1186 lines
38 KiB
Python
1186 lines
38 KiB
Python
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from sympy.assumptions import Q
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from sympy.core.expr import Expr
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from sympy.core.add import Add
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from sympy.core.function import Function
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from sympy.core.kind import NumberKind, UndefinedKind
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from sympy.core.numbers import I, Integer, oo, pi, Rational
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from sympy.core.singleton import S
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from sympy.core.symbol import Symbol, symbols
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from sympy.functions.elementary.complexes import Abs
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import cos, sin
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from sympy.matrices.common import (ShapeError, NonSquareMatrixError,
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_MinimalMatrix, _CastableMatrix, MatrixShaping, MatrixProperties,
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MatrixOperations, MatrixArithmetic, MatrixSpecial, MatrixKind)
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from sympy.matrices.matrices import MatrixCalculus
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from sympy.matrices import (Matrix, diag, eye,
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matrix_multiply_elementwise, ones, zeros, SparseMatrix, banded,
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MutableDenseMatrix, MutableSparseMatrix, ImmutableDenseMatrix,
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ImmutableSparseMatrix)
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from sympy.polys.polytools import Poly
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from sympy.utilities.iterables import flatten
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from sympy.testing.pytest import raises, XFAIL
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from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray as Array
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from sympy.abc import x, y, z
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# classes to test the basic matrix classes
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class ShapingOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixShaping):
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pass
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def eye_Shaping(n):
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return ShapingOnlyMatrix(n, n, lambda i, j: int(i == j))
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def zeros_Shaping(n):
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return ShapingOnlyMatrix(n, n, lambda i, j: 0)
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class PropertiesOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixProperties):
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pass
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def eye_Properties(n):
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return PropertiesOnlyMatrix(n, n, lambda i, j: int(i == j))
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def zeros_Properties(n):
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return PropertiesOnlyMatrix(n, n, lambda i, j: 0)
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class OperationsOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixOperations):
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pass
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def eye_Operations(n):
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return OperationsOnlyMatrix(n, n, lambda i, j: int(i == j))
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def zeros_Operations(n):
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return OperationsOnlyMatrix(n, n, lambda i, j: 0)
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class ArithmeticOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixArithmetic):
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pass
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def eye_Arithmetic(n):
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return ArithmeticOnlyMatrix(n, n, lambda i, j: int(i == j))
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def zeros_Arithmetic(n):
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return ArithmeticOnlyMatrix(n, n, lambda i, j: 0)
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class SpecialOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixSpecial):
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pass
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class CalculusOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixCalculus):
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pass
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def test__MinimalMatrix():
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x = _MinimalMatrix(2, 3, [1, 2, 3, 4, 5, 6])
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assert x.rows == 2
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assert x.cols == 3
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assert x[2] == 3
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assert x[1, 1] == 5
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assert list(x) == [1, 2, 3, 4, 5, 6]
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assert list(x[1, :]) == [4, 5, 6]
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assert list(x[:, 1]) == [2, 5]
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assert list(x[:, :]) == list(x)
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assert x[:, :] == x
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assert _MinimalMatrix(x) == x
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assert _MinimalMatrix([[1, 2, 3], [4, 5, 6]]) == x
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assert _MinimalMatrix(([1, 2, 3], [4, 5, 6])) == x
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assert _MinimalMatrix([(1, 2, 3), (4, 5, 6)]) == x
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assert _MinimalMatrix(((1, 2, 3), (4, 5, 6))) == x
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assert not (_MinimalMatrix([[1, 2], [3, 4], [5, 6]]) == x)
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def test_kind():
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assert Matrix([[1, 2], [3, 4]]).kind == MatrixKind(NumberKind)
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assert Matrix([[0, 0], [0, 0]]).kind == MatrixKind(NumberKind)
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assert Matrix(0, 0, []).kind == MatrixKind(NumberKind)
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assert Matrix([[x]]).kind == MatrixKind(NumberKind)
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assert Matrix([[1, Matrix([[1]])]]).kind == MatrixKind(UndefinedKind)
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assert SparseMatrix([[1]]).kind == MatrixKind(NumberKind)
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assert SparseMatrix([[1, Matrix([[1]])]]).kind == MatrixKind(UndefinedKind)
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# ShapingOnlyMatrix tests
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def test_vec():
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m = ShapingOnlyMatrix(2, 2, [1, 3, 2, 4])
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m_vec = m.vec()
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assert m_vec.cols == 1
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for i in range(4):
