559 lines
18 KiB
Python
559 lines
18 KiB
Python
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"""Tests for hermite 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 as herm
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from numpy.polynomial.polynomial import polyval
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from numpy.testing import (
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assert_,
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assert_almost_equal,
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assert_equal,
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assert_raises,
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)
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H0 = np.array([1])
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H1 = np.array([0, 2])
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H2 = np.array([-2, 0, 4])
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H3 = np.array([0, -12, 0, 8])
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H4 = np.array([12, 0, -48, 0, 16])
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H5 = np.array([0, 120, 0, -160, 0, 32])
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H6 = np.array([-120, 0, 720, 0, -480, 0, 64])
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H7 = np.array([0, -1680, 0, 3360, 0, -1344, 0, 128])
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H8 = np.array([1680, 0, -13440, 0, 13440, 0, -3584, 0, 256])
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H9 = np.array([0, 30240, 0, -80640, 0, 48384, 0, -9216, 0, 512])
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Hlist = [H0, H1, H2, H3, H4, H5, H6, H7, H8, H9]
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def trim(x):
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return herm.hermtrim(x, tol=1e-6)
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class TestConstants:
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def test_hermdomain(self):
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assert_equal(herm.hermdomain, [-1, 1])
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def test_hermzero(self):
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assert_equal(herm.hermzero, [0])
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def test_hermone(self):
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assert_equal(herm.hermone, [1])
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def test_hermx(self):
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assert_equal(herm.hermx, [0, .5])
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class TestArithmetic:
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x = np.linspace(-3, 3, 100)
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def test_hermadd(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 = herm.hermadd([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_hermsub(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 = herm.hermsub([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_hermmulx(self):
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assert_equal(herm.hermmulx([0]), [0])
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assert_equal(herm.hermmulx([1]), [0, .5])
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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, .5]
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assert_equal(herm.hermmulx(ser), tgt)
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def test_hermmul(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 = herm.hermval(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 = herm.hermval(self.x, pol2)
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pol3 = herm.hermmul(pol1, pol2)
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val3 = herm.hermval(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_hermdiv(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 = herm.hermadd(ci, cj)
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quo, rem = herm.hermdiv(tgt, ci)
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res = herm.hermadd(herm.hermmul(quo, ci), rem)
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assert_equal(trim(res), trim(tgt), err_msg=msg)
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def test_hermpow(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(herm.hermmul, [c] * j, np.array([1]))
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res = herm.hermpow(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([2.5, 1., .75])
