598 lines
23 KiB
Python
598 lines
23 KiB
Python
import pytest
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import numpy as np
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from numpy import arange, array, eye, copy, sqrt
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from numpy.testing import (assert_equal, assert_array_equal,
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assert_array_almost_equal, assert_allclose)
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from pytest import raises as assert_raises
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from scipy.fft import fft
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from scipy.special import comb
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from scipy.linalg import (toeplitz, hankel, circulant, hadamard, leslie, dft,
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companion, kron, block_diag,
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helmert, hilbert, invhilbert, pascal, invpascal,
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fiedler, fiedler_companion, eigvals,
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convolution_matrix)
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from numpy.linalg import cond
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class TestToeplitz:
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def test_basic(self):
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y = toeplitz([1, 2, 3])
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assert_array_equal(y, [[1, 2, 3], [2, 1, 2], [3, 2, 1]])
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y = toeplitz([1, 2, 3], [1, 4, 5])
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assert_array_equal(y, [[1, 4, 5], [2, 1, 4], [3, 2, 1]])
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def test_complex_01(self):
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data = (1.0 + arange(3.0)) * (1.0 + 1.0j)
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x = copy(data)
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t = toeplitz(x)
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# Calling toeplitz should not change x.
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assert_array_equal(x, data)
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# According to the docstring, x should be the first column of t.
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col0 = t[:, 0]
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assert_array_equal(col0, data)
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assert_array_equal(t[0, 1:], data[1:].conj())
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def test_scalar_00(self):
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"""Scalar arguments still produce a 2D array."""
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t = toeplitz(10)
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assert_array_equal(t, [[10]])
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t = toeplitz(10, 20)
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assert_array_equal(t, [[10]])
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def test_scalar_01(self):
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c = array([1, 2, 3])
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t = toeplitz(c, 1)
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assert_array_equal(t, [[1], [2], [3]])
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def test_scalar_02(self):
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c = array([1, 2, 3])
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t = toeplitz(c, array(1))
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assert_array_equal(t, [[1], [2], [3]])
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def test_scalar_03(self):
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c = array([1, 2, 3])
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t = toeplitz(c, array([1]))
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assert_array_equal(t, [[1], [2], [3]])
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def test_scalar_04(self):
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r = array([10, 2, 3])
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t = toeplitz(1, r)
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assert_array_equal(t, [[1, 2, 3]])
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class TestHankel:
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def test_basic(self):
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y = hankel([1, 2, 3])
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assert_array_equal(y, [[1, 2, 3], [2, 3, 0], [3, 0, 0]])
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y = hankel([1, 2, 3], [3, 4, 5])
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assert_array_equal(y, [[1, 2, 3], [2, 3, 4], [3, 4, 5]])
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class TestCirculant:
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def test_basic(self):
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y = circulant([1, 2, 3])
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assert_array_equal(y, [[1, 3, 2], [2, 1, 3], [3, 2, 1]])
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class TestHadamard:
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def test_basic(self):
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y = hadamard(1)
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assert_array_equal(y, [[1]])
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y = hadamard(2, dtype=float)
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assert_array_equal(y, [[1.0, 1.0], [1.0, -1.0]])
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y = hadamard(4)
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assert_array_equal(y, [[1, 1, 1, 1],
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[1, -1, 1, -1],
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[1, 1, -1, -1],
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[1, -1, -1, 1]])
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assert_raises(ValueError, hadamard, 0)
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assert_raises(ValueError, hadamard, 5)
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class TestLeslie:
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def test_bad_shapes(self):
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assert_raises(ValueError, leslie, [[1, 1], [2, 2]], [3, 4, 5])
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assert_raises(ValueError, leslie, [3, 4, 5], [[1, 1], [2, 2]])
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assert_raises(ValueError, leslie, [1, 2], [1, 2])
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assert_raises(ValueError, leslie, [1], [])
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def test_basic(self):
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a = leslie([1, 2, 3], [0.25, 0.5])
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expected = array([[1.0, 2.0, 3.0],
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[0.25, 0.0, 0.0],
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[0.0, 0.5, 0.0]])
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assert_array_equal(a, expected)
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class TestCompanion:
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def test_bad_shapes(self):
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assert_raises(ValueError, companion, [[1, 1], [2, 2]])
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assert_raises(ValueError, companion, [0, 4, 5])
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assert_raises(ValueError, companion, [1])
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assert_raises(ValueError, companion, [])
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def test_basic(self):