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assert m_vec[i] == i + 1
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def test_todok():
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a, b, c, d = symbols('a:d')
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m1 = MutableDenseMatrix([[a, b], [c, d]])
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m2 = ImmutableDenseMatrix([[a, b], [c, d]])
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m3 = MutableSparseMatrix([[a, b], [c, d]])
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m4 = ImmutableSparseMatrix([[a, b], [c, d]])
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assert m1.todok() == m2.todok() == m3.todok() == m4.todok() == \
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{(0, 0): a, (0, 1): b, (1, 0): c, (1, 1): d}
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def test_tolist():
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lst = [[S.One, S.Half, x*y, S.Zero], [x, y, z, x**2], [y, -S.One, z*x, 3]]
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flat_lst = [S.One, S.Half, x*y, S.Zero, x, y, z, x**2, y, -S.One, z*x, 3]
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m = ShapingOnlyMatrix(3, 4, flat_lst)
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assert m.tolist() == lst
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def test_todod():
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m = ShapingOnlyMatrix(3, 2, [[S.One, 0], [0, S.Half], [x, 0]])
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dict = {0: {0: S.One}, 1: {1: S.Half}, 2: {0: x}}
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assert m.todod() == dict
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def test_row_col_del():
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e = ShapingOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
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raises(IndexError, lambda: e.row_del(5))
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raises(IndexError, lambda: e.row_del(-5))
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raises(IndexError, lambda: e.col_del(5))
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raises(IndexError, lambda: e.col_del(-5))
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assert e.row_del(2) == e.row_del(-1) == Matrix([[1, 2, 3], [4, 5, 6]])
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assert e.col_del(2) == e.col_del(-1) == Matrix([[1, 2], [4, 5], [7, 8]])
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assert e.row_del(1) == e.row_del(-2) == Matrix([[1, 2, 3], [7, 8, 9]])
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assert e.col_del(1) == e.col_del(-2) == Matrix([[1, 3], [4, 6], [7, 9]])
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def test_get_diag_blocks1():
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a = Matrix([[1, 2], [2, 3]])
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b = Matrix([[3, x], [y, 3]])
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c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
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assert a.get_diag_blocks() == [a]
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assert b.get_diag_blocks() == [b]
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assert c.get_diag_blocks() == [c]
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def test_get_diag_blocks2():
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a = Matrix([[1, 2], [2, 3]])
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b = Matrix([[3, x], [y, 3]])
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c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
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A, B, C, D = diag(a, b, b), diag(a, b, c), diag(a, c, b), diag(c, c, b)
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A = ShapingOnlyMatrix(A.rows, A.cols, A)
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B = ShapingOnlyMatrix(B.rows, B.cols, B)
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C = ShapingOnlyMatrix(C.rows, C.cols, C)
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D = ShapingOnlyMatrix(D.rows, D.cols, D)
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assert A.get_diag_blocks() == [a, b, b]
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assert B.get_diag_blocks() == [a, b, c]
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assert C.get_diag_blocks() == [a, c, b]
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assert D.get_diag_blocks() == [c, c, b]
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def test_shape():
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m = ShapingOnlyMatrix(1, 2, [0, 0])
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assert m.shape == (1, 2)
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def test_reshape():
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m0 = eye_Shaping(3)
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assert m0.reshape(1, 9) == Matrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
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m1 = ShapingOnlyMatrix(3, 4, lambda i, j: i + j)
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assert m1.reshape(
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4, 3) == Matrix(((0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)))
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assert m1.reshape(2, 6) == Matrix(((0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)))
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def test_row_col():
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m = ShapingOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
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assert m.row(0) == Matrix(1, 3, [1, 2, 3])
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assert m.col(0) == Matrix(3, 1, [1, 4, 7])
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def test_row_join():
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assert eye_Shaping(3).row_join(Matrix([7, 7, 7])) == \
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Matrix([[1, 0, 0, 7],
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[0, 1, 0, 7],
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[0, 0, 1, 7]])
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def test_col_join():
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assert eye_Shaping(3).col_join(Matrix([[7, 7, 7]])) == \
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Matrix([[1, 0, 0],
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[0, 1, 0],
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[0, 0, 1],
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[7, 7, 7]])
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def test_row_insert():
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r4 = Matrix([[4, 4, 4]])
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for i in range(-4, 5):
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l = [1, 0, 0]
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l.insert(i, 4)
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assert flatten(eye_Shaping(3).row_insert(i, r4).col(0).tolist()) == l
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def test_col_insert():
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c4 = Matrix([4, 4, 4])
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for i in range(-4, 5):
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l = [0, 0, 0]
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l.insert(i, 4)
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assert flatten(zeros_Shaping(3).col_insert(i, c4).row(0).tolist()) == l
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# issue 13643