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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_hermval(self):
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# check empty input
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assert_equal(herm.hermval([], [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 Hlist]
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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 = herm.hermval(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(herm.hermval(x, [1]).shape, dims)
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assert_equal(herm.hermval(x, [1, 0]).shape, dims)
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assert_equal(herm.hermval(x, [1, 0, 0]).shape, dims)
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def test_hermval2d(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, herm.hermval2d, x1, x2[:2], self.c2d)
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# test values
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tgt = y1 * y2
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res = herm.hermval2d(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 = herm.hermval2d(z, z, self.c2d)
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assert_(res.shape == (2, 3))
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def test_hermval3d(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, herm.hermval3d, 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 = herm.hermval3d(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 = herm.hermval3d(z, z, z, self.c3d)
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assert_(res.shape == (2, 3))
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def test_hermgrid2d(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 = herm.hermgrid2d(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 = herm.hermgrid2d(z, z, self.c2d)
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assert_(res.shape == (2, 3) * 2)
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def test_hermgrid3d(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 = herm.hermgrid3d(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 = herm.hermgrid3d(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_hermint(self):
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# check exceptions
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assert_raises(TypeError, herm.hermint, [0], .5)
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assert_raises(ValueError, herm.hermint, [0], -1)
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assert_raises(ValueError, herm.hermint, [0], 1, [0, 0])
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assert_raises(ValueError, herm.hermint, [0], lbnd=[0])
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assert_raises(ValueError, herm.hermint, [0], scl=[0])
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assert_raises(TypeError, herm.hermint, [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 = herm.hermint([0], m=i, k=k)
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assert_almost_equal(res, [0, .5])
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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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hermpol = herm.poly2herm(pol)
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hermint = herm.hermint(hermpol, m=1, k=[i])
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res = herm.herm2poly(hermint)
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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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hermpol = herm.poly2herm(pol)
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hermint = herm.hermint(hermpol, m=1, k=[i], lbnd=-1)
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assert_almost_equal(herm.hermval(-1, hermint), 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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hermpol = herm.poly2herm(pol)
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hermint = herm.hermint(hermpol, m=1, k=[i], scl=2)
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res = herm.herm2poly(hermint)
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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 = herm.hermint(tgt, m=1)
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res = herm.hermint(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 = herm.hermint(tgt, m=1, k=[k])
