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556 lines
18 KiB
556 lines
18 KiB
"""Tests for hermite_e module.
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"""
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from functools import reduce
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import numpy as np
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import numpy.polynomial.hermite_e as herme
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from numpy.polynomial.polynomial import polyval
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from numpy.testing import (
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assert_almost_equal, assert_raises, assert_equal, assert_,
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)
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He0 = np.array([1])
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He1 = np.array([0, 1])
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He2 = np.array([-1, 0, 1])
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He3 = np.array([0, -3, 0, 1])
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He4 = np.array([3, 0, -6, 0, 1])
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He5 = np.array([0, 15, 0, -10, 0, 1])
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He6 = np.array([-15, 0, 45, 0, -15, 0, 1])
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He7 = np.array([0, -105, 0, 105, 0, -21, 0, 1])
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He8 = np.array([105, 0, -420, 0, 210, 0, -28, 0, 1])
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He9 = np.array([0, 945, 0, -1260, 0, 378, 0, -36, 0, 1])
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Helist = [He0, He1, He2, He3, He4, He5, He6, He7, He8, He9]
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def trim(x):
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return herme.hermetrim(x, tol=1e-6)
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class TestConstants:
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def test_hermedomain(self):
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assert_equal(herme.hermedomain, [-1, 1])
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def test_hermezero(self):
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assert_equal(herme.hermezero, [0])
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def test_hermeone(self):
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assert_equal(herme.hermeone, [1])
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def test_hermex(self):
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assert_equal(herme.hermex, [0, 1])
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class TestArithmetic:
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x = np.linspace(-3, 3, 100)
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def test_hermeadd(self):
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for i in range(5):
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for j in range(5):
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msg = f"At i={i}, j={j}"
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tgt = np.zeros(max(i, j) + 1)
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tgt[i] += 1
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tgt[j] += 1
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res = herme.hermeadd([0]*i + [1], [0]*j + [1])
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assert_equal(trim(res), trim(tgt), err_msg=msg)
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def test_hermesub(self):
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for i in range(5):
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for j in range(5):
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msg = f"At i={i}, j={j}"
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tgt = np.zeros(max(i, j) + 1)
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tgt[i] += 1
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tgt[j] -= 1
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res = herme.hermesub([0]*i + [1], [0]*j + [1])
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assert_equal(trim(res), trim(tgt), err_msg=msg)
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def test_hermemulx(self):
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assert_equal(herme.hermemulx([0]), [0])
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assert_equal(herme.hermemulx([1]), [0, 1])
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for i in range(1, 5):
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ser = [0]*i + [1]
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tgt = [0]*(i - 1) + [i, 0, 1]
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assert_equal(herme.hermemulx(ser), tgt)
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def test_hermemul(self):
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# check values of result
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for i in range(5):
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pol1 = [0]*i + [1]
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val1 = herme.hermeval(self.x, pol1)
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for j in range(5):
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msg = f"At i={i}, j={j}"
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pol2 = [0]*j + [1]
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val2 = herme.hermeval(self.x, pol2)
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pol3 = herme.hermemul(pol1, pol2)
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val3 = herme.hermeval(self.x, pol3)
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assert_(len(pol3) == i + j + 1, msg)
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assert_almost_equal(val3, val1*val2, err_msg=msg)
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def test_hermediv(self):
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for i in range(5):
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for j in range(5):
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msg = f"At i={i}, j={j}"
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ci = [0]*i + [1]
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cj = [0]*j + [1]
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tgt = herme.hermeadd(ci, cj)
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quo, rem = herme.hermediv(tgt, ci)
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res = herme.hermeadd(herme.hermemul(quo, ci), rem)
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assert_equal(trim(res), trim(tgt), err_msg=msg)
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def test_hermepow(self):
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for i in range(5):
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for j in range(5):
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msg = f"At i={i}, j={j}"
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c = np.arange(i + 1)
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tgt = reduce(herme.hermemul, [c]*j, np.array([1]))
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res = herme.hermepow(c, j)
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assert_equal(trim(res), trim(tgt), err_msg=msg)
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class TestEvaluation:
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# coefficients of 1 + 2*x + 3*x**2
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c1d = np.array([4., 2., 3.])
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c2d = np.einsum('i,j->ij', c1d, c1d)
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c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d)
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# some random values in [-1, 1)
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x = np.random.random((3, 5))*2 - 1
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y = polyval(x, [1., 2., 3.])