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c = companion([1, 2, 3])
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expected = array([
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[-2.0, -3.0],
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[1.0, 0.0]])
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assert_array_equal(c, expected)
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c = companion([2.0, 5.0, -10.0])
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expected = array([
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[-2.5, 5.0],
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[1.0, 0.0]])
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assert_array_equal(c, expected)
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class TestBlockDiag:
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def test_basic(self):
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x = block_diag(eye(2), [[1, 2], [3, 4], [5, 6]], [[1, 2, 3]])
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assert_array_equal(x, [[1, 0, 0, 0, 0, 0, 0],
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[0, 1, 0, 0, 0, 0, 0],
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[0, 0, 1, 2, 0, 0, 0],
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[0, 0, 3, 4, 0, 0, 0],
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[0, 0, 5, 6, 0, 0, 0],
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[0, 0, 0, 0, 1, 2, 3]])
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def test_dtype(self):
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x = block_diag([[1.5]])
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assert_equal(x.dtype, float)
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x = block_diag([[True]])
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assert_equal(x.dtype, bool)
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def test_mixed_dtypes(self):
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actual = block_diag([[1]], [[1j]])
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desired = np.array([[1, 0], [0, 1j]])
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assert_array_equal(actual, desired)
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def test_scalar_and_1d_args(self):
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a = block_diag(1)
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assert_equal(a.shape, (1, 1))
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assert_array_equal(a, [[1]])
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a = block_diag([2, 3], 4)
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assert_array_equal(a, [[2, 3, 0], [0, 0, 4]])
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def test_bad_arg(self):
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assert_raises(ValueError, block_diag, [[[1]]])
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def test_no_args(self):
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a = block_diag()
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assert_equal(a.ndim, 2)
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assert_equal(a.nbytes, 0)
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def test_empty_matrix_arg(self):
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# regression test for gh-4596: check the shape of the result
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# for empty matrix inputs. Empty matrices are no longer ignored
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# (gh-4908) it is viewed as a shape (1, 0) matrix.
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a = block_diag([[1, 0], [0, 1]],
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[],
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[[2, 3], [4, 5], [6, 7]])
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assert_array_equal(a, [[1, 0, 0, 0],
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[0, 1, 0, 0],
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[0, 0, 0, 0],
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[0, 0, 2, 3],
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[0, 0, 4, 5],
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[0, 0, 6, 7]])
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def test_zerosized_matrix_arg(self):
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# test for gh-4908: check the shape of the result for
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# zero-sized matrix inputs, i.e. matrices with shape (0,n) or (n,0).
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# note that [[]] takes shape (1,0)
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a = block_diag([[1, 0], [0, 1]],
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[[]],
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[[2, 3], [4, 5], [6, 7]],
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np.zeros([0, 2], dtype='int32'))
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assert_array_equal(a, [[1, 0, 0, 0, 0, 0],
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[0, 1, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0],
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[0, 0, 2, 3, 0, 0],
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[0, 0, 4, 5, 0, 0],
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[0, 0, 6, 7, 0, 0]])
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class TestKron:
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def test_basic(self):
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a = kron(array([[1, 2], [3, 4]]), array([[1, 1, 1]]))
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assert_array_equal(a, array([[1, 1, 1, 2, 2, 2],
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[3, 3, 3, 4, 4, 4]]))
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m1 = array([[1, 2], [3, 4]])
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m2 = array([[10], [11]])
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a = kron(m1, m2)
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expected = array([[10, 20],
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[11, 22],
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[30, 40],
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[33, 44]])
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assert_array_equal(a, expected)
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class TestHelmert:
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def test_orthogonality(self):
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for n in range(1, 7):
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H = helmert(n, full=True)
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Id = np.eye(n)
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assert_allclose(H.dot(H.T), Id, atol=1e-12)
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assert_allclose(H.T.dot(H), Id, atol=1e-12)
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def test_subspace(self):
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for n in range(2, 7):
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H_full = helmert(n, full=True)
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H_partial = helmert(n)
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for U in H_full[1:, :].T, H_partial.T:
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C = np.eye(n) - np.full((n, n), 1 / n)
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assert_allclose(U.dot(U.T), C)
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assert_allclose(U.T.dot(U), np.eye(n-1), atol=1e-12)
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class TestHilbert:
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def test_basic(self):
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h3 = array([[1.0, 1/2., 1/3.],
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[1/2., 1/3., 1/4.],
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[1/3., 1/4., 1/5.]])