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assert eye_Shaping(6).col_insert(3, Matrix([[2, 2], [2, 2], [2, 2], [2, 2], [2, 2], [2, 2]])) == \
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Matrix([[1, 0, 0, 2, 2, 0, 0, 0],
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[0, 1, 0, 2, 2, 0, 0, 0],
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[0, 0, 1, 2, 2, 0, 0, 0],
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[0, 0, 0, 2, 2, 1, 0, 0],
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[0, 0, 0, 2, 2, 0, 1, 0],
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[0, 0, 0, 2, 2, 0, 0, 1]])
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def test_extract():
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m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
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assert m.extract([0, 1, 3], [0, 1]) == Matrix(3, 2, [0, 1, 3, 4, 9, 10])
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assert m.extract([0, 3], [0, 0, 2]) == Matrix(2, 3, [0, 0, 2, 9, 9, 11])
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assert m.extract(range(4), range(3)) == m
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raises(IndexError, lambda: m.extract([4], [0]))
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raises(IndexError, lambda: m.extract([0], [3]))
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def test_hstack():
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m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
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m2 = ShapingOnlyMatrix(3, 4, lambda i, j: i*3 + j)
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assert m == m.hstack(m)
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assert m.hstack(m, m, m) == ShapingOnlyMatrix.hstack(m, m, m) == Matrix([
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[0, 1, 2, 0, 1, 2, 0, 1, 2],
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[3, 4, 5, 3, 4, 5, 3, 4, 5],
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[6, 7, 8, 6, 7, 8, 6, 7, 8],
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[9, 10, 11, 9, 10, 11, 9, 10, 11]])
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raises(ShapeError, lambda: m.hstack(m, m2))
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assert Matrix.hstack() == Matrix()
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# test regression #12938
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M1 = Matrix.zeros(0, 0)
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M2 = Matrix.zeros(0, 1)
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M3 = Matrix.zeros(0, 2)
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M4 = Matrix.zeros(0, 3)
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m = ShapingOnlyMatrix.hstack(M1, M2, M3, M4)
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assert m.rows == 0 and m.cols == 6
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def test_vstack():
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m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
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m2 = ShapingOnlyMatrix(3, 4, lambda i, j: i*3 + j)
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assert m == m.vstack(m)
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assert m.vstack(m, m, m) == ShapingOnlyMatrix.vstack(m, m, m) == Matrix([
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[0, 1, 2],
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[3, 4, 5],
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[6, 7, 8],
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[9, 10, 11],
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[0, 1, 2],
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[3, 4, 5],
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[6, 7, 8],
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[9, 10, 11],
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[0, 1, 2],
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[3, 4, 5],
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[6, 7, 8],
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[9, 10, 11]])
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raises(ShapeError, lambda: m.vstack(m, m2))
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assert Matrix.vstack() == Matrix()
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# PropertiesOnlyMatrix tests
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def test_atoms():
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m = PropertiesOnlyMatrix(2, 2, [1, 2, x, 1 - 1/x])
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assert m.atoms() == {S.One, S(2), S.NegativeOne, x}
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assert m.atoms(Symbol) == {x}
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def test_free_symbols():
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assert PropertiesOnlyMatrix([[x], [0]]).free_symbols == {x}
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def test_has():
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A = PropertiesOnlyMatrix(((x, y), (2, 3)))
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assert A.has(x)
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assert not A.has(z)
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assert A.has(Symbol)
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A = PropertiesOnlyMatrix(((2, y), (2, 3)))
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assert not A.has(x)
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def test_is_anti_symmetric():
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x = symbols('x')
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assert PropertiesOnlyMatrix(2, 1, [1, 2]).is_anti_symmetric() is False
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m = PropertiesOnlyMatrix(3, 3, [0, x**2 + 2*x + 1, y, -(x + 1)**2, 0, x*y, -y, -x*y, 0])
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assert m.is_anti_symmetric() is True
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assert m.is_anti_symmetric(simplify=False) is False
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assert m.is_anti_symmetric(simplify=lambda x: x) is False
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m = PropertiesOnlyMatrix(3, 3, [x.expand() for x in m])
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assert m.is_anti_symmetric(simplify=False) is True
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m = PropertiesOnlyMatrix(3, 3, [x.expand() for x in [S.One] + list(m)[1:]])
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|
|
assert m.is_anti_symmetric() is False
|
||
|
|
|
||
|
|
|
||
|
|
def test_diagonal_symmetrical():
|
||
|
|
m = PropertiesOnlyMatrix(2, 2, [0, 1, 1, 0])
|
||
|
|
assert not m.is_diagonal()
|
||
|
|
assert m.is_symmetric()
|
||
|
|
assert m.is_symmetric(simplify=False)
|
||
|
|
|
||
|
|
m = PropertiesOnlyMatrix(2, 2, [1, 0, 0, 1])
|
||
|
|
assert m.is_diagonal()
|
||
|
|
|
||
|
|
m = PropertiesOnlyMatrix(3, 3, diag(1, 2, 3))
|
||
|
|
assert m.is_diagonal()
|
||
|
|
assert m.is_symmetric()
|
||
|
|
|
||
|
|
m = PropertiesOnlyMatrix(3, 3, [1, 0, 0, 0, 2, 0, 0, 0, 3])
|
||
|
|
assert m == diag(1, 2, 3)
|
||
|
|
|
||
|
|
m = PropertiesOnlyMatrix(2, 3, zeros(2, 3))
|
||
|
|
assert not m.is_symmetric()
|
||
|
|
assert m.is_diagonal()
|
||
|
|
|
||
|
|
m = PropertiesOnlyMatrix(((5, 0), (0, 6), (0, 0)))
|
||
|
|
assert m.is_diagonal()
|
||
|
|
|
||
|
|
m = PropertiesOnlyMatrix(((5, 0, 0), (0, 6, 0)))
|
||
|
|
assert m.is_diagonal()
|
||
|
|
|
||
|
|
m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2, 2, 0, y, 0, 3])
|
||
|
|
assert m.is_symmetric()
|
||
|
|
assert not m.is_symmetric(simplify=False)
|
||
|
|
assert m.expand().is_symmetric(simplify=False)
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_hermitian():