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def test_hermeval(self):
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#check empty input
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assert_equal(herme.hermeval([], [1]).size, 0)
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#check normal input)
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x = np.linspace(-1, 1)
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y = [polyval(x, c) for c in Helist]
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for i in range(10):
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msg = f"At i={i}"
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tgt = y[i]
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res = herme.hermeval(x, [0]*i + [1])
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assert_almost_equal(res, tgt, err_msg=msg)
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#check that shape is preserved
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for i in range(3):
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dims = [2]*i
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x = np.zeros(dims)
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assert_equal(herme.hermeval(x, [1]).shape, dims)
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assert_equal(herme.hermeval(x, [1, 0]).shape, dims)
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assert_equal(herme.hermeval(x, [1, 0, 0]).shape, dims)
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def test_hermeval2d(self):
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x1, x2, x3 = self.x
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y1, y2, y3 = self.y
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#test exceptions
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assert_raises(ValueError, herme.hermeval2d, x1, x2[:2], self.c2d)
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#test values
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tgt = y1*y2
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res = herme.hermeval2d(x1, x2, self.c2d)
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assert_almost_equal(res, tgt)
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#test shape
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z = np.ones((2, 3))
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res = herme.hermeval2d(z, z, self.c2d)
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assert_(res.shape == (2, 3))
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def test_hermeval3d(self):
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x1, x2, x3 = self.x
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y1, y2, y3 = self.y
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#test exceptions
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assert_raises(ValueError, herme.hermeval3d, x1, x2, x3[:2], self.c3d)
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#test values
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tgt = y1*y2*y3
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res = herme.hermeval3d(x1, x2, x3, self.c3d)
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assert_almost_equal(res, tgt)
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#test shape
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z = np.ones((2, 3))
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res = herme.hermeval3d(z, z, z, self.c3d)
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assert_(res.shape == (2, 3))
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def test_hermegrid2d(self):
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x1, x2, x3 = self.x
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y1, y2, y3 = self.y
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#test values
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tgt = np.einsum('i,j->ij', y1, y2)
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res = herme.hermegrid2d(x1, x2, self.c2d)
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assert_almost_equal(res, tgt)
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#test shape
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z = np.ones((2, 3))
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res = herme.hermegrid2d(z, z, self.c2d)
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assert_(res.shape == (2, 3)*2)
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def test_hermegrid3d(self):
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x1, x2, x3 = self.x
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y1, y2, y3 = self.y
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#test values
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tgt = np.einsum('i,j,k->ijk', y1, y2, y3)
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res = herme.hermegrid3d(x1, x2, x3, self.c3d)
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assert_almost_equal(res, tgt)
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#test shape
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z = np.ones((2, 3))
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res = herme.hermegrid3d(z, z, z, self.c3d)
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assert_(res.shape == (2, 3)*3)
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class TestIntegral:
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def test_hermeint(self):
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# check exceptions
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assert_raises(TypeError, herme.hermeint, [0], .5)
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assert_raises(ValueError, herme.hermeint, [0], -1)
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assert_raises(ValueError, herme.hermeint, [0], 1, [0, 0])
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assert_raises(ValueError, herme.hermeint, [0], lbnd=[0])
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assert_raises(ValueError, herme.hermeint, [0], scl=[0])
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assert_raises(TypeError, herme.hermeint, [0], axis=.5)
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# test integration of zero polynomial
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for i in range(2, 5):
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k = [0]*(i - 2) + [1]
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res = herme.hermeint([0], m=i, k=k)
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assert_almost_equal(res, [0, 1])
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# check single integration with integration constant
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for i in range(5):
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scl = i + 1
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pol = [0]*i + [1]
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tgt = [i] + [0]*i + [1/scl]
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hermepol = herme.poly2herme(pol)
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hermeint = herme.hermeint(hermepol, m=1, k=[i])
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res = herme.herme2poly(hermeint)
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assert_almost_equal(trim(res), trim(tgt))
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# check single integration with integration constant and lbnd
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for i in range(5):
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scl = i + 1
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pol = [0]*i + [1]
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hermepol = herme.poly2herme(pol)
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hermeint = herme.hermeint(hermepol, m=1, k=[i], lbnd=-1)
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assert_almost_equal(herme.hermeval(-1, hermeint), i)
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# check single integration with integration constant and scaling
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for i in range(5):
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scl = i + 1
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pol = [0]*i + [1]
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tgt = [i] + [0]*i + [2/scl]
