624 lines
20 KiB
Python
624 lines
20 KiB
Python
"""Tests for chebyshev 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.chebyshev as cheb
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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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def trim(x):
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return cheb.chebtrim(x, tol=1e-6)
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T0 = [1]
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T1 = [0, 1]
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T2 = [-1, 0, 2]
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T3 = [0, -3, 0, 4]
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T4 = [1, 0, -8, 0, 8]
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T5 = [0, 5, 0, -20, 0, 16]
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T6 = [-1, 0, 18, 0, -48, 0, 32]
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T7 = [0, -7, 0, 56, 0, -112, 0, 64]
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T8 = [1, 0, -32, 0, 160, 0, -256, 0, 128]
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T9 = [0, 9, 0, -120, 0, 432, 0, -576, 0, 256]
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Tlist = [T0, T1, T2, T3, T4, T5, T6, T7, T8, T9]
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class TestPrivate:
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def test__cseries_to_zseries(self):
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for i in range(5):
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inp = np.array([2] + [1] * i, np.double)
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tgt = np.array([.5] * i + [2] + [.5] * i, np.double)
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res = cheb._cseries_to_zseries(inp)
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assert_equal(res, tgt)
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def test__zseries_to_cseries(self):
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for i in range(5):
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inp = np.array([.5] * i + [2] + [.5] * i, np.double)
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tgt = np.array([2] + [1] * i, np.double)
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res = cheb._zseries_to_cseries(inp)
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assert_equal(res, tgt)
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class TestConstants:
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def test_chebdomain(self):
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assert_equal(cheb.chebdomain, [-1, 1])
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def test_chebzero(self):
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assert_equal(cheb.chebzero, [0])
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def test_chebone(self):
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assert_equal(cheb.chebone, [1])
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def test_chebx(self):
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assert_equal(cheb.chebx, [0, 1])
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class TestArithmetic:
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def test_chebadd(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 = cheb.chebadd([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_chebsub(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 = cheb.chebsub([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_chebmulx(self):
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assert_equal(cheb.chebmulx([0]), [0])
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assert_equal(cheb.chebmulx([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) + [.5, 0, .5]
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assert_equal(cheb.chebmulx(ser), tgt)
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def test_chebmul(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(i + j + 1)
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tgt[i + j] += .5
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tgt[abs(i - j)] += .5
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res = cheb.chebmul([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_chebdiv(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 = cheb.chebadd(ci, cj)
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quo, rem = cheb.chebdiv(tgt, ci)
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res = cheb.chebadd(cheb.chebmul(quo, ci), rem)
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assert_equal(trim(res), trim(tgt), err_msg=msg)
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def test_chebpow(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(cheb.chebmul, [c] * j, np.array([1]))
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res = cheb.chebpow(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, 2., 1.5])
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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_chebval(self):