|
||
|
|
a = PropertiesOnlyMatrix([[1, I], [-I, 1]])
|
||
|
|
assert a.is_hermitian
|
||
|
|
a = PropertiesOnlyMatrix([[2*I, I], [-I, 1]])
|
||
|
|
assert a.is_hermitian is False
|
||
|
|
a = PropertiesOnlyMatrix([[x, I], [-I, 1]])
|
||
|
|
assert a.is_hermitian is None
|
||
|
|
a = PropertiesOnlyMatrix([[x, 1], [-I, 1]])
|
||
|
|
assert a.is_hermitian is False
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_Identity():
|
||
|
|
assert eye_Properties(3).is_Identity
|
||
|
|
assert not PropertiesOnlyMatrix(zeros(3)).is_Identity
|
||
|
|
assert not PropertiesOnlyMatrix(ones(3)).is_Identity
|
||
|
|
# issue 6242
|
||
|
|
assert not PropertiesOnlyMatrix([[1, 0, 0]]).is_Identity
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_symbolic():
|
||
|
|
a = PropertiesOnlyMatrix([[x, x], [x, x]])
|
||
|
|
assert a.is_symbolic() is True
|
||
|
|
a = PropertiesOnlyMatrix([[1, 2, 3, 4], [5, 6, 7, 8]])
|
||
|
|
assert a.is_symbolic() is False
|
||
|
|
a = PropertiesOnlyMatrix([[1, 2, 3, 4], [5, 6, x, 8]])
|
||
|
|
assert a.is_symbolic() is True
|
||
|
|
a = PropertiesOnlyMatrix([[1, x, 3]])
|
||
|
|
assert a.is_symbolic() is True
|
||
|
|
a = PropertiesOnlyMatrix([[1, 2, 3]])
|
||
|
|
assert a.is_symbolic() is False
|
||
|
|
a = PropertiesOnlyMatrix([[1], [x], [3]])
|
||
|
|
assert a.is_symbolic() is True
|
||
|
|
a = PropertiesOnlyMatrix([[1], [2], [3]])
|
||
|
|
assert a.is_symbolic() is False
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_upper():
|
||
|
|
a = PropertiesOnlyMatrix([[1, 2, 3]])
|
||
|
|
assert a.is_upper is True
|
||
|
|
a = PropertiesOnlyMatrix([[1], [2], [3]])
|
||
|
|
assert a.is_upper is False
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_lower():
|
||
|
|
a = PropertiesOnlyMatrix([[1, 2, 3]])
|
||
|
|
assert a.is_lower is False
|
||
|
|
a = PropertiesOnlyMatrix([[1], [2], [3]])
|
||
|
|
assert a.is_lower is True
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_square():
|
||
|
|
m = PropertiesOnlyMatrix([[1], [1]])
|
||
|
|
m2 = PropertiesOnlyMatrix([[2, 2], [2, 2]])
|
||
|
|
assert not m.is_square
|
||
|
|
assert m2.is_square
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_symmetric():
|
||
|
|
m = PropertiesOnlyMatrix(2, 2, [0, 1, 1, 0])
|
||
|
|
assert m.is_symmetric()
|
||
|
|
m = PropertiesOnlyMatrix(2, 2, [0, 1, 0, 1])
|
||
|
|
assert not m.is_symmetric()
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_hessenberg():
|
||
|
|
A = PropertiesOnlyMatrix([[3, 4, 1], [2, 4, 5], [0, 1, 2]])
|
||
|
|
assert A.is_upper_hessenberg
|
||
|
|
A = PropertiesOnlyMatrix(3, 3, [3, 2, 0, 4, 4, 1, 1, 5, 2])
|
||
|
|
assert A.is_lower_hessenberg
|
||
|
|
A = PropertiesOnlyMatrix(3, 3, [3, 2, -1, 4, 4, 1, 1, 5, 2])
|
||
|
|
assert A.is_lower_hessenberg is False
|
||
|
|
assert A.is_upper_hessenberg is False
|
||
|
|
|
||
|
|
A = PropertiesOnlyMatrix([[3, 4, 1], [2, 4, 5], [3, 1, 2]])
|
||
|
|
assert not A.is_upper_hessenberg
|
||
|
|
|
||
|
|
|
||
|
|
def test_is_zero():
|
||
|
|
assert PropertiesOnlyMatrix(0, 0, []).is_zero_matrix
|
||
|
|
assert PropertiesOnlyMatrix([[0, 0], [0, 0]]).is_zero_matrix
|
||
|
|
assert PropertiesOnlyMatrix(zeros(3, 4)).is_zero_matrix
|
||
|
|
assert not PropertiesOnlyMatrix(eye(3)).is_zero_matrix
|
||
|
|
assert PropertiesOnlyMatrix([[x, 0], [0, 0]]).is_zero_matrix == None
|
||
|
|
assert PropertiesOnlyMatrix([[x, 1], [0, 0]]).is_zero_matrix == False
|
||
|
|
a = Symbol('a', nonzero=True)
|
||
|
|
assert PropertiesOnlyMatrix([[a, 0], [0, 0]]).is_zero_matrix == False
|
||
|
|
|
||
|
|
|
||
|
|
def test_values():
|
||
|
|
assert set(PropertiesOnlyMatrix(2, 2, [0, 1, 2, 3]
|
||
|
|
).values()) == {1, 2, 3}
|
||
|
|
x = Symbol('x', real=True)
|
||
|
|
assert set(PropertiesOnlyMatrix(2, 2, [x, 0, 0, 1]
|
||
|
|
).values()) == {x, 1}
|
||
|
|
|
||
|
|
|
||
|
|
# OperationsOnlyMatrix tests
|
||
|
|
def test_applyfunc():
|
||
|
|
m0 = OperationsOnlyMatrix(eye(3))
|
||
|
|
assert m0.applyfunc(lambda x: 2*x) == eye(3)*2
|
||
|
|
assert m0.applyfunc(lambda x: 0) == zeros(3)
|
||
|
|
assert m0.applyfunc(lambda x: 1) == ones(3)
|
||
|
|
|
||
|
|
|
||
|
|
def test_adjoint():
|
||
|
|
dat = [[0, I], [1, 0]]
|
||
|
|
ans = OperationsOnlyMatrix([[0, 1], [-I, 0]])
|
||
|
|
assert ans.adjoint() == Matrix(dat)
|
||
|
|
|
||
|
|
|
||
|
|
def test_as_real_imag():
|
||
|
|
m1 = OperationsOnlyMatrix(2, 2, [1, 2, 3, 4])
|
||
|
|
m3 = OperationsOnlyMatrix(2, 2,
|
||
|
|
[1 + S.ImaginaryUnit, 2 + 2*S.ImaginaryUnit,
|
||
|
|
3 + 3*S.ImaginaryUnit, 4 + 4*S.ImaginaryUnit])
|
||
|
|
|
||
|
|
a, b = m3.as_real_imag()
|
||
|
|
assert a == m1
|
||
|
|
assert b == m1
|
||
|
|
|
||
|
|
|
||
|
|
def test_conjugate():
|
||
|
|
M = OperationsOnlyMatrix([[0, I, 5],
|
||
|
|
[1, 2, 0]])
|
||
|
|
|
||
|
|
assert M.T == Matrix([[0, 1],
|
||
|
|
[I, 2],
|
||
|
|
[5, 0]])
|
||
|
|
|
||
|
|
assert M.C == Matrix([[0, -I, 5],
|
||
|
|
[1, 2, 0]])
|
||
|
|
assert M.C == M.conjugate()
|
||
|
|
|
||
|
|
assert M.H == M.T.C
|
||
|
|
assert M.H == Matrix([[ 0, 1],
|
||
|
|
[-I, 2],
|
||
|
|
[ 5, 0]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_doit():
|
||
|
|
a = OperationsOnlyMatrix([[Add(x, x, evaluate=False)]])
|
||
|
|
assert a[0] != 2*x
|
||
|
|
assert a.doit() == Matrix([[2*x]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_evalf():
|
||
|
|
a = OperationsOnlyMatrix(2, 1, [sqrt(5), 6])
|
||
|
|
assert all(a.evalf()[i] == a[i].evalf() for i in range(2))
|
||
|
|
assert all(a.evalf(2)[i] == a[i].evalf(2) for i in range(2))
|
||
|
|
assert all(a.n(2)[i] == a[i].n(2) for i in range(2))
|
||
|
|
|
||
|
|
|
||
|
|
def test_expand():
|
||
|
|
m0 = OperationsOnlyMatrix([[x*(x + y), 2], [((x + y)*y)*x, x*(y + x*(x + y))]])
|
||
|
|
# Test if expand() returns a matrix
|
||
|
|
m1 = m0.expand()
|
||
|
|
assert m1 == Matrix(
|
||
|
|
[[x*y + x**2, 2], [x*y**2 + y*x**2, x*y + y*x**2 + x**3]])
|
||
|
|
|
||
|
|
a = Symbol('a', real=True)
|
||
|
|
|
||
|
|
assert OperationsOnlyMatrix(1, 1, [exp(I*a)]).expand(complex=True) == \
|
||
|
|
Matrix([cos(a) + I*sin(a)])
|
||
|
|
|
||
|
|
|
||
|
|
def test_refine():
|
||
|
|
m0 = OperationsOnlyMatrix([[Abs(x)**2, sqrt(x**2)],
|
||
|
|
[sqrt(x**2)*Abs(y)**2, sqrt(y**2)*Abs(x)**2]])
|
||
|
|
m1 = m0.refine(Q.real(x) & Q.real(y))
|
||
|
|
assert m1 == Matrix([[x**2, Abs(x)], [y**2*Abs(x), x**2*Abs(y)]])
|
||
|
|
|
||
|
|
m1 = m0.refine(Q.positive(x) & Q.positive(y))
|
||
|
|
assert m1 == Matrix([[x**2, x], [x*y**2, x**2*y]])
|
||
|
|
|
||
|
|
m1 = m0.refine(Q.negative(x) & Q.negative(y))
|
||
|
|
assert m1 == Matrix([[x**2, -x], [-x*y**2, -x**2*y]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_replace():
|
||
|
|
F, G = symbols('F, G', cls=Function)
|
||
|
|
K = OperationsOnlyMatrix(2, 2, lambda i, j: G(i+j))
|
||
|
|
M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
|
||
|
|
N = M.replace(F, G)
|
||
|
|
assert N == K
|
||
|
|
|
||
|
|
|
||
|
|
def test_replace_map():
|
||
|
|
F, G = symbols('F, G', cls=Function)
|
||
|
|
K = OperationsOnlyMatrix(2, 2, [(G(0), {F(0): G(0)}), (G(1), {F(1): G(1)}), (G(1), {F(1) \
|
||
|
|
: G(1)}), (G(2), {F(2): G(2)})])
|
||
|
|
M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
|
||
|
|
N = M.replace(F, G, True)
|
||
|
|
assert N == K
|
||
|
|
|
||
|
|
|
||
|
|
def test_rot90():
|
||
|
|
A = Matrix([[1, 2], [3, 4]])
|
||
|
|
assert A == A.rot90(0) == A.rot90(4)
|
||
|
|
assert A.rot90(2) == A.rot90(-2) == A.rot90(6) == Matrix(((4, 3), (2, 1)))
|
||
|
|
assert A.rot90(3) == A.rot90(-1) == A.rot90(7) == Matrix(((2, 4), (1, 3)))
|
||
|
|
assert A.rot90() == A.rot90(-7) == A.rot90(-3) == Matrix(((3, 1), (4, 2)))
|
||
|
|
|
||
|
|
def test_simplify():
|
||
|
|
n = Symbol('n')
|
||
|
|
f = Function('f')
|
||
|
|
|
||
|
|
M = OperationsOnlyMatrix([[ 1/x + 1/y, (x + x*y) / x ],
|
||
|
|
[ (f(x) + y*f(x))/f(x), 2 * (1/n - cos(n * pi)/n) / pi ]])
|
||
|
|
assert M.simplify() == Matrix([[ (x + y)/(x * y), 1 + y ],
|
||
