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599 lines
23 KiB
Python
599 lines
23 KiB
Python
"""Tests for functions in special_matrices.py."""
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from __future__ import division, print_function, absolute_import
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import numpy as np
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from numpy import arange, add, 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._lib.six import xrange
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from scipy import fftpack
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from scipy.special import comb
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from scipy.linalg import (toeplitz, hankel, circulant, hadamard, leslie,
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companion, tri, triu, tril, kron, block_diag,
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helmert, hilbert, invhilbert, pascal, invpascal, dft)
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from numpy.linalg import cond
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def get_mat(n):
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data = arange(n)
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data = add.outer(data,data)
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return data
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class TestTri(object):
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def test_basic(self):
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assert_equal(tri(4),array([[1,0,0,0],
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[1,1,0,0],
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[1,1,1,0],
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[1,1,1,1]]))
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assert_equal(tri(4,dtype='f'),array([[1,0,0,0],
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[1,1,0,0],
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[1,1,1,0],
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[1,1,1,1]],'f'))
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def test_diag(self):
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assert_equal(tri(4,k=1),array([[1,1,0,0],
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[1,1,1,0],
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[1,1,1,1],
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[1,1,1,1]]))
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assert_equal(tri(4,k=-1),array([[0,0,0,0],
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[1,0,0,0],
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[1,1,0,0],
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[1,1,1,0]]))
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def test_2d(self):
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assert_equal(tri(4,3),array([[1,0,0],
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[1,1,0],
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[1,1,1],
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[1,1,1]]))
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assert_equal(tri(3,4),array([[1,0,0,0],
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[1,1,0,0],
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[1,1,1,0]]))
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def test_diag2d(self):
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assert_equal(tri(3,4,k=2),array([[1,1,1,0],
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[1,1,1,1],
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[1,1,1,1]]))
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assert_equal(tri(4,3,k=-2),array([[0,0,0],
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[0,0,0],
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[1,0,0],
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[1,1,0]]))
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class TestTril(object):
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def test_basic(self):
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a = (100*get_mat(5)).astype('l')
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b = a.copy()
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for k in range(5):
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for l in range(k+1,5):
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b[k,l] = 0
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assert_equal(tril(a),b)
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def test_diag(self):
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a = (100*get_mat(5)).astype('f')
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b = a.copy()
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for k in range(5):
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for l in range(k+3,5):
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b[k,l] = 0
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assert_equal(tril(a,k=2),b)
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b = a.copy()
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for k in range(5):
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for l in range(max((k-1,0)),5):
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b[k,l] = 0
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assert_equal(tril(a,k=-2),b)
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class TestTriu(object):
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def test_basic(self):
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a = (100*get_mat(5)).astype('l')
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b = a.copy()
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for k in range(5):
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for l in range(k+1,5):
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b[l,k] = 0
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assert_equal(triu(a),b)
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def test_diag(self):
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a = (100*get_mat(5)).astype('f')
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b = a.copy()
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for k in range(5):
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for l in range(max((k-1,0)),5):
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b[l,k] = 0
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assert_equal(triu(a,k=2),b)
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b = a.copy()
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for k in range(5):
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for l in range(k+3,5):
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b[l,k] = 0
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assert_equal(triu(a,k=-2),b)