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res = herm.hermint(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 = herm.hermint(tgt, m=1, k=[k], lbnd=-1)
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res = herm.hermint(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 = herm.hermint(tgt, m=1, k=[k], scl=2)
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res = herm.hermint(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_hermint_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([herm.hermint(c) for c in c2d.T]).T
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res = herm.hermint(c2d, axis=0)
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assert_almost_equal(res, tgt)
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tgt = np.vstack([herm.hermint(c) for c in c2d])
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res = herm.hermint(c2d, axis=1)
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assert_almost_equal(res, tgt)
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tgt = np.vstack([herm.hermint(c, k=3) for c in c2d])
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res = herm.hermint(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_hermder(self):
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# check exceptions
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assert_raises(TypeError, herm.hermder, [0], .5)
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assert_raises(ValueError, herm.hermder, [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 = herm.hermder(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 = herm.hermder(herm.hermint(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 = herm.hermder(herm.hermint(tgt, m=j, scl=2), m=j, scl=.5)
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||
|
|
assert_almost_equal(trim(res), trim(tgt))
|
||
|
|
|
||
|
|
def test_hermder_axis(self):
|
||
|
|
# check that axis keyword works
|
||
|
|
c2d = np.random.random((3, 4))
|
||
|
|
|
||
|
|
tgt = np.vstack([herm.hermder(c) for c in c2d.T]).T
|
||
|
|
res = herm.hermder(c2d, axis=0)
|
||
|
|
assert_almost_equal(res, tgt)
|
||
|
|
|
||
|
|
tgt = np.vstack([herm.hermder(c) for c in c2d])
|
||
|
|
res = herm.hermder(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_hermvander(self):
|
||
|
|
# check for 1d x
|
||
|
|
x = np.arange(3)
|
||
|
|
v = herm.hermvander(x, 3)
|
||
|
|
assert_(v.shape == (3, 4))
|
||
|
|
for i in range(4):
|
||
|
|
coef = [0] * i + [1]
|
||
|
|
assert_almost_equal(v[..., i], herm.hermval(x, coef))
|
||
|
|
|
||
|
|
# check for 2d x
|
||
|
|
x = np.array([[1, 2], [3, 4], [5, 6]])
|
||
|
|
v = herm.hermvander(x, 3)
|
||
|
|
assert_(v.shape == (3, 2, 4))
|
||
|
|
for i in range(4):
|
||
|
|
coef = [0] * i + [1]
|
||
|
|
assert_almost_equal(v[..., i], herm.hermval(x, coef))
|
||
|
|
|
||
|
|
def test_hermvander2d(self):
|
||
|
|
# also tests hermval2d for non-square coefficient array
|
||
|
|
x1, x2, x3 = self.x
|
||
|
|
c = np.random.random((2, 3))
|
||
|
|
van = herm.hermvander2d(x1, x2, [1, 2])
|
||
|
|
tgt = herm.hermval2d(x1, x2, c)
|
||
|
|
res = np.dot(van, c.flat)
|
||
|
|
assert_almost_equal(res, tgt)
|
||
|
|
|
||
|
|
# check shape
|
||
|
|
van = herm.hermvander2d([x1], [x2], [1, 2])
|
||
|
|
assert_(van.shape == (1, 5, 6))
|
||
|
|
|
||
|
|
def test_hermvander3d(self):
|
||
|
|
# also tests hermval3d for non-square coefficient array
|
||
|
|
x1, x2, x3 = self.x
|
||
|
|
c = np.random.random((2, 3, 4))
|
||
|
|
van = herm.hermvander3d(x1, x2, x3, [1, 2, 3])
|
||
|
|
tgt = herm.hermval3d(x1, x2, x3, c)
|
||
|
|
res = np.dot(van, c.flat)
|
||
|
|
assert_almost_equal(res, tgt)
|
||
|
|
|
||
|
|
# check shape
|
||
|
|
van = herm.hermvander3d([x1], [x2], [x3], [1, 2, 3])
|
||
|
|
assert_(van.shape == (1, 5, 24))
|
||
|
|
|
||
|
|
|
||
|
|
class TestFitting:
|
||
|
|
|
||
|
|
def test_hermfit(self):
|
||
|
|
def f(x):
|
||
|
|
return x * (x - 1) * (x - 2)
|
||
|
|
|
||
|
|
def f2(x):
|
||
|
|
return x**4 + x**2 + 1
|
||
|
|
|
||
|
|
# Test exceptions
|
||
|
|
assert_raises(ValueError, herm.hermfit, [1], [1], -1)
|
||
|
|
assert_raises(TypeError, herm.hermfit, [[1]], [1], 0)
|
||
|
|
assert_raises(TypeError, herm.hermfit, [], [1], 0)
|
||
|
|
assert_raises(TypeError, herm.hermfit, [1], [[[1]]], 0)
|
||
|
|
assert_raises(TypeError, herm.hermfit, [1, 2], [1], 0)
|
||
|
|
assert_raises(TypeError, herm.hermfit, [1], [1, 2], 0)
|
||
|
|
assert_raises(TypeError, herm.hermfit, [1], [1], 0, w=[[1]])
|
||
|
|
assert_raises(TypeError, herm.hermfit, [1], [1], 0, w=[1, 1])
