, [1], [1], -1) assert_raises(TypeError, herme.hermefit, [[1]], [1], 0) assert_raises(TypeError, herme.hermefit, [], [1], 0) assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0) assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0) assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0) assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]]) assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1]) assert_raises(ValueError, herme.hermefit, [1], [1], [-1,]) assert_raises(ValueError, herme.hermefit, [1], [1], [2, -1, 6]) assert_raises(TypeError, herme.hermefit, [1], [1], []) # Test fit x = np.linspace(0, 2) y = f(x) # coef3 = herme.hermefit(x, y, 3) assert_equal(len(coef3), 4) assert_almost_equal(herme.hermeval(x, coef3), y) coef3 = herme.hermefit(x, y, [0, 1, 2, 3]) assert_equal(len(coef3), 4) assert_almost_equal(herme.hermeval(x, coef3), y) # coef4 = herme.hermefit(x, y, 4) assert_equal(len(coef4), 5) assert_almost_equal(herme.hermeval(x, coef4), y) coef4 = herme.hermefit(x, y, [0, 1, 2, 3, 4]) assert_equal(len(coef4), 5) assert_almost_equal(herme.hermeval(x, coef4), y) # check things still work if deg is not in strict increasing coef4 = herme.hermefit(x, y, [2, 3, 4, 1, 0]) assert_equal(len(coef4), 5) assert_almost_equal(herme.hermeval(x, coef4), y) # coef2d = herme.hermefit(x, np.array([y, y]).T, 3) assert_almost_equal(coef2d, np.array([coef3, coef3]).T) coef2d = herme.hermefit(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 = herme.hermefit(x, yw, 3, w=w) assert_almost_equal(wcoef3, coef3) wcoef3 = herme.hermefit(x, yw, [0, 1, 2, 3], w=w) assert_almost_equal(wcoef3, coef3) # wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, 3, w=w) assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) wcoef2d = herme.hermefit(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(herme.hermefit(x, x, 1), [0, 1]) assert_almost_equal(herme.hermefit(x, x, [0, 1]), [0, 1]) # test fitting only even Legendre polynomials x = np.linspace(-1, 1) y = f2(x) coef1 = herme.hermefit(x, y, 4) assert_almost_equal(herme.hermeval(x, coef1), y) coef2 = herme.hermefit(x, y, [0, 2, 4]) assert_almost_equal(herme.hermeval(x, coef2), y) assert_almost_equal(coef1, coef2) class TestCompanion: def test_raises(self): assert_raises(ValueError, herme.hermecompanion, []) assert_raises(ValueError, herme.hermecompanion, [1]) def test_dimensions(self): for i in range(1, 5): coef = [0]*i + [1] assert_(herme.hermecompanion(coef).shape == (i, i)) def test_linear_root(self): assert_(herme.hermecompanion([1, 2])[0, 0] == -.5) class TestGauss: def test_100(self): x, w = herme.hermegauss(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 = herme.hermevander(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.sqrt(2*np.pi) assert_almost_equal(w.sum(), tgt) class TestMisc: def test_hermefromroots(self): res = herme.hermefromroots([]) 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 = herme.hermefromroots(roots) res = herme.hermeval(roots, pol) tgt = 0 assert_(len(pol) == i + 1) assert_almost_equal(herme.herme2poly(pol)[-1], 1) assert_almost_equal(res, tgt) def test_hermeroots(self): assert_almost_equal(herme.hermeroots([1]), []) assert_almost_equal(herme.hermeroots([1, 1]), [-1]) for i in range(2, 5): tgt = np.linspace(-1, 1, i) res = herme.hermeroots(herme.hermefromroots(tgt)) assert_almost_equal(trim(res), trim(tgt)) def test_hermetrim(self): coef = [2, -1, 1, 0] # Test exceptions assert_raises(ValueError, herme.hermetrim, coef, -1) # Test results assert_equal(herme.hermetrim(coef), coef[:-1]) assert_equal(herme.hermetrim(coef, 1), coef[:-3]) assert_equal(herme.hermetrim(coef, 2), [0]) def test_hermeline(self): assert_equal(herme.hermeline(3, 4), [3, 4]) def test_herme2poly(self): for i in range(10): assert_almost_equal(herme.herme2poly([0]*i + [1]), Helist[i]) def test_poly2herme(self): for i in range(10): assert_almost_equal(herme.poly2herme(Helist[i]), [0]*i + [1]) def test_weight(self): x = np.linspace(-5, 5, 11) tgt = np.exp(-.5*x**2) res = herme.hermeweight(x) assert_almost_equal(res, tgt)