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# check empty input
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assert_equal(cheb.chebval([], [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 Tlist]
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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 = cheb.chebval(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(cheb.chebval(x, [1]).shape, dims)
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assert_equal(cheb.chebval(x, [1, 0]).shape, dims)
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assert_equal(cheb.chebval(x, [1, 0, 0]).shape, dims)
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def test_chebval2d(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, cheb.chebval2d, x1, x2[:2], self.c2d)
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# test values
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tgt = y1 * y2
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res = cheb.chebval2d(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 = cheb.chebval2d(z, z, self.c2d)
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assert_(res.shape == (2, 3))
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def test_chebval3d(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, cheb.chebval3d, 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 = cheb.chebval3d(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 = cheb.chebval3d(z, z, z, self.c3d)
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assert_(res.shape == (2, 3))
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def test_chebgrid2d(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 = cheb.chebgrid2d(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 = cheb.chebgrid2d(z, z, self.c2d)
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assert_(res.shape == (2, 3) * 2)
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def test_chebgrid3d(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 = cheb.chebgrid3d(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 = cheb.chebgrid3d(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_chebint(self):
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# check exceptions
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assert_raises(TypeError, cheb.chebint, [0], .5)
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assert_raises(ValueError, cheb.chebint, [0], -1)
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assert_raises(ValueError, cheb.chebint, [0], 1, [0, 0])
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assert_raises(ValueError, cheb.chebint, [0], lbnd=[0])
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assert_raises(ValueError, cheb.chebint, [0], scl=[0])
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assert_raises(TypeError, cheb.chebint, [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 = cheb.chebint([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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chebpol = cheb.poly2cheb(pol)
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chebint = cheb.chebint(chebpol, m=1, k=[i])
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res = cheb.cheb2poly(chebint)
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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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chebpol = cheb.poly2cheb(pol)
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chebint = cheb.chebint(chebpol, m=1, k=[i], lbnd=-1)
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assert_almost_equal(cheb.chebval(-1, chebint), 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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chebpol = cheb.poly2cheb(pol)
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chebint = cheb.chebint(chebpol, m=1, k=[i], scl=2)
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res = cheb.cheb2poly(chebint)
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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 = cheb.chebint(tgt, m=1)
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res = cheb.chebint(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 = cheb.chebint(tgt, m=1, k=[k])
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res = cheb.chebint(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 = cheb.chebint(tgt, m=1, k=[k], lbnd=-1)
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res = cheb.chebint(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 = cheb.chebint(tgt, m=1, k=[k], scl=2)
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res = cheb.chebint(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_chebint_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([cheb.chebint(c) for c in c2d.T]).T
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res = cheb.chebint(c2d, axis=0)
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assert_almost_equal(res, tgt)