|
|
[ 1 + y, 2*((1 - 1*cos(pi*n))/(pi*n)) ]])
|
||
|
|
eq = (1 + x)**2
|
||
|
|
M = OperationsOnlyMatrix([[eq]])
|
||
|
|
assert M.simplify() == Matrix([[eq]])
|
||
|
|
assert M.simplify(ratio=oo) == Matrix([[eq.simplify(ratio=oo)]])
|
||
|
|
|
||
|
|
# https://github.com/sympy/sympy/issues/19353
|
||
|
|
m = Matrix([[30, 2], [3, 4]])
|
||
|
|
assert (1/(m.trace())).simplify() == Rational(1, 34)
|
||
|
|
|
||
|
|
|
||
|
|
def test_subs():
|
||
|
|
assert OperationsOnlyMatrix([[1, x], [x, 4]]).subs(x, 5) == Matrix([[1, 5], [5, 4]])
|
||
|
|
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs([[x, -1], [y, -2]]) == \
|
||
|
|
Matrix([[-1, 2], [-3, 4]])
|
||
|
|
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs([(x, -1), (y, -2)]) == \
|
||
|
|
Matrix([[-1, 2], [-3, 4]])
|
||
|
|
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs({x: -1, y: -2}) == \
|
||
|
|
Matrix([[-1, 2], [-3, 4]])
|
||
|
|
assert OperationsOnlyMatrix([[x*y]]).subs({x: y - 1, y: x - 1}, simultaneous=True) == \
|
||
|
|
Matrix([[(x - 1)*(y - 1)]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_trace():
|
||
|
|
M = OperationsOnlyMatrix([[1, 0, 0],
|
||
|
|
[0, 5, 0],
|
||
|
|
[0, 0, 8]])
|
||
|
|
assert M.trace() == 14
|
||
|
|
|
||
|
|
|
||
|
|
def test_xreplace():
|
||
|
|
assert OperationsOnlyMatrix([[1, x], [x, 4]]).xreplace({x: 5}) == \
|
||
|
|
Matrix([[1, 5], [5, 4]])
|
||
|
|
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).xreplace({x: -1, y: -2}) == \
|
||
|
|
Matrix([[-1, 2], [-3, 4]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_permute():
|
||
|
|
a = OperationsOnlyMatrix(3, 4, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])
|
||
|
|
|
||
|
|
raises(IndexError, lambda: a.permute([[0, 5]]))
|
||
|
|
raises(ValueError, lambda: a.permute(Symbol('x')))
|
||
|
|
b = a.permute_rows([[0, 2], [0, 1]])
|
||
|
|
assert a.permute([[0, 2], [0, 1]]) == b == Matrix([
|
||
|
|
[5, 6, 7, 8],
|
||
|
|
[9, 10, 11, 12],
|
||
|
|
[1, 2, 3, 4]])
|
||
|
|
|
||
|
|
b = a.permute_cols([[0, 2], [0, 1]])
|
||
|
|
assert a.permute([[0, 2], [0, 1]], orientation='cols') == b ==\
|
||
|
|
Matrix([
|
||
|
|
[ 2, 3, 1, 4],
|
||
|
|
[ 6, 7, 5, 8],
|
||
|
|
[10, 11, 9, 12]])
|
||
|
|
|
||
|
|
b = a.permute_cols([[0, 2], [0, 1]], direction='backward')
|
||
|
|
assert a.permute([[0, 2], [0, 1]], orientation='cols', direction='backward') == b ==\
|
||
|
|
Matrix([
|
||
|
|
[ 3, 1, 2, 4],
|
||
|
|
[ 7, 5, 6, 8],
|
||
|
|
[11, 9, 10, 12]])
|
||
|
|
|
||
|
|
assert a.permute([1, 2, 0, 3]) == Matrix([
|
||
|
|
[5, 6, 7, 8],
|
||
|
|
[9, 10, 11, 12],
|
||
|
|
[1, 2, 3, 4]])
|
||
|
|
|
||
|
|
from sympy.combinatorics import Permutation
|
||
|
|
assert a.permute(Permutation([1, 2, 0, 3])) == Matrix([
|
||
|
|
[5, 6, 7, 8],
|
||
|
|
[9, 10, 11, 12],
|
||
|
|
[1, 2, 3, 4]])
|
||
|
|
|
||
|
|
def test_upper_triangular():
|
||
|
|
|
||
|
|
A = OperationsOnlyMatrix([
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1]
|
||
|
|
])
|
||
|
|
|
||
|
|
R = A.upper_triangular(2)
|
||
|
|
assert R == OperationsOnlyMatrix([
|
||
|
|
[0, 0, 1, 1],
|
||
|
|
[0, 0, 0, 1],
|
||
|
|
[0, 0, 0, 0],
|
||
|
|
[0, 0, 0, 0]
|
||
|
|
])
|
||
|
|
|
||
|
|
R = A.upper_triangular(-2)
|
||
|
|
assert R == OperationsOnlyMatrix([
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[0, 1, 1, 1]
|
||
|
|
])
|
||
|
|
|
||
|
|
R = A.upper_triangular()
|
||
|
|
assert R == OperationsOnlyMatrix([
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[0, 1, 1, 1],
|
||
|
|
[0, 0, 1, 1],
|
||
|
|
[0, 0, 0, 1]
|
||
|
|
])
|
||
|
|
|
||
|
|
def test_lower_triangular():
|
||
|
|
A = OperationsOnlyMatrix([
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1]
|
||
|
|
])
|
||
|
|
|
||
|
|
L = A.lower_triangular()
|
||
|
|
assert L == ArithmeticOnlyMatrix([
|
||
|
|
[1, 0, 0, 0],
|
||
|
|
[1, 1, 0, 0],
|
||
|
|
[1, 1, 1, 0],
|
||
|
|
[1, 1, 1, 1]])
|
||
|
|
|
||
|
|
L = A.lower_triangular(2)
|
||
|
|
assert L == ArithmeticOnlyMatrix([
|
||
|
|
[1, 1, 1, 0],
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1],
|
||
|
|
[1, 1, 1, 1]
|
||
|
|
])
|
||
|
|
|
||
|
|
L = A.lower_triangular(-2)
|
||
|
|
assert L == ArithmeticOnlyMatrix([
|
||
|
|
[0, 0, 0, 0],
|
||
|
|
[0, 0, 0, 0],
|
||
|
|
[1, 0, 0, 0],
|
||
|
|
[1, 1, 0, 0]
|
||
|
|
])
|
||
|
|
|
||
|
|
|
||
|
|
# ArithmeticOnlyMatrix tests
|
||
|
|
def test_abs():
|
||
|
|
m = ArithmeticOnlyMatrix([[1, -2], [x, y]])
|
||
|
|
assert abs(m) == ArithmeticOnlyMatrix([[1, 2], [Abs(x), Abs(y)]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_add():
|
||
|
|
m = ArithmeticOnlyMatrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
|
||
|
|
assert m + m == ArithmeticOnlyMatrix([[2, 4, 6], [2*x, 2*y, 2*x], [4*y, -100, 2*z*x]])
|
||
|
|
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
|
||
|
|
raises(ShapeError, lambda: m + n)
|
||
|
|
|
||
|
|
|
||
|
|
def test_multiplication():
|
||
|
|
a = ArithmeticOnlyMatrix((
|
||
|
|
(1, 2),
|
||
|
|
(3, 1),
|
||
|
|
(0, 6),
|
||
|
|
))
|
||
|
|
|
||
|
|
b = ArithmeticOnlyMatrix((
|
||
|
|
(1, 2),
|
||
|
|
(3, 0),
|
||
|
|
))
|
||
|
|
|
||
|
|
raises(ShapeError, lambda: b*a)
|
||
|
|
raises(TypeError, lambda: a*{})
|
||
|
|
|
||
|
|
c = a*b
|
||
|
|
assert c[0, 0] == 7
|
||
|
|
assert c[0, 1] == 2
|
||
|
|
assert c[1, 0] == 6
|
||
|
|
assert c[1, 1] == 6
|
||
|
|
assert c[2, 0] == 18
|
||
|
|
assert c[2, 1] == 0
|
||
|
|
|
||
|
|
try:
|
||
|
|
eval('c = a @ b')
|
||
|
|
except SyntaxError:
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
assert c[0, 0] == 7
|
||
|
|
assert c[0, 1] == 2
|
||
|
|
assert c[1, 0] == 6
|
||
|
|
assert c[1, 1] == 6
|
||
|
|
assert c[2, 0] == 18
|
||
|
|
assert c[2, 1] == 0
|
||
|
|
|
||
|
|
h = a.multiply_elementwise(c)
|
||
|
|
assert h == matrix_multiply_elementwise(a, c)
|
||
|
|
assert h[0, 0] == 7
|
||
|
|
assert h[0, 1] == 4
|
||
|
|
assert h[1, 0] == 18
|
||
|
|
assert h[1, 1] == 6
|
||
|
|
assert h[2, 0] == 0
|
||
|
|
assert h[2, 1] == 0
|
||
|
|
raises(ShapeError, lambda: a.multiply_elementwise(b))
|
||
|
|
|
||
|
|
c = b * Symbol("x")
|
||
|
|
assert isinstance(c, ArithmeticOnlyMatrix)
|
||
|
|
assert c[0, 0] == x
|
||
|
|
assert c[0, 1] == 2*x
|
||
|
|
assert c[1, 0] == 3*x
|
||
|
|
assert c[1, 1] == 0
|
||
|
|
|
||
|
|
c2 = x * b
|
||
|
|
assert c == c2
|
||
|
|
|
||
|
|
c = 5 * b
|
||
|
|
assert isinstance(c, ArithmeticOnlyMatrix)
|
||
|
|
assert c[0, 0] == 5
|
||
|
|
assert c[0, 1] == 2*5
|
||
|
|
assert c[1, 0] == 3*5
|
||
|
|
assert c[1, 1] == 0
|
||
|
|
|
||
|
|
try:
|
||
|
|
eval('c = 5 @ b')
|
||
|
|
except SyntaxError:
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
assert isinstance(c, ArithmeticOnlyMatrix)
|
||
|
|
assert c[0, 0] == 5
|
||
|
|
assert c[0, 1] == 2*5
|
||
|
|
assert c[1, 0] == 3*5
|
||
|
|
assert c[1, 1] == 0
|
||
|
|
|
||
|
|
# https://github.com/sympy/sympy/issues/22353
|
||
|
|
A = Matrix(ones(3, 1))
|
||
|
|
_h = -Rational(1, 2)
|
||
|
|
B = Matrix([_h, _h, _h])
|
||
|
|
assert A.multiply_elementwise(B) == Matrix([
|
||
|
|
[_h],
|
||
|
|
[_h],
|
||
|
|
[_h]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_matmul():
|
||
|
|
a = Matrix([[1, 2], [3, 4]])
|
||
|
|
|
||
|
|
assert a.__matmul__(2) == NotImplemented
|
||
|
|
|
||
|
|
assert a.__rmatmul__(2) == NotImplemented
|
||
|
|
|
||
|
|
#This is done this way because @ is only supported in Python 3.5+
|
||
|
|
#To check 2@a case
|
||
|
|
try:
|
||
|
|
eval('2 @ a')
|
||
|
|
except SyntaxError:
|
||
|
|
pass
|
||
|
|
except TypeError: #TypeError is raised in case of NotImplemented is returned
|
||
|
|
pass
|
||
|
|
|
||
|
|
#Check a@2 case
|
||
|
|
try:
|
||
|
|
eval('a @ 2')
|
||
|
|
except SyntaxError:
|
||
|
|
pass
|
||
|
|
except TypeError: #TypeError is raised in case of NotImplemented is returned
|
||
|
|
pass
|
||
|
|
|
||
|
|
|
||
|
|
def test_non_matmul():
|
||
|
|
"""
|
||
|
|
Test that if explicitly specified as non-matrix, mul reverts
|
||
|
|
to scalar multiplication.