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hermepol = herme.poly2herme(pol)
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hermeint = herme.hermeint(hermepol, m=1, k=[i], scl=2)
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res = herme.herme2poly(hermeint)
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assert_almost_equal(trim(res), trim(tgt))
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# check multiple integrations with default k
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for i in range(5):
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for j in range(2, 5):
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pol = [0]*i + [1]
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tgt = pol[:]
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for k in range(j):
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tgt = herme.hermeint(tgt, m=1)
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res = herme.hermeint(pol, m=j)
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assert_almost_equal(trim(res), trim(tgt))
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# check multiple integrations with defined k
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for i in range(5):
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for j in range(2, 5):
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pol = [0]*i + [1]
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tgt = pol[:]
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for k in range(j):
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tgt = herme.hermeint(tgt, m=1, k=[k])
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res = herme.hermeint(pol, m=j, k=list(range(j)))
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assert_almost_equal(trim(res), trim(tgt))
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# check multiple integrations with lbnd
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for i in range(5):
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for j in range(2, 5):
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pol = [0]*i + [1]
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tgt = pol[:]
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for k in range(j):
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tgt = herme.hermeint(tgt, m=1, k=[k], lbnd=-1)
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res = herme.hermeint(pol, m=j, k=list(range(j)), lbnd=-1)
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assert_almost_equal(trim(res), trim(tgt))
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# check multiple integrations with scaling
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for i in range(5):
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for j in range(2, 5):
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pol = [0]*i + [1]
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tgt = pol[:]
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for k in range(j):
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tgt = herme.hermeint(tgt, m=1, k=[k], scl=2)
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res = herme.hermeint(pol, m=j, k=list(range(j)), scl=2)
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assert_almost_equal(trim(res), trim(tgt))
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def test_hermeint_axis(self):
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# check that axis keyword works
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c2d = np.random.random((3, 4))
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tgt = np.vstack([herme.hermeint(c) for c in c2d.T]).T
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res = herme.hermeint(c2d, axis=0)
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assert_almost_equal(res, tgt)
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tgt = np.vstack([herme.hermeint(c) for c in c2d])
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res = herme.hermeint(c2d, axis=1)
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assert_almost_equal(res, tgt)
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tgt = np.vstack([herme.hermeint(c, k=3) for c in c2d])
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res = herme.hermeint(c2d, k=3, axis=1)
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assert_almost_equal(res, tgt)
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class TestDerivative:
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def test_hermeder(self):
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# check exceptions
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assert_raises(TypeError, herme.hermeder, [0], .5)
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assert_raises(ValueError, herme.hermeder, [0], -1)
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# check that zeroth derivative does nothing
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for i in range(5):
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tgt = [0]*i + [1]
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res = herme.hermeder(tgt, m=0)
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assert_equal(trim(res), trim(tgt))
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# check that derivation is the inverse of integration
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for i in range(5):
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for j in range(2, 5):
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tgt = [0]*i + [1]
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res = herme.hermeder(herme.hermeint(tgt, m=j), m=j)
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assert_almost_equal(trim(res), trim(tgt))
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# check derivation with scaling
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for i in range(5):
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for j in range(2, 5):
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tgt = [0]*i + [1]
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res = herme.hermeder(
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herme.hermeint(tgt, m=j, scl=2), m=j, scl=.5)
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assert_almost_equal(trim(res), trim(tgt))
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def test_hermeder_axis(self):
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# check that axis keyword works
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c2d = np.random.random((3, 4))
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tgt = np.vstack([herme.hermeder(c) for c in c2d.T]).T
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res = herme.hermeder(c2d, axis=0)
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assert_almost_equal(res, tgt)
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tgt = np.vstack([herme.hermeder(c) for c in c2d])
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res = herme.hermeder(c2d, axis=1)
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assert_almost_equal(res, tgt)
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class TestVander:
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# some random values in [-1, 1)
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x = np.random.random((3, 5))*2 - 1
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def test_hermevander(self):
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# check for 1d x
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x = np.arange(3)
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v = herme.hermevander(x, 3)
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assert_(v.shape == (3, 4))
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for i in range(4):
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coef = [0]*i + [1]
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assert_almost_equal(v[..., i], herme.hermeval(x, coef))
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# check for 2d x
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x = np.array([[1, 2], [3, 4], [5, 6]])
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v = herme.hermevander(x, 3)
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assert_(v.shape == (3, 2, 4))