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tgt = np.vstack([cheb.chebint(c) for c in c2d])
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res = cheb.chebint(c2d, axis=1)
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assert_almost_equal(res, tgt)
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tgt = np.vstack([cheb.chebint(c, k=3) for c in c2d])
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res = cheb.chebint(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_chebder(self):
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# check exceptions
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assert_raises(TypeError, cheb.chebder, [0], .5)
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assert_raises(ValueError, cheb.chebder, [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 = cheb.chebder(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 = cheb.chebder(cheb.chebint(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 = cheb.chebder(cheb.chebint(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_chebder_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([cheb.chebder(c) for c in c2d.T]).T
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res = cheb.chebder(c2d, axis=0)
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assert_almost_equal(res, tgt)
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tgt = np.vstack([cheb.chebder(c) for c in c2d])
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res = cheb.chebder(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_chebvander(self):
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# check for 1d x
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x = np.arange(3)
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v = cheb.chebvander(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], cheb.chebval(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 = cheb.chebvander(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], cheb.chebval(x, coef))
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def test_chebvander2d(self):
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# also tests chebval2d 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 = cheb.chebvander2d(x1, x2, [1, 2])
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tgt = cheb.chebval2d(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 = cheb.chebvander2d([x1], [x2], [1, 2])
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assert_(van.shape == (1, 5, 6))
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def test_chebvander3d(self):
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# also tests chebval3d 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 = cheb.chebvander3d(x1, x2, x3, [1, 2, 3])
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tgt = cheb.chebval3d(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 = cheb.chebvander3d([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_chebfit(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, cheb.chebfit, [1], [1], -1)
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assert_raises(TypeError, cheb.chebfit, [[1]], [1], 0)
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assert_raises(TypeError, cheb.chebfit, [], [1], 0)
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assert_raises(TypeError, cheb.chebfit, [1], [[[1]]], 0)
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assert_raises(TypeError, cheb.chebfit, [1, 2], [1], 0)
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assert_raises(TypeError, cheb.chebfit, [1], [1, 2], 0)
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assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[[1]])
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assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[1, 1])
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assert_raises(ValueError, cheb.chebfit, [1], [1], [-1,])
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assert_raises(ValueError, cheb.chebfit, [1], [1], [2, -1, 6])
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assert_raises(TypeError, cheb.chebfit, [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 = cheb.chebfit(x, y, 3)
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assert_equal(len(coef3), 4)
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assert_almost_equal(cheb.chebval(x, coef3), y)
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coef3 = cheb.chebfit(x, y, [0, 1, 2, 3])
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assert_equal(len(coef3), 4)
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|
assert_almost_equal(cheb.chebval(x, coef3), y)
|
|
#
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|
coef4 = cheb.chebfit(x, y, 4)
|
|
assert_equal(len(coef4), 5)
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|
assert_almost_equal(cheb.chebval(x, coef4), y)
|
|
coef4 = cheb.chebfit(x, y, [0, 1, 2, 3, 4])
|
|
assert_equal(len(coef4), 5)
|
|
assert_almost_equal(cheb.chebval(x, coef4), y)
|
|
# check things still work if deg is not in strict increasing
|
|
