Пример #1
0
    """ Merge explicit MatrixBase arguments

    >>> from sympy import MatrixSymbol, eye, Matrix, MatAdd, pprint
    >>> from sympy.matrices.expressions.matadd import merge_explicit
    >>> A = MatrixSymbol('A', 2, 2)
    >>> B = eye(2)
    >>> C = Matrix([[1, 2], [3, 4]])
    >>> X = MatAdd(A, B, C)
    >>> pprint(X)
        [1  0]   [1  2]
    A + [    ] + [    ]
        [0  1]   [3  4]
    >>> pprint(merge_explicit(X))
        [2  2]
    A + [    ]
        [3  5]
    """
    groups = sift(matadd.args, lambda arg: isinstance(arg, MatrixBase))
    if len(groups[True]) > 1:
        return MatAdd(*(groups[False] + [reduce(add, groups[True])]))
    else:
        return matadd


rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)), unpack, flatten,
         glom(matrix_of, factor_of,
              combine), merge_explicit, sort(default_sort_key))

canonicalize = exhaust(
    condition(lambda x: isinstance(x, MatAdd), do_one(*rules)))
Пример #2
0
    >>> from sympy.matrices.expressions.matadd import merge_explicit
    >>> A = MatrixSymbol('A', 2, 2)
    >>> B = eye(2)
    >>> C = Matrix([[1, 2], [3, 4]])
    >>> X = MatAdd(A, B, C)
    >>> pprint(X)
        [1  0]   [1  2]
    A + [    ] + [    ]
        [0  1]   [3  4]
    >>> pprint(merge_explicit(X))
        [2  2]
    A + [    ]
        [3  5]
    """
    groups = sift(matadd.args, lambda arg: isinstance(arg, MatrixBase))
    if len(groups[True]) > 1:
        return MatAdd(*(groups[False] + [reduce(add, groups[True])]))
    else:
        return matadd


rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)),
         unpack,
         flatten,
         glom(matrix_of, factor_of, combine),
         merge_explicit,
         sort(default_sort_key))

canonicalize = exhaust(condition(lambda x: isinstance(x, MatAdd),
                                 do_one(*rules)))
Пример #3
0
def canonicalize(x):
    """Canonicalize the Hadamard product ``x`` with mathematical properties.

    Examples
    ========

    >>> from sympy.matrices.expressions import MatrixSymbol, HadamardProduct
    >>> from sympy.matrices.expressions import OneMatrix, ZeroMatrix
    >>> from sympy.matrices.expressions.hadamard import canonicalize
    >>> from sympy import init_printing
    >>> init_printing(use_unicode=False)

    >>> A = MatrixSymbol('A', 2, 2)
    >>> B = MatrixSymbol('B', 2, 2)
    >>> C = MatrixSymbol('C', 2, 2)

    Hadamard product associativity:

    >>> X = HadamardProduct(A, HadamardProduct(B, C))
    >>> X
    A.*(B.*C)
    >>> canonicalize(X)
    A.*B.*C

    Hadamard product commutativity:

    >>> X = HadamardProduct(A, B)
    >>> Y = HadamardProduct(B, A)
    >>> X
    A.*B
    >>> Y
    B.*A
    >>> canonicalize(X)
    A.*B
    >>> canonicalize(Y)
    A.*B

    Hadamard product identity:

    >>> X = HadamardProduct(A, OneMatrix(2, 2))
    >>> X
    A.*1
    >>> canonicalize(X)
    A

    Absorbing element of Hadamard product:

    >>> X = HadamardProduct(A, ZeroMatrix(2, 2))
    >>> X
    A.*0
    >>> canonicalize(X)
    0

    Rewriting to Hadamard Power

    >>> X = HadamardProduct(A, A, A)
    >>> X
    A.*A.*A
    >>> canonicalize(X)
     .3
    A

    Notes
    =====

    As the Hadamard product is associative, nested products can be flattened.

    The Hadamard product is commutative so that factors can be sorted for
    canonical form.

    A matrix of only ones is an identity for Hadamard product,
    so every matrices of only ones can be removed.

    Any zero matrix will make the whole product a zero matrix.