|
||
|
|
assert_raises(ValueError, herm.hermfit, [1], [1], [-1,])
|
||
|
|
assert_raises(ValueError, herm.hermfit, [1], [1], [2, -1, 6])
|
||
|
|
assert_raises(TypeError, herm.hermfit, [1], [1], [])
|
||
|
|
|
||
|
|
# Test fit
|
||
|
|
x = np.linspace(0, 2)
|
||
|
|
y = f(x)
|
||
|
|
#
|
||
|
|
coef3 = herm.hermfit(x, y, 3)
|
||
|
|
assert_equal(len(coef3), 4)
|
||
|
|
assert_almost_equal(herm.hermval(x, coef3), y)
|
||
|
|
coef3 = herm.hermfit(x, y, [0, 1, 2, 3])
|
||
|
|
assert_equal(len(coef3), 4)
|
||
|
|
assert_almost_equal(herm.hermval(x, coef3), y)
|
||
|
|
#
|
||
|
|
coef4 = herm.hermfit(x, y, 4)
|
||
|
|
assert_equal(len(coef4), 5)
|
||
|
|
assert_almost_equal(herm.hermval(x, coef4), y)
|
||
|
|
coef4 = herm.hermfit(x, y, [0, 1, 2, 3, 4])
|
||
|
|
assert_equal(len(coef4), 5)
|
||
|
|
assert_almost_equal(herm.hermval(x, coef4), y)
|
||
|
|
# check things still work if deg is not in strict increasing
|
||
|
|
coef4 = herm.hermfit(x, y, [2, 3, 4, 1, 0])
|
||
|
|
assert_equal(len(coef4), 5)
|
||
|
|
assert_almost_equal(herm.hermval(x, coef4), y)
|
||
|
|
#
|
||
|
|
coef2d = herm.hermfit(x, np.array([y, y]).T, 3)
|
||
|
|
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
|
||
|
|
coef2d = herm.hermfit(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 = herm.hermfit(x, yw, 3, w=w)
|
||
|
|
assert_almost_equal(wcoef3, coef3)
|
||
|
|
wcoef3 = herm.hermfit(x, yw, [0, 1, 2, 3], w=w)
|
||
|
|
assert_almost_equal(wcoef3, coef3)
|
||
|
|
#
|
||
|
|
wcoef2d = herm.hermfit(x, np.array([yw, yw]).T, 3, w=w)
|
||
|
|
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
|
||
|
|
wcoef2d = herm.hermfit(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(herm.hermfit(x, x, 1), [0, .5])
|
||
|
|
assert_almost_equal(herm.hermfit(x, x, [0, 1]), [0, .5])
|
||
|
|
# test fitting only even Legendre polynomials
|
||
|
|
x = np.linspace(-1, 1)
|
||
|
|
y = f2(x)
|
||
|
|
coef1 = herm.hermfit(x, y, 4)
|
||
|
|
assert_almost_equal(herm.hermval(x, coef1), y)
|
||
|
|
coef2 = herm.hermfit(x, y, [0, 2, 4])
|
||
|
|
assert_almost_equal(herm.hermval(x, coef2), y)
|
||
|
|
assert_almost_equal(coef1, coef2)
|
||
|
|
|
||
|
|
|
||
|
|
class TestCompanion:
|
||
|
|
|
||
|
|
def test_raises(self):
|
||
|
|
assert_raises(ValueError, herm.hermcompanion, [])
|
||
|
|
assert_raises(ValueError, herm.hermcompanion, [1])
|
||
|
|
|
||
|
|
def test_dimensions(self):
|
||
|
|
for i in range(1, 5):
|
||
|
|
coef = [0] * i + [1]
|
||
|
|
assert_(herm.hermcompanion(coef).shape == (i, i))
|
||
|
|
|
||
|
|
def test_linear_root(self):
|
||
|
|
assert_(herm.hermcompanion([1, 2])[0, 0] == -.25)
|
||
|
|
|
||
|
|
|
||
|
|
class TestGauss:
|
||
|
|
|
||
|
|
def test_100(self):
|
||
|
|
x, w = herm.hermgauss(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 = herm.hermvander(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(np.pi)
|
||
|
|
assert_almost_equal(w.sum(), tgt)
|
||
|
|
|
||
|
|
|
||
|
|
class TestMisc:
|
||
|
|
|
||
|
|
def test_hermfromroots(self):
|
||
|
|
res = herm.hermfromroots([])
|
||
|
|
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 = herm.hermfromroots(roots)
|
||
|
|
res = herm.hermval(roots, pol)
|
||
|
|
tgt = 0
|
||
|
|
assert_(len(pol) == i + 1)
|
||
|
|
assert_almost_equal(herm.herm2poly(pol)[-1], 1)
|
||
|
|
assert_almost_equal(res, tgt)
|
||
|
|
|
||
|
|
def test_hermroots(self):
|
||
|
|
assert_almost_equal(herm.hermroots([1]), [])
|
||
|
|
assert_almost_equal(herm.hermroots([1, 1]), [-.5])
|
||
|
|
for i in range(2, 5):
|
||
|
|
tgt = np.linspace(-1, 1, i)
|
||
|
|
res = herm.hermroots(herm.hermfromroots(tgt))
|
||
|
|
assert_almost_equal(trim(res), trim(tgt))
|
||
|
|
|
||
|
|
def test_hermtrim(self):
|
||
|
|
coef = [2, -1, 1, 0]
|
||
|
|
|
||
|
|
# Test exceptions
|
||
|
|
assert_raises(ValueError, herm.hermtrim, coef, -1)
|
||
|
|
|
||
|
|
# Test results
|
||
|
|
assert_equal(herm.hermtrim(coef), coef[:-1])
|
||
|
|
assert_equal(herm.hermtrim(coef, 1), coef[:-3])
|
||
|
|
assert_equal(herm.hermtrim(coef, 2), [0])
|
||
|
|
|
||
|
|
def test_hermline(self):
|
||
|
|
assert_equal(herm.hermline(3, 4), [3, 2])
|
||
|
|
|
||
|
|
def test_herm2poly(self):
|
||
|
|
for i in range(10):
|
||
|
|
assert_almost_equal(herm.herm2poly([0] * i + [1]), Hlist[i])
|
||
|
|
|
||
|
|
def test_poly2herm(self):
|
||
|
|
for i in range(10):
|
||
|
|
assert_almost_equal(herm.poly2herm(Hlist[i]), [0] * i + [1])
|
||
|
|
|
||
|
|
def test_weight(self):
|
||
|
|
x = np.linspace(-5, 5, 11)
|
||
|
|
tgt = np.exp(-x**2)
|
||
|
|
res = herm.hermweight(x)
|
||
|
|
assert_almost_equal(res, tgt)
|