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assert_array_almost_equal(hilbert(3), h3)
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assert_array_equal(hilbert(1), [[1.0]])
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h0 = hilbert(0)
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assert_equal(h0.shape, (0, 0))
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class TestInvHilbert:
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def test_basic(self):
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invh1 = array([[1]])
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assert_array_equal(invhilbert(1, exact=True), invh1)
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assert_array_equal(invhilbert(1), invh1)
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invh2 = array([[4, -6],
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[-6, 12]])
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assert_array_equal(invhilbert(2, exact=True), invh2)
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assert_array_almost_equal(invhilbert(2), invh2)
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invh3 = array([[9, -36, 30],
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[-36, 192, -180],
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[30, -180, 180]])
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assert_array_equal(invhilbert(3, exact=True), invh3)
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assert_array_almost_equal(invhilbert(3), invh3)
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invh4 = array([[16, -120, 240, -140],
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[-120, 1200, -2700, 1680],
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[240, -2700, 6480, -4200],
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[-140, 1680, -4200, 2800]])
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assert_array_equal(invhilbert(4, exact=True), invh4)
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assert_array_almost_equal(invhilbert(4), invh4)
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invh5 = array([[25, -300, 1050, -1400, 630],
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[-300, 4800, -18900, 26880, -12600],
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[1050, -18900, 79380, -117600, 56700],
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[-1400, 26880, -117600, 179200, -88200],
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[630, -12600, 56700, -88200, 44100]])
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assert_array_equal(invhilbert(5, exact=True), invh5)
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assert_array_almost_equal(invhilbert(5), invh5)
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invh17 = array([
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[289, -41616, 1976760, -46124400, 629598060, -5540462928,
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33374693352, -143034400080, 446982500250, -1033026222800,
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1774926873720, -2258997839280, 2099709530100, -1384423866000,
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613101997800, -163493866080, 19835652870],
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[-41616, 7990272, -426980160, 10627061760, -151103534400,
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1367702848512, -8410422724704, 36616806420480, -115857864064800,
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270465047424000, -468580694662080, 600545887119360,
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-561522320049600, 372133135180800, -165537539406000,
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44316454993920, -5395297580640],
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[1976760, -426980160, 24337869120, -630981792000, 9228108708000,
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-85267724461920, 532660105897920, -2348052711713280,
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7504429831470000, -17664748409880000, 30818191841236800,
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-39732544853164800, 37341234283298400, -24857330514030000,
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11100752642520000, -2982128117299200, 364182586693200],
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[-46124400, 10627061760, -630981792000, 16826181120000,
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-251209625940000, 2358021022156800, -14914482965141760,
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66409571644416000, -214015221119700000, 507295338950400000,
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-890303319857952000, 1153715376477081600, -1089119333262870000,
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727848632044800000, -326170262829600000, 87894302404608000,
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-10763618673376800],
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[629598060, -151103534400, 9228108708000,
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-251209625940000, 3810012660090000, -36210360321495360,
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231343968720664800, -1038687206500944000, 3370739732635275000,
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-8037460526495400000, 14178080368737885600, -18454939322943942000,
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17489975175339030000, -11728977435138600000, 5272370630081100000,
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-1424711708039692800, 174908803442373000],
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[-5540462928, 1367702848512, -85267724461920, 2358021022156800,
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-36210360321495360, 347619459086355456, -2239409617216035264,