    Duplicate elements can be collected and rewritten as HadamardPower

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
    """
    # Associativity
    rule = condition(
            lambda x: isinstance(x, HadamardProduct),
            flatten
        )
    fun = exhaust(rule)
    x = fun(x)

    # Identity
    fun = condition(
            lambda x: isinstance(x, HadamardProduct),
            rm_id(lambda x: isinstance(x, OneMatrix))
        )
    x = fun(x)

    # Absorbing by Zero Matrix
    def absorb(x):
        if any(isinstance(c, ZeroMatrix) for c in x.args):
            return ZeroMatrix(*x.shape)
        else:
            return x
    fun = condition(
            lambda x: isinstance(x, HadamardProduct),
            absorb
        )
    x = fun(x)

    # Rewriting with HadamardPower
    if isinstance(x, HadamardProduct):
        from collections import Counter
        tally = Counter(x.args)

        new_arg = []
        for base, exp in tally.items():
            if exp == 1:
                new_arg.append(base)
            else:
                new_arg.append(HadamardPower(base, exp))

        x = HadamardProduct(*new_arg)

    # Commutativity
    fun = condition(
            lambda x: isinstance(x, HadamardProduct),
            sort(default_sort_key)
        )
    x = fun(x)

    # Unpacking
    x = unpack(x)
    return x
Пример #4
0
    def doit(self, **ignored):
        return canonicalize(self)


def validate(*args):
    if not all(arg.is_Matrix for arg in args):
        raise TypeError("Mix of Matrix and Scalar symbols")
    A = args[0]
    for B in args[1:]:
        if A.shape != B.shape:
            raise ShapeError("Matrices %s and %s are not aligned" % (A, B))


factor_of = lambda arg: arg.as_coeff_mmul()[0]
matrix_of = lambda arg: unpack(arg.as_coeff_mmul()[1])


def combine(cnt, mat):
    from .matmul import MatMul
    if cnt == 1:
        return mat
    else:
        return cnt * mat


rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)), unpack, flatten,
         glom(matrix_of, factor_of, combine), sort(default_sort_key))

canonicalize = exhaust(
    condition(lambda x: isinstance(x, MatAdd), do_one(*rules)))
Пример #5
0
def canonicalize(x):
    """Canonicalize the Hadamard product ``x`` with mathematical properties.

    Examples
    ========

    >>> from sympy.matrices.expressions import MatrixSymbol, HadamardProduct
    >>> from sympy.matrices.expressions import OneMatrix, ZeroMatrix
    >>> from sympy.matrices.expressions.hadamard import canonicalize

    >>> A = MatrixSymbol('A', 2, 2)
    >>> B = MatrixSymbol('B', 2, 2)
    >>> C = MatrixSymbol('C', 2, 2)

    Hadamard product associativity:

    >>> X = HadamardProduct(A, HadamardProduct(B, C))
    >>> X
    A.*(B.*C)
    >>> canonicalize(X)
    A.*B.*C

    Hadamard product commutativity:

    >>> X = HadamardProduct(A, B)
    >>> Y = HadamardProduct(B, A)
    >>> X
    A.*B
    >>> Y
    B.*A
    >>> canonicalize(X)
    A.*B
    >>> canonicalize(Y)
    A.*B

    Hadamard product identity:

    >>> X = HadamardProduct(A, OneMatrix(2, 2))
    >>> X
    A.*OneMatrix(2, 2)
    >>> canonicalize(X)
    A

    Absorbing element of Hadamard product:

    >>> X = HadamardProduct(A, ZeroMatrix(2, 2))
    >>> X
    A.*0
    >>> canonicalize(X)
    0

    Rewriting to Hadamard Power

    >>> X = HadamardProduct(A, A, A)
    >>> X
    A.*A.*A
    >>> canonicalize(X)
    A.**3

    Notes
    =====

    As the Hadamard product is associative, nested products can be flattened.

    The Hadamard product is commutative so that factors can be sorted for
    canonical form.

    A matrix of only ones is an identity for Hadamard product,
    so every matrices of only ones can be removed.

    Any zero matrix will make the whole product a zero matrix.

    Duplicate elements can be collected and rewritten as HadamardPower

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
    """
    from sympy.core.compatibility import default_sort_key

    # Associativity
    rule = condition(
            lambda x: isinstance(x, HadamardProduct),
            flatten
        )
    fun = exhaust(rule)
    x = fun(x)

    # Identity
    fun = condition(
            lambda x: isinstance(x, HadamardProduct),
            rm_id(lambda x: isinstance(x, OneMatrix))
        )
    x = fun(x)

    # Absorbing by Zero Matrix
    def absorb(x):
        if any(isinstance(c, ZeroMatrix) for c in x.args):
            return ZeroMatrix(*x.shape)
        else:
            return x
    fun = condition(
            lambda x: isinstance(x, HadamardProduct),
            absorb
        )
    x = fun(x)

    # Rewriting with HadamardPower
    if isinstance(x, HadamardProduct):
        from collections import Counter
        tally = Counter(x.args)

        new_arg = []
        for base, exp in tally.items():
            if exp == 1:
                new_arg.append(base)
            else:
                new_arg.append(HadamardPower(base, exp))

        x = HadamardProduct(*new_arg)