|
||
|
|
"""
|
||
|
|
class foo(Expr):
|
||
|
|
is_Matrix=False
|
||
|
|
is_MatrixLike=False
|
||
|
|
shape = (1, 1)
|
||
|
|
|
||
|
|
A = Matrix([[1, 2], [3, 4]])
|
||
|
|
b = foo()
|
||
|
|
assert b*A == Matrix([[b, 2*b], [3*b, 4*b]])
|
||
|
|
assert A*b == Matrix([[b, 2*b], [3*b, 4*b]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_power():
|
||
|
|
raises(NonSquareMatrixError, lambda: Matrix((1, 2))**2)
|
||
|
|
|
||
|
|
A = ArithmeticOnlyMatrix([[2, 3], [4, 5]])
|
||
|
|
assert (A**5)[:] == (6140, 8097, 10796, 14237)
|
||
|
|
A = ArithmeticOnlyMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
|
||
|
|
assert (A**3)[:] == (290, 262, 251, 448, 440, 368, 702, 954, 433)
|
||
|
|
assert A**0 == eye(3)
|
||
|
|
assert A**1 == A
|
||
|
|
assert (ArithmeticOnlyMatrix([[2]]) ** 100)[0, 0] == 2**100
|
||
|
|
assert ArithmeticOnlyMatrix([[1, 2], [3, 4]])**Integer(2) == ArithmeticOnlyMatrix([[7, 10], [15, 22]])
|
||
|
|
A = Matrix([[1,2],[4,5]])
|
||
|
|
assert A.pow(20, method='cayley') == A.pow(20, method='multiply')
|
||
|
|
|
||
|
|
def test_neg():
|
||
|
|
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
|
||
|
|
assert -n == ArithmeticOnlyMatrix(1, 2, [-1, -2])
|
||
|
|
|
||
|
|
|
||
|
|
def test_sub():
|
||
|
|
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
|
||
|
|
assert n - n == ArithmeticOnlyMatrix(1, 2, [0, 0])
|
||
|
|
|
||
|
|
|
||
|
|
def test_div():
|
||
|
|
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
|
||
|
|
assert n/2 == ArithmeticOnlyMatrix(1, 2, [S.Half, S(2)/2])
|
||
|
|
|
||
|
|
# SpecialOnlyMatrix tests
|
||
|
|
def test_eye():
|
||
|
|
assert list(SpecialOnlyMatrix.eye(2, 2)) == [1, 0, 0, 1]
|
||
|
|
assert list(SpecialOnlyMatrix.eye(2)) == [1, 0, 0, 1]
|
||
|
|
assert type(SpecialOnlyMatrix.eye(2)) == SpecialOnlyMatrix
|
||
|
|
assert type(SpecialOnlyMatrix.eye(2, cls=Matrix)) == Matrix
|
||
|
|
|
||
|
|
|
||
|
|
def test_ones():
|
||
|
|
assert list(SpecialOnlyMatrix.ones(2, 2)) == [1, 1, 1, 1]
|
||
|
|
assert list(SpecialOnlyMatrix.ones(2)) == [1, 1, 1, 1]
|
||
|
|
assert SpecialOnlyMatrix.ones(2, 3) == Matrix([[1, 1, 1], [1, 1, 1]])
|
||
|
|
assert type(SpecialOnlyMatrix.ones(2)) == SpecialOnlyMatrix
|
||
|
|
assert type(SpecialOnlyMatrix.ones(2, cls=Matrix)) == Matrix
|
||
|
|
|
||
|
|
|
||
|
|
def test_zeros():
|
||
|
|
assert list(SpecialOnlyMatrix.zeros(2, 2)) == [0, 0, 0, 0]
|
||
|
|
assert list(SpecialOnlyMatrix.zeros(2)) == [0, 0, 0, 0]
|
||
|
|
assert SpecialOnlyMatrix.zeros(2, 3) == Matrix([[0, 0, 0], [0, 0, 0]])
|
||
|
|
assert type(SpecialOnlyMatrix.zeros(2)) == SpecialOnlyMatrix
|
||
|
|
assert type(SpecialOnlyMatrix.zeros(2, cls=Matrix)) == Matrix
|
||
|
|
|
||
|
|
|
||
|
|
def test_diag_make():
|
||
|
|
diag = SpecialOnlyMatrix.diag
|
||
|
|
a = Matrix([[1, 2], [2, 3]])
|
||
|
|
b = Matrix([[3, x], [y, 3]])
|
||
|
|
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
|
||
|
|
assert diag(a, b, b) == Matrix([
|
||
|
|
[1, 2, 0, 0, 0, 0],
|
||
|
|
[2, 3, 0, 0, 0, 0],
|
||
|
|
[0, 0, 3, x, 0, 0],
|
||
|
|
[0, 0, y, 3, 0, 0],
|
||
|
|
[0, 0, 0, 0, 3, x],
|
||
|
|
[0, 0, 0, 0, y, 3],
|
||
|
|
])
|
||
|
|
assert diag(a, b, c) == Matrix([
|
||
|
|
[1, 2, 0, 0, 0, 0, 0],
|
||
|
|
[2, 3, 0, 0, 0, 0, 0],
|
||
|
|
[0, 0, 3, x, 0, 0, 0],
|
||
|
|
[0, 0, y, 3, 0, 0, 0],
|
||
|
|
[0, 0, 0, 0, 3, x, 3],
|
||
|
|
[0, 0, 0, 0, y, 3, z],
|
||
|
|
[0, 0, 0, 0, x, y, z],
|
||
|
|
])
|
||
|
|
assert diag(a, c, b) == Matrix([
|
||
|
|
[1, 2, 0, 0, 0, 0, 0],
|
||
|
|
[2, 3, 0, 0, 0, 0, 0],
|
||
|
|
[0, 0, 3, x, 3, 0, 0],
|
||
|
|
[0, 0, y, 3, z, 0, 0],
|
||
|
|
[0, 0, x, y, z, 0, 0],
|
||
|
|
[0, 0, 0, 0, 0, 3, x],
|
||
|
|
[0, 0, 0, 0, 0, y, 3],
|
||
|
|
])
|
||
|
|
a = Matrix([x, y, z])
|
||
|
|
b = Matrix([[1, 2], [3, 4]])
|
||
|
|
c = Matrix([[5, 6]])
|
||
|
|
# this "wandering diagonal" is what makes this
|
||
|
|
# a block diagonal where each block is independent
|
||
|
|
# of the others
|
||
|
|
assert diag(a, 7, b, c) == Matrix([
|
||
|
|
[x, 0, 0, 0, 0, 0],
|
||
|
|
[y, 0, 0, 0, 0, 0],
|
||
|
|
[z, 0, 0, 0, 0, 0],
|
||
|
|
[0, 7, 0, 0, 0, 0],
|
||
|
|
[0, 0, 1, 2, 0, 0],
|
||
|
|
[0, 0, 3, 4, 0, 0],
|
||
|
|
[0, 0, 0, 0, 5, 6]])
|
||
|
|
raises(ValueError, lambda: diag(a, 7, b, c, rows=5))
|
||
|
|
assert diag(1) == Matrix([[1]])
|
||
|
|
assert diag(1, rows=2) == Matrix([[1, 0], [0, 0]])
|
||
|
|
assert diag(1, cols=2) == Matrix([[1, 0], [0, 0]])
|
||
|
|
assert diag(1, rows=3, cols=2) == Matrix([[1, 0], [0, 0], [0, 0]])