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for i in range(4):
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coef = [0]*i + [1]
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assert_almost_equal(v[..., i], herme.hermeval(x, coef))
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def test_hermevander2d(self):
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# also tests hermeval2d for non-square coefficient array
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x1, x2, x3 = self.x
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c = np.random.random((2, 3))
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van = herme.hermevander2d(x1, x2, [1, 2])
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tgt = herme.hermeval2d(x1, x2, c)
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res = np.dot(van, c.flat)
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assert_almost_equal(res, tgt)
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# check shape
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van = herme.hermevander2d([x1], [x2], [1, 2])
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assert_(van.shape == (1, 5, 6))
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def test_hermevander3d(self):
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# also tests hermeval3d for non-square coefficient array
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x1, x2, x3 = self.x
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c = np.random.random((2, 3, 4))
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van = herme.hermevander3d(x1, x2, x3, [1, 2, 3])
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tgt = herme.hermeval3d(x1, x2, x3, c)
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res = np.dot(van, c.flat)
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assert_almost_equal(res, tgt)
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# check shape
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van = herme.hermevander3d([x1], [x2], [x3], [1, 2, 3])
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assert_(van.shape == (1, 5, 24))
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class TestFitting:
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def test_hermefit(self):
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def f(x):
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return x*(x - 1)*(x - 2)
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def f2(x):
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return x**4 + x**2 + 1
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# Test exceptions
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assert_raises(ValueError, herme.hermefit, [1], [1], -1)
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assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
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assert_raises(TypeError, herme.hermefit, [], [1], 0)
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assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
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assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
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assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
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assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
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assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1])
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assert_raises(ValueError, herme.hermefit, [1], [1], [-1,])
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assert_raises(ValueError, herme.hermefit, [1], [1], [2, -1, 6])
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assert_raises(TypeError, herme.hermefit, [1], [1], [])
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# Test fit
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x = np.linspace(0, 2)
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y = f(x)
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#
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coef3 = herme.hermefit(x, y, 3)
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assert_equal(len(coef3), 4)
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assert_almost_equal(herme.hermeval(x, coef3), y)
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coef3 = herme.hermefit(x, y, [0, 1, 2, 3])
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assert_equal(len(coef3), 4)
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assert_almost_equal(herme.hermeval(x, coef3), y)
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#
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coef4 = herme.hermefit(x, y, 4)
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assert_equal(len(coef4), 5)
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assert_almost_equal(herme.hermeval(x, coef4), y)
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coef4 = herme.hermefit(x, y, [0, 1, 2, 3, 4])
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assert_equal(len(coef4), 5)
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assert_almost_equal(herme.hermeval(x, coef4), y)
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# check things still work if deg is not in strict increasing
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coef4 = herme.hermefit(x, y, [2, 3, 4, 1, 0])
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assert_equal(len(coef4), 5)
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assert_almost_equal(herme.hermeval(x, coef4), y)
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#
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coef2d = herme.hermefit(x, np.array([y, y]).T, 3)
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assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
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coef2d = herme.hermefit(x, np.array([y, y]).T, [0, 1, 2, 3])
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assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
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# test weighting
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w = np.zeros_like(x)
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yw = y.copy()
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w[1::2] = 1
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y[0::2] = 0
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wcoef3 = herme.hermefit(x, yw, 3, w=w)
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assert_almost_equal(wcoef3, coef3)
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wcoef3 = herme.hermefit(x, yw, [0, 1, 2, 3], w=w)
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assert_almost_equal(wcoef3, coef3)
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#
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wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, 3, w=w)
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assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
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wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w)
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assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
|
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# test scaling with complex values x points whose square
|
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# is zero when summed.
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x = [1, 1j, -1, -1j]
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assert_almost_equal(herme.hermefit(x, x, 1), [0, 1])
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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)
|
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assert_almost_equal(coef1, coef2)
|
|
|
|
|
|
class TestCompanion:
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|
|
|
def test_raises(self):
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assert_raises(ValueError, herme.hermecompanion, [])
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assert_raises(ValueError, herme.hermecompanion, [1])
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|
|
|
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)
|
|
|