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10124803292907663360, -33052510749726468000,
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79217210949138662400, -140362995650505067440,
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183420385176741672960, -174433352415381259200,
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117339159519533952000, -52892422160973595200,
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14328529177999196160, -1763080738699119840],
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[33374693352, -8410422724704, 532660105897920,
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-14914482965141760, 231343968720664800, -2239409617216035264,
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14527452132196331328, -66072377044391477760,
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216799987176909536400, -521925895055522958000,
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928414062734059661760, -1217424500995626443520,
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1161358898976091015200, -783401860847777371200,
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354015418167362952000, -96120549902411274240,
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11851820521255194480],
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[-143034400080, 36616806420480, -2348052711713280,
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66409571644416000, -1038687206500944000, 10124803292907663360,
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-66072377044391477760, 302045152202932469760,
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-995510145200094810000, 2405996923185123840000,
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-4294704507885446054400, 5649058909023744614400,
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-5403874060541811254400, 3654352703663101440000,
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-1655137020003255360000, 450325202737117593600,
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-55630994283442749600],
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[446982500250, -115857864064800, 7504429831470000,
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-214015221119700000, 3370739732635275000, -33052510749726468000,
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216799987176909536400, -995510145200094810000,
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3293967392206196062500, -7988661659013106500000,
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14303908928401362270000, -18866974090684772052000,
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18093328327706957325000, -12263364009096700500000,
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5565847995255512250000, -1517208935002984080000,
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187754605706619279900],
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[-1033026222800, 270465047424000, -17664748409880000,
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507295338950400000, -8037460526495400000, 79217210949138662400,
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-521925895055522958000, 2405996923185123840000,
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-7988661659013106500000, 19434404971634224000000,
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-34894474126569249192000, 46141453390504792320000,
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-44349976506971935800000, 30121928988527376000000,
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-13697025107665828500000, 3740200989399948902400,
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-463591619028689580000],
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[1774926873720, -468580694662080,
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30818191841236800, -890303319857952000, 14178080368737885600,
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-140362995650505067440, 928414062734059661760,
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-4294704507885446054400, 14303908928401362270000,
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-34894474126569249192000, 62810053427824648545600,
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-83243376594051600326400, 80177044485212743068000,
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-54558343880470209780000, 24851882355348879230400,
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-6797096028813368678400, 843736746632215035600],
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[-2258997839280, 600545887119360, -39732544853164800,
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1153715376477081600, -18454939322943942000, 183420385176741672960,
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-1217424500995626443520, 5649058909023744614400,
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-18866974090684772052000, 46141453390504792320000,
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-83243376594051600326400, 110552468520163390156800,
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-106681852579497947388000, 72720410752415168870400,
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-33177973900974346080000, 9087761081682520473600,
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-1129631016152221783200],
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[2099709530100, -561522320049600, 37341234283298400,
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-1089119333262870000, 17489975175339030000,
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-174433352415381259200, 1161358898976091015200,
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-5403874060541811254400, 18093328327706957325000,
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-44349976506971935800000, 80177044485212743068000,