|
||
|
|
assert diag(*[2, 3]) == Matrix([
|
||
|
|
[2, 0],
|
||
|
|
[0, 3]])
|
||
|
|
assert diag(Matrix([2, 3])) == Matrix([
|
||
|
|
[2],
|
||
|
|
[3]])
|
||
|
|
assert diag([1, [2, 3], 4], unpack=False) == \
|
||
|
|
diag([[1], [2, 3], [4]], unpack=False) == Matrix([
|
||
|
|
[1, 0],
|
||
|
|
[2, 3],
|
||
|
|
[4, 0]])
|
||
|
|
assert type(diag(1)) == SpecialOnlyMatrix
|
||
|
|
assert type(diag(1, cls=Matrix)) == Matrix
|
||
|
|
assert Matrix.diag([1, 2, 3]) == Matrix.diag(1, 2, 3)
|
||
|
|
assert Matrix.diag([1, 2, 3], unpack=False).shape == (3, 1)
|
||
|
|
assert Matrix.diag([[1, 2, 3]]).shape == (3, 1)
|
||
|
|
assert Matrix.diag([[1, 2, 3]], unpack=False).shape == (1, 3)
|
||
|
|
assert Matrix.diag([[[1, 2, 3]]]).shape == (1, 3)
|
||
|
|
# kerning can be used to move the starting point
|
||
|
|
assert Matrix.diag(ones(0, 2), 1, 2) == Matrix([
|
||
|
|
[0, 0, 1, 0],
|
||
|
|
[0, 0, 0, 2]])
|
||
|
|
assert Matrix.diag(ones(2, 0), 1, 2) == Matrix([
|
||
|
|
[0, 0],
|
||
|
|
[0, 0],
|
||
|
|
[1, 0],
|
||
|
|
[0, 2]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_diagonal():
|
||
|
|
m = Matrix(3, 3, range(9))
|
||
|
|
d = m.diagonal()
|
||
|
|
assert d == m.diagonal(0)
|
||
|
|
assert tuple(d) == (0, 4, 8)
|
||
|
|
assert tuple(m.diagonal(1)) == (1, 5)
|
||
|
|
assert tuple(m.diagonal(-1)) == (3, 7)
|
||
|
|
assert tuple(m.diagonal(2)) == (2,)
|
||
|
|
assert type(m.diagonal()) == type(m)
|
||
|
|
s = SparseMatrix(3, 3, {(1, 1): 1})
|
||
|
|
assert type(s.diagonal()) == type(s)
|
||
|
|
assert type(m) != type(s)
|
||
|
|
raises(ValueError, lambda: m.diagonal(3))
|
||
|
|
raises(ValueError, lambda: m.diagonal(-3))
|
||
|
|
raises(ValueError, lambda: m.diagonal(pi))
|
||
|
|
M = ones(2, 3)
|
||
|
|
assert banded({i: list(M.diagonal(i))
|
||
|
|
for i in range(1-M.rows, M.cols)}) == M
|
||
|
|
|
||
|
|
|
||
|
|
def test_jordan_block():
|
||
|
|
assert SpecialOnlyMatrix.jordan_block(3, 2) == SpecialOnlyMatrix.jordan_block(3, eigenvalue=2) \
|
||
|
|
== SpecialOnlyMatrix.jordan_block(size=3, eigenvalue=2) \
|
||
|
|
== SpecialOnlyMatrix.jordan_block(3, 2, band='upper') \
|
||
|
|
== SpecialOnlyMatrix.jordan_block(
|
||
|
|
size=3, eigenval=2, eigenvalue=2) \
|
||
|
|
== Matrix([
|
||
|
|
[2, 1, 0],
|
||
|
|
[0, 2, 1],
|
||
|
|
[0, 0, 2]])
|
||
|
|
|
||
|
|
assert SpecialOnlyMatrix.jordan_block(3, 2, band='lower') == Matrix([
|
||
|
|
[2, 0, 0],
|
||
|
|
[1, 2, 0],
|
||
|
|
[0, 1, 2]])
|
||
|
|
# missing eigenvalue
|
||
|
|
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(2))
|
||
|
|
# non-integral size
|
||
|
|
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(3.5, 2))
|
||
|
|
# size not specified
|
||
|
|
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(eigenvalue=2))
|
||
|
|
# inconsistent eigenvalue
|
||
|
|
raises(ValueError,
|
||
|
|
lambda: SpecialOnlyMatrix.jordan_block(
|
||
|
|
eigenvalue=2, eigenval=4))
|
||
|
|
|
||
|
|
# Using alias keyword
|
||
|
|
assert SpecialOnlyMatrix.jordan_block(size=3, eigenvalue=2) == \
|
||
|
|
SpecialOnlyMatrix.jordan_block(size=3, eigenval=2)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonalize():
|
||
|
|
m = Matrix([[1, 2], [3, 4]])
|
||
|
|
assert m.orthogonalize(Matrix([[2], [1]])) == [Matrix([[2], [1]])]
|
||
|
|
assert m.orthogonalize(Matrix([[2], [1]]), normalize=True) == \
|
||
|
|
[Matrix([[2*sqrt(5)/5], [sqrt(5)/5]])]
|
||
|
|
assert m.orthogonalize(Matrix([[1], [2]]), Matrix([[-1], [4]])) == \
|
||
|
|
[Matrix([[1], [2]]), Matrix([[Rational(-12, 5)], [Rational(6, 5)]])]
|
||
|
|
assert m.orthogonalize(Matrix([[0], [0]]), Matrix([[-1], [4]])) == \
|
||
|
|
[Matrix([[-1], [4]])]
|
||
|
|
assert m.orthogonalize(Matrix([[0], [0]])) == []
|
||
|
|
|
||
|
|
n = Matrix([[9, 1, 9], [3, 6, 10], [8, 5, 2]])
|
||
|
|
vecs = [Matrix([[-5], [1]]), Matrix([[-5], [2]]), Matrix([[-5], [-2]])]
|
||
|
|
assert n.orthogonalize(*vecs) == \
|
||
|
|
[Matrix([[-5], [1]]), Matrix([[Rational(5, 26)], [Rational(25, 26)]])]
|
||
|
|
|
||
|
|
vecs = [Matrix([0, 0, 0]), Matrix([1, 2, 3]), Matrix([1, 4, 5])]
|
||
|
|
raises(ValueError, lambda: Matrix.orthogonalize(*vecs, rankcheck=True))
|
||
|
|
|
||
|
|
vecs = [Matrix([1, 2, 3]), Matrix([4, 5, 6]), Matrix([7, 8, 9])]
|
||
|
|
raises(ValueError, lambda: Matrix.orthogonalize(*vecs, rankcheck=True))
|
||
|
|
|
||
|
|
def test_wilkinson():
|
||
|
|
|
||
|
|
wminus, wplus = Matrix.wilkinson(1)
|
||
|
|
assert wminus == Matrix([
|
||
|
|
[-1, 1, 0],
|
||
|
|
[1, 0, 1],
|