coef4 = cheb.chebfit(x, y, [2, 3, 4, 1, 0])
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|
assert_equal(len(coef4), 5)
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|
assert_almost_equal(cheb.chebval(x, coef4), y)
|
|
#
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|
coef2d = cheb.chebfit(x, np.array([y, y]).T, 3)
|
|
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
|
|
coef2d = cheb.chebfit(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()
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w[1::2] = 1
|
|
y[0::2] = 0
|
|
wcoef3 = cheb.chebfit(x, yw, 3, w=w)
|
|
assert_almost_equal(wcoef3, coef3)
|
|
wcoef3 = cheb.chebfit(x, yw, [0, 1, 2, 3], w=w)
|
|
assert_almost_equal(wcoef3, coef3)
|
|
#
|
|
wcoef2d = cheb.chebfit(x, np.array([yw, yw]).T, 3, w=w)
|
|
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
|
|
wcoef2d = cheb.chebfit(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(cheb.chebfit(x, x, 1), [0, 1])
|
|
assert_almost_equal(cheb.chebfit(x, x, [0, 1]), [0, 1])
|
|
# test fitting only even polynomials
|
|
x = np.linspace(-1, 1)
|
|
y = f2(x)
|
|
coef1 = cheb.chebfit(x, y, 4)
|
|
assert_almost_equal(cheb.chebval(x, coef1), y)
|
|
coef2 = cheb.chebfit(x, y, [0, 2, 4])
|
|
assert_almost_equal(cheb.chebval(x, coef2), y)
|
|
assert_almost_equal(coef1, coef2)
|
|
|
|
|
|
class TestInterpolate:
|
|
|
|
def f(self, x):
|
|
return x * (x - 1) * (x - 2)
|
|
|
|
def test_raises(self):
|
|
assert_raises(ValueError, cheb.chebinterpolate, self.f, -1)
|
|
assert_raises(TypeError, cheb.chebinterpolate, self.f, 10.)
|
|
|
|
def test_dimensions(self):
|
|
for deg in range(1, 5):
|
|
assert_(cheb.chebinterpolate(self.f, deg).shape == (deg + 1,))
|
|
|
|
def test_approximation(self):
|
|
|
|
def powx(x, p):
|
|
return x**p
|
|
|
|
x = np.linspace(-1, 1, 10)
|
|
for deg in range(10):
|
|
for p in range(deg + 1):
|
|
c = cheb.chebinterpolate(powx, deg, (p,))
|
|
assert_almost_equal(cheb.chebval(x, c), powx(x, p), decimal=12)
|
|
|
|
|
|
class TestCompanion:
|
|
|
|
def test_raises(self):
|
|
assert_raises(ValueError, cheb.chebcompanion, [])
|
|
assert_raises(ValueError, cheb.chebcompanion, [1])
|
|
|
|
def test_dimensions(self):
|
|
for i in range(1, 5):
|
|
coef = [0] * i + [1]
|
|
assert_(cheb.chebcompanion(coef).shape == (i, i))
|
|
|
|
def test_linear_root(self):
|
|
assert_(cheb.chebcompanion([1, 2])[0, 0] == -.5)
|
|
|
|
|
|
class TestGauss:
|
|
|
|
def test_100(self):
|
|
x, w = cheb.chebgauss(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 = cheb.chebvander(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.pi
|
|
assert_almost_equal(w.sum(), tgt)
|
|
|
|
|
|
class TestMisc:
|
|
|
|
def test_chebfromroots(self):
|
|
res = cheb.chebfromroots([])
|
|
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])
|
|
tgt = [0] * i + [1]
|
|
res = cheb.chebfromroots(roots) * 2**(i - 1)
|
|
assert_almost_equal(trim(res), trim(tgt))
|
|
|
|
def test_chebroots(self):
|
|
assert_almost_equal(cheb.chebroots([1]), [])
|
|
assert_almost_equal(cheb.chebroots([1, 2]), [-.5])
|
|
for i in range(2, 5):
|
|
tgt = np.linspace(-1, 1, i)
|
|
res = cheb.chebroots(cheb.chebfromroots(tgt))
|
|
assert_almost_equal(trim(res), trim(tgt))
|
|
|
|
def test_chebtrim(self):
|
|
coef = [2, -1, 1, 0]
|
|
|
|
# Test exceptions
|
|
assert_raises(ValueError, cheb.chebtrim, coef, -1)
|
|
|
|
# Test results
|
|
assert_equal(cheb.chebtrim(coef), coef[:-1])
|
|
assert_equal(cheb.chebtrim(coef, 1), coef[:-3])
|
|
assert_equal(cheb.chebtrim(coef, 2), [0])
|
|
|
|
def test_chebline(self):
|
|
assert_equal(cheb.chebline(3, 4), [3, 4])
|
|
|
|
def test_cheb2poly(self):
|
|
for i in range(10):
|
|
assert_almost_equal(cheb.cheb2poly([0] * i + [1]), Tlist[i])
|
|
|
|
def test_poly2cheb(self):
|
|
for i in range(10):
|
|
assert_almost_equal(cheb.poly2cheb(Tlist[i]), [0] * i + [1])
|
|
|
|
def test_weight(self):
|
|
x = np.linspace(-1, 1, 11)[1:-1]
|
|
tgt = 1. / (np.sqrt(1 + x) * np.sqrt(1 - x))
|
|
res = cheb.chebweight(x)
|
|
assert_almost_equal(res, tgt)
|
|
|
|
def test_chebpts1(self):
|
|
# test exceptions
|
|
assert_raises(ValueError, cheb.chebpts1, 1.5)
|
|
assert_raises(ValueError, cheb.chebpts1, 0)
|
|
|
|
# test points
|
|
tgt = [0]
|
|
assert_almost_equal(cheb.chebpts1(1), tgt)
|
|
tgt = [-0.70710678118654746, 0.70710678118654746]
|
|
assert_almost_equal(cheb.chebpts1(2), tgt)
|
|
tgt = [-0.86602540378443871, 0, 0.86602540378443871]
|
|
assert_almost_equal(cheb.chebpts1(3), tgt)
|
|
tgt = [-0.9238795325, -0.3826834323, 0.3826834323, 0.9238795325]
|
|
assert_almost_equal(cheb.chebpts1(4), tgt)
|
|
|
|
def test_chebpts2(self):
|
|
# test exceptions
|
|
assert_raises(ValueError, cheb.chebpts2, 1.5)
|
|
assert_raises(ValueError, cheb.chebpts2, 1)
|
|
|
|
# test points
|
|
tgt = [-1, 1]
|
|
assert_almost_equal(cheb.chebpts2(2), tgt)
|
|
tgt = [-1, 0, 1]
|
|
assert_almost_equal(cheb.chebpts2(3), tgt)
|
|
tgt = [-1, -0.5, .5, 1]
|
|
assert_almost_equal(cheb.chebpts2(4), tgt)
|
|
tgt = [-1.0, -0.707106781187, 0, 0.707106781187, 1.0]
|
|
assert_almost_equal(cheb.chebpts2(5), tgt)
|