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-106681852579497947388000, 103125790826848015808400,
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-70409051543137015800000, 32171029219823375700000,
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-8824053728865840192000, 1098252376814660067000],
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[-1384423866000, 372133135180800,
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-24857330514030000, 727848632044800000, -11728977435138600000,
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117339159519533952000, -783401860847777371200,
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3654352703663101440000, -12263364009096700500000,
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30121928988527376000000, -54558343880470209780000,
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72720410752415168870400, -70409051543137015800000,
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48142941226076592000000, -22027500987368499000000,
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6049545098753157120000, -753830033789944188000],
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[613101997800, -165537539406000,
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11100752642520000, -326170262829600000, 5272370630081100000,
|
|
-52892422160973595200, 354015418167362952000,
|
|
-1655137020003255360000, 5565847995255512250000,
|
|
-13697025107665828500000, 24851882355348879230400,
|
|
-33177973900974346080000, 32171029219823375700000,
|
|
-22027500987368499000000, 10091416708498869000000,
|
|
-2774765838662800128000, 346146444087219270000],
|
|
[-163493866080, 44316454993920, -2982128117299200,
|
|
87894302404608000, -1424711708039692800,
|
|
14328529177999196160, -96120549902411274240,
|
|
450325202737117593600, -1517208935002984080000,
|
|
3740200989399948902400, -6797096028813368678400,
|
|
9087761081682520473600, -8824053728865840192000,
|
|
6049545098753157120000, -2774765838662800128000,
|
|
763806510427609497600, -95382575704033754400],
|
|
[19835652870, -5395297580640, 364182586693200, -10763618673376800,
|
|
174908803442373000, -1763080738699119840, 11851820521255194480,
|
|
-55630994283442749600, 187754605706619279900,
|
|
-463591619028689580000, 843736746632215035600,
|
|
-1129631016152221783200, 1098252376814660067000,
|
|
-753830033789944188000, 346146444087219270000,
|
|
-95382575704033754400, 11922821963004219300]
|
|
])
|
|
assert_array_equal(invhilbert(17, exact=True), invh17)
|
|
assert_allclose(invhilbert(17), invh17.astype(float), rtol=1e-12)
|
|
|
|
def test_inverse(self):
|
|
for n in range(1, 10):
|
|
a = hilbert(n)
|
|
b = invhilbert(n)
|
|
# The Hilbert matrix is increasingly badly conditioned,
|
|
# so take that into account in the test
|
|
c = cond(a)
|
|
assert_allclose(a.dot(b), eye(n), atol=1e-15*c, rtol=1e-15*c)
|
|
|
|
|
|
class TestPascal:
|
|
|
|
cases = [
|
|
(1, array([[1]]), array([[1]])),
|
|
(2, array([[1, 1],
|
|
[1, 2]]),
|
|
array([[1, 0],
|
|
[1, 1]])),
|
|
(3, array([[1, 1, 1],
|
|
[1, 2, 3],
|
|
[1, 3, 6]]),
|
|
array([[1, 0, 0],
|
|
[1, 1, 0],
|
|
[1, 2, 1]])),
|
|
(4, array([[1, 1, 1, 1],
|
|
[1, 2, 3, 4],
|
|
[1, 3, 6, 10],
|
|
[1, 4, 10, 20]]),
|
|
array([[1, 0, 0, 0],
|
|
[1, 1, 0, 0],
|
|
[1, 2, 1, 0],
|
|
[1, 3, 3, 1]])),
|
|
]
|
|
|
|
def check_case(self, n, sym, low):
|
|
assert_array_equal(pascal(n), sym)
|
|
assert_array_equal(pascal(n, kind='lower'), low)
|
|
assert_array_equal(pascal(n, kind='upper'), low.T)
|
|
assert_array_almost_equal(pascal(n, exact=False), sym)
|
|
assert_array_almost_equal(pascal(n, exact=False, kind='lower'), low)
|
|
assert_array_almost_equal(pascal(n, exact=False, kind='upper'), low.T)
|
|
|
|
def test_cases(self):
|
|
for n, sym, low in self.cases:
|
|
self.check_case(n, sym, low)
|
|
|
|
def test_big(self):
|
|
p = pascal(50)
|
|
assert p[-1, -1] == comb(98, 49, exact=True)
|
|
|
|
def test_threshold(self):
|
|
# Regression test. An early version of `pascal` returned an
|
|
# array of type np.uint64 for n=35, but that data type is too small
|
|
# to hold p[-1, -1]. The second assert_equal below would fail
|
|
# because p[-1, -1] overflowed.
|
|
p = pascal(34)
|
|
assert_equal(2*p.item(-1, -2), p.item(-1, -1), err_msg="n = 34")
|
|
p = pascal(35)
|
|
assert_equal(2.*p.item(-1, -2), 1.*p.item(-1, -1), err_msg="n = 35")
|
|
|
|
|
|
def test_invpascal():
|
|
|
|
def check_invpascal(n, kind, exact):
|
|
ip = invpascal(n, kind=kind, exact=exact)
|
|
p = pascal(n, kind=kind, exact=exact)
|
|
# Matrix-multiply ip and p, and check that we get the identity matrix.
|
|
# We can't use the simple expression e = ip.dot(p), because when
|
|
# n < 35 and exact is True, p.dtype is np.uint64 and ip.dtype is
|
|
# np.int64. The product of those dtypes is np.float64, which loses
|
|
# precision when n is greater than 18. Instead we'll cast both to
|
|
# object arrays, and then multiply.
|
|
e = ip.astype(object).dot(p.astype(object))
|
|
assert_array_equal(e, eye(n), err_msg="n=%d kind=%r exact=%r" %
|
|
(n, kind, exact))
|
|
|
|
kinds = ['symmetric', 'lower', 'upper']
|
|
|
|
ns = [1, 2, 5, 18]
|
|
for n in ns:
|
|
for kind in kinds:
|
|
for exact in [True, False]:
|
|
check_invpascal(n, kind, exact)
|
|
|
|
ns = [19, 34, 35, 50]
|
|
for n in ns:
|
|
for kind in kinds:
|
|
check_invpascal(n, kind, True)
|
|
|
|
|
|
def test_dft():
|
|
m = dft(2)
|
|
expected = array([[1.0, 1.0], [1.0, -1.0]])
|
|
assert_array_almost_equal(m, expected)
|
|
m = dft(2, scale='n')
|
|
assert_array_almost_equal(m, expected/2.0)
|
|
m = dft(2, scale='sqrtn')
|
|
assert_array_almost_equal(m, expected/sqrt(2.0))
|
|
|
|
x = array([0, 1, 2, 3, 4, 5, 0, 1])
|
|
m = dft(8)
|
|
mx = m.dot(x)
|
|
fx = fft(x)
|
|
assert_array_almost_equal(mx, fx)
|
|
|
|
|
|
def test_fiedler():
|
|
f = fiedler([])
|
|
assert_equal(f.size, 0)
|
|
f = fiedler([123.])