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class TestToeplitz(object):
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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(object):
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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(object):
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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(object):
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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], [1,-1,1,-1], [1,1,-1,-1], [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(object):
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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([
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[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(object):
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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(object):
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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.ones((n, n)) / 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(object):
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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(object):
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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, 1367702848512,
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-8410422724704, 36616806420480, -115857864064800, 270465047424000,
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-468580694662080, 600545887119360, -561522320049600, 372133135180800,
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-165537539406000, 44316454993920, -5395297580640],
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[1976760, -426980160, 24337869120, -630981792000, 9228108708000,
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-85267724461920, 532660105897920, -2348052711713280, 7504429831470000,
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-17664748409880000, 30818191841236800, -39732544853164800,
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37341234283298400, -24857330514030000, 11100752642520000,
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-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, 79217210949138662400,
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-140362995650505067440, 183420385176741672960, -174433352415381259200,
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117339159519533952000, -52892422160973595200, 14328529177999196160,
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-1763080738699119840],
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[33374693352, -8410422724704, 532660105897920,
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-14914482965141760, 231343968720664800, -2239409617216035264,
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14527452132196331328, -66072377044391477760, 216799987176909536400,
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-521925895055522958000, 928414062734059661760, -1217424500995626443520,
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1161358898976091015200, -783401860847777371200, 354015418167362952000,
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-96120549902411274240, 11851820521255194480],
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[-143034400080, 36616806420480, -2348052711713280, 66409571644416000,
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-1038687206500944000, 10124803292907663360, -66072377044391477760,
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302045152202932469760, -995510145200094810000, 2405996923185123840000,
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-4294704507885446054400, 5649058909023744614400,
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|
-5403874060541811254400, 3654352703663101440000,
|
|
-1655137020003255360000, 450325202737117593600, -55630994283442749600],
|
|
[446982500250, -115857864064800, 7504429831470000, -214015221119700000,
|
|
3370739732635275000, -33052510749726468000, 216799987176909536400,
|
|
-995510145200094810000, 3293967392206196062500,
|
|
-7988661659013106500000, 14303908928401362270000,
|
|
-18866974090684772052000, 18093328327706957325000,
|
|
-12263364009096700500000, 5565847995255512250000,
|
|
-1517208935002984080000, 187754605706619279900],
|
|
[-1033026222800, 270465047424000, -17664748409880000,
|
|
507295338950400000, -8037460526495400000, 79217210949138662400,
|
|
-521925895055522958000, 2405996923185123840000,
|
|
-7988661659013106500000, 19434404971634224000000,
|
|
-34894474126569249192000, 46141453390504792320000,
|
|
-44349976506971935800000, 30121928988527376000000,
|
|
-13697025107665828500000, 3740200989399948902400,
|
|
-463591619028689580000],
|
|
[1774926873720, -468580694662080,
|
|
30818191841236800, -890303319857952000, 14178080368737885600,
|
|
-140362995650505067440, 928414062734059661760, -4294704507885446054400,
|
|
14303908928401362270000, -34894474126569249192000,
|
|
62810053427824648545600, -83243376594051600326400,
|
|
80177044485212743068000, -54558343880470209780000,
|
|
24851882355348879230400, -6797096028813368678400, 843736746632215035600],
|
|
[-2258997839280, 600545887119360, -39732544853164800,
|
|
1153715376477081600, -18454939322943942000, 183420385176741672960,
|
|
-1217424500995626443520, 5649058909023744614400,
|
|
-18866974090684772052000, 46141453390504792320000,
|
|
-83243376594051600326400, 110552468520163390156800,
|
|
-106681852579497947388000, 72720410752415168870400,
|
|
-33177973900974346080000, 9087761081682520473600,
|
|
-1129631016152221783200],
|
|
[2099709530100, -561522320049600, 37341234283298400,
|
|
-1089119333262870000, 17489975175339030000, -174433352415381259200,
|
|
1161358898976091015200, -5403874060541811254400,
|
|
18093328327706957325000, -44349976506971935800000,
|
|
80177044485212743068000, -106681852579497947388000,
|
|
103125790826848015808400, -70409051543137015800000,
|
|
32171029219823375700000, -8824053728865840192000,
|
|
1098252376814660067000],
|
|
[-1384423866000, 372133135180800,
|
|
-24857330514030000, 727848632044800000, -11728977435138600000,
|
|
117339159519533952000, -783401860847777371200, 3654352703663101440000,
|
|
-12263364009096700500000, 30121928988527376000000,
|
|
-54558343880470209780000, 72720410752415168870400,
|
|
-70409051543137015800000, 48142941226076592000000,
|
|
-22027500987368499000000, 6049545098753157120000,
|
|
-753830033789944188000],
|
|
[613101997800, -165537539406000,
|
|
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 xrange(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(object):
|
|
|
|
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_equal(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), 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 = fftpack.fft(x)
|
|
assert_array_almost_equal(mx, fx)
|
|
|