||
|
|
[0, 1, 1]])
|
||
|
|
assert wplus == Matrix([
|
||
|
|
[1, 1, 0],
|
||
|
|
[1, 0, 1],
|
||
|
|
[0, 1, 1]])
|
||
|
|
|
||
|
|
wminus, wplus = Matrix.wilkinson(3)
|
||
|
|
assert wminus == Matrix([
|
||
|
|
[-3, 1, 0, 0, 0, 0, 0],
|
||
|
|
[1, -2, 1, 0, 0, 0, 0],
|
||
|
|
[0, 1, -1, 1, 0, 0, 0],
|
||
|
|
[0, 0, 1, 0, 1, 0, 0],
|
||
|
|
[0, 0, 0, 1, 1, 1, 0],
|
||
|
|
[0, 0, 0, 0, 1, 2, 1],
|
||
|
|
|
||
|
|
[0, 0, 0, 0, 0, 1, 3]])
|
||
|
|
|
||
|
|
assert wplus == Matrix([
|
||
|
|
[3, 1, 0, 0, 0, 0, 0],
|
||
|
|
[1, 2, 1, 0, 0, 0, 0],
|
||
|
|
[0, 1, 1, 1, 0, 0, 0],
|
||
|
|
[0, 0, 1, 0, 1, 0, 0],
|
||
|
|
[0, 0, 0, 1, 1, 1, 0],
|
||
|
|
[0, 0, 0, 0, 1, 2, 1],
|
||
|
|
[0, 0, 0, 0, 0, 1, 3]])
|
||
|
|
|
||
|
|
|
||
|
|
# CalculusOnlyMatrix tests
|
||
|
|
@XFAIL
|
||
|
|
def test_diff():
|
||
|
|
x, y = symbols('x y')
|
||
|
|
m = CalculusOnlyMatrix(2, 1, [x, y])
|
||
|
|
# TODO: currently not working as ``_MinimalMatrix`` cannot be sympified:
|
||
|
|
assert m.diff(x) == Matrix(2, 1, [1, 0])
|
||
|
|
|
||
|
|
|
||
|
|
def test_integrate():
|
||
|
|
x, y = symbols('x y')
|
||
|
|
m = CalculusOnlyMatrix(2, 1, [x, y])
|
||
|
|
assert m.integrate(x) == Matrix(2, 1, [x**2/2, y*x])
|
||
|
|
|
||
|
|
|
||
|
|
def test_jacobian2():
|
||
|
|
rho, phi = symbols("rho,phi")
|
||
|
|
X = CalculusOnlyMatrix(3, 1, [rho*cos(phi), rho*sin(phi), rho**2])
|
||
|
|
Y = CalculusOnlyMatrix(2, 1, [rho, phi])
|
||
|
|
J = Matrix([
|
||
|
|
[cos(phi), -rho*sin(phi)],
|
||
|
|
[sin(phi), rho*cos(phi)],
|
||
|
|
[ 2*rho, 0],
|
||
|
|
])
|
||
|
|
assert X.jacobian(Y) == J
|
||
|
|
|
||
|
|
m = CalculusOnlyMatrix(2, 2, [1, 2, 3, 4])
|
||
|
|
m2 = CalculusOnlyMatrix(4, 1, [1, 2, 3, 4])
|
||
|
|
raises(TypeError, lambda: m.jacobian(Matrix([1, 2])))
|
||
|
|
raises(TypeError, lambda: m2.jacobian(m))
|
||
|
|
|
||
|
|
|
||
|
|
def test_limit():
|
||
|
|
x, y = symbols('x y')
|
||
|
|
m = CalculusOnlyMatrix(2, 1, [1/x, y])
|
||
|
|
assert m.limit(x, 5) == Matrix(2, 1, [Rational(1, 5), y])
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_13774():
|
||
|
|
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
|
||
|
|
v = [1, 1, 1]
|
||
|
|
raises(TypeError, lambda: M*v)
|
||
|
|
raises(TypeError, lambda: v*M)
|
||
|
|
|
||
|
|
def test_companion():
|
||
|
|
x = Symbol('x')
|
||
|
|
y = Symbol('y')
|
||
|
|
raises(ValueError, lambda: Matrix.companion(1))
|
||
|
|
raises(ValueError, lambda: Matrix.companion(Poly([1], x)))
|
||
|
|
raises(ValueError, lambda: Matrix.companion(Poly([2, 1], x)))
|
||
|
|
raises(ValueError, lambda: Matrix.companion(Poly(x*y, [x, y])))
|
||
|
|
|
||
|
|
c0, c1, c2 = symbols('c0:3')
|
||
|
|
assert Matrix.companion(Poly([1, c0], x)) == Matrix([-c0])
|
||
|
|
assert Matrix.companion(Poly([1, c1, c0], x)) == \
|
||
|
|
Matrix([[0, -c0], [1, -c1]])
|
||
|
|
assert Matrix.companion(Poly([1, c2, c1, c0], x)) == \
|
||
|
|
Matrix([[0, 0, -c0], [1, 0, -c1], [0, 1, -c2]])
|
||
|
|
|
||
|
|
def test_issue_10589():
|
||
|
|
x, y, z = symbols("x, y z")
|
||
|
|
M1 = Matrix([x, y, z])
|
||
|
|
M1 = M1.subs(zip([x, y, z], [1, 2, 3]))
|
||
|
|
assert M1 == Matrix([[1], [2], [3]])
|
||
|
|
|
||
|
|
M2 = Matrix([[x, x, x, x, x], [x, x, x, x, x], [x, x, x, x, x]])
|
||
|
|
M2 = M2.subs(zip([x], [1]))
|
||
|
|
assert M2 == Matrix([[1, 1, 1, 1, 1], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1]])
|
||
|
|
|
||
|
|
def test_rmul_pr19860():
|
||
|
|
class Foo(ImmutableDenseMatrix):
|
||
|
|
_op_priority = MutableDenseMatrix._op_priority + 0.01
|
||
|
|
|
||
|
|
a = Matrix(2, 2, [1, 2, 3, 4])
|
||
|
|
b = Foo(2, 2, [1, 2, 3, 4])
|
||
|
|
|
||
|
|
# This would throw a RecursionError: maximum recursion depth
|
||
|
|
# since b always has higher priority even after a.as_mutable()
|
||
|
|
c = a*b
|
||
|
|
|
||
|
|
assert isinstance(c, Foo)
|
||
|
|
assert c == Matrix([[7, 10], [15, 22]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_18956():
|
||
|
|
A = Array([[1, 2], [3, 4]])
|
||
|
|
B = Matrix([[1,2],[3,4]])
|
||
|
|
raises(TypeError, lambda: B + A)
|
||
|
|
raises(TypeError, lambda: A + B)
|
||
|
|
|
||
|
|
|
||
|
|
def test__eq__():
|
||
|
|
class My(object):
|
||
|
|
def __iter__(self):
|
||
|
|
yield 1
|
||
|
|
yield 2
|
||
|
|
return
|
||
|
|
def __getitem__(self, i):
|
||
|
|
return list(self)[i]
|
||
|
|
a = Matrix(2, 1, [1, 2])
|
||
|
|
assert a != My()
|
||
|
|
class My_sympy(My):
|
||
|
|
def _sympy_(self):
|
||
|
|
return Matrix(self)
|
||
|
|
assert a == My_sympy()
|