|
|
assert_array_equal(f, np.array([[0.]]))
|
|
f = fiedler(np.arange(1, 7))
|
|
des = np.array([[0, 1, 2, 3, 4, 5],
|
|
[1, 0, 1, 2, 3, 4],
|
|
[2, 1, 0, 1, 2, 3],
|
|
[3, 2, 1, 0, 1, 2],
|
|
[4, 3, 2, 1, 0, 1],
|
|
[5, 4, 3, 2, 1, 0]])
|
|
assert_array_equal(f, des)
|
|
|
|
|
|
def test_fiedler_companion():
|
|
fc = fiedler_companion([])
|
|
assert_equal(fc.size, 0)
|
|
fc = fiedler_companion([1.])
|
|
assert_equal(fc.size, 0)
|
|
fc = fiedler_companion([1., 2.])
|
|
assert_array_equal(fc, np.array([[-2.]]))
|
|
fc = fiedler_companion([1e-12, 2., 3.])
|
|
assert_array_almost_equal(fc, companion([1e-12, 2., 3.]))
|
|
with assert_raises(ValueError):
|
|
fiedler_companion([0, 1, 2])
|
|
fc = fiedler_companion([1., -16., 86., -176., 105.])
|
|
assert_array_almost_equal(eigvals(fc),
|
|
np.array([7., 5., 3., 1.]))
|
|
|
|
|
|
class TestConvolutionMatrix:
|
|
"""
|
|
Test convolution_matrix vs. numpy.convolve for various parameters.
|
|
"""
|
|
|
|
def create_vector(self, n, cpx):
|
|
"""Make a complex or real test vector of length n."""
|
|
x = np.linspace(-2.5, 2.2, n)
|
|
if cpx:
|
|
x = x + 1j*np.linspace(-1.5, 3.1, n)
|
|
return x
|
|
|
|
def test_bad_n(self):
|
|
# n must be a positive integer
|
|
with pytest.raises(ValueError, match='n must be a positive integer'):
|
|
convolution_matrix([1, 2, 3], 0)
|
|
|
|
def test_bad_first_arg(self):
|
|
# first arg must be a 1d array, otherwise ValueError
|
|
with pytest.raises(ValueError, match='one-dimensional'):
|
|
convolution_matrix(1, 4)
|
|
|
|
def test_empty_first_arg(self):
|
|
# first arg must have at least one value
|
|
with pytest.raises(ValueError, match=r'len\(a\)'):
|
|
convolution_matrix([], 4)
|
|
|
|
def test_bad_mode(self):
|
|
# mode must be in ('full', 'valid', 'same')
|
|
with pytest.raises(ValueError, match='mode.*must be one of'):
|
|
convolution_matrix((1, 1), 4, mode='invalid argument')
|
|
|
|
@pytest.mark.parametrize('cpx', [False, True])
|
|
@pytest.mark.parametrize('na', [1, 2, 9])
|
|
@pytest.mark.parametrize('nv', [1, 2, 9])
|
|
@pytest.mark.parametrize('mode', [None, 'full', 'valid', 'same'])
|
|
def test_against_numpy_convolve(self, cpx, na, nv, mode):
|
|
a = self.create_vector(na, cpx)
|
|
v = self.create_vector(nv, cpx)
|
|
if mode is None:
|
|
y1 = np.convolve(v, a)
|
|
A = convolution_matrix(a, nv)
|
|
else:
|
|
y1 = np.convolve(v, a, mode)
|
|
A = convolution_matrix(a, nv, mode)
|
|
y2 = A @ v
|
|
assert_array_almost_equal(y1, y2)
|