    # Commutativity
    fun = condition(
            lambda x: isinstance(x, HadamardProduct),
            sort(default_sort_key)
        )
    x = fun(x)

    # Unpacking
    x = unpack(x)
    return x
Пример #6
0
    >>> from sympy.matrices.expressions.matadd import merge_explicit
    >>> A = MatrixSymbol('A', 2, 2)
    >>> B = eye(2)
    >>> C = Matrix([[1, 2], [3, 4]])
    >>> X = MatAdd(A, B, C)
    >>> pprint(X)
        [1  0]   [1  2]
    A + [    ] + [    ]
        [0  1]   [3  4]
    >>> pprint(merge_explicit(X))
        [2  2]
    A + [    ]
        [3  5]
    """
    groups = sift(matadd.args, lambda arg: isinstance(arg, MatrixBase))
    if len(groups[True]) > 1:
        return MatAdd(*(groups[False] + [reduce(add, groups[True])]))
    else:
        return matadd


rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)),
         unpack,
         flatten,
         glom(matrix_of, factor_of, combine),
         merge_explicit,
         sort(default_sort_key))

canonicalize = exhaust(condition(lambda x: isinstance(x, MatAdd),
                                 do_one(*rules)))
Пример #7
0
        return MatAdd(*[Trace(arg) for arg in self.args]).doit()

    def doit(self, **ignored):
        return canonicalize(self)

def validate(*args):
    if not all(arg.is_Matrix for arg in args):
        raise TypeError("Mix of Matrix and Scalar symbols")
    A = args[0]
    for B in args[1:]:
        if A.shape != B.shape:
            raise ShapeError("Matrices %s and %s are not aligned"%(A, B))

factor_of = lambda arg: arg.as_coeff_mmul()[0]
matrix_of = lambda arg: unpack(arg.as_coeff_mmul()[1])
def combine(cnt, mat):
    from matmul import MatMul
    if cnt == 1:
        return mat
    else:
        return cnt * mat

rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)),
         unpack,
         flatten,
         glom(matrix_of, factor_of, combine),
         sort(str))

canonicalize = exhaust(condition(lambda x: isinstance(x, MatAdd),
                                 do_one(*rules)))
Пример #8
0
    # there has to be some bug deep inside the core, I absolutely need this
    # function to correctly perform addition
    def recreate_args(args):
        return [a.func(*a.args) for a in args]

    # Adapted from sympy.matrices.expressions.matadd.py
    groups = sift(vecadd.args, lambda arg: isinstance(arg, (Vector)))
    if len(groups[True]) > 1:
        return VecAdd(*(recreate_args(groups[False]) +
                        [reduce(add, recreate_args(groups[True]))]))
        # return VecAdd(*(groups[False] + [reduce(add, groups[True])]))
    else:
        return vecadd


rules = (merge_explicit, sort(default_sort_key))

canonicalize = exhaust(
    condition(lambda x: isinstance(x, VecAdd), do_one(*rules)))


class VecMul(VectorExpr, Mul):
    """ A product of Vector expressions.

    VecMul inherits from and operates like SymPy Mul.
    """
    is_VecMul = True
    is_commutative = True

    def __new__(cls, *args, **kwargs):
        evaluate = kwargs.get('evaluate', True)
Пример #9
0
    def doit(self, **ignored):
        return canonicalize(self)


def validate(*args):
    if not all(arg.is_Matrix for arg in args):
        raise TypeError("Mix of Matrix and Scalar symbols")
    A = args[0]
    for B in args[1:]:
        if A.shape != B.shape:
            raise ShapeError("Matrices %s and %s are not aligned" % (A, B))


factor_of = lambda arg: arg.as_coeff_mmul()[0]
matrix_of = lambda arg: unpack(arg.as_coeff_mmul()[1])


def combine(cnt, mat):
    from matmul import MatMul
    if cnt == 1:
        return mat
    else:
        return cnt * mat


rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)), unpack, flatten,
         glom(matrix_of, factor_of, combine), sort(str))

canonicalize = exhaust(
    condition(lambda x: isinstance(x, MatAdd), do_one(*rules)))
Пример #10
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        return super(HadamardProduct, cls).__new__(cls, *args)

    @property
    def shape(self):
        return self.args[0].shape

    def _entry(self, i, j):
        return Mul(*[arg._entry(i, j) for arg in self.args])

    def _eval_transpose(self):
        from transpose import Transpose
        return HadamardProduct(*[Transpose(arg) for arg in self.args])

    def doit(self, **ignored):
        return canonicalize(self)

def validate(*args):
    if not all(arg.is_Matrix for arg in args):
        raise TypeError("Mix of Matrix and Scalar symbols")
    A = args[0]
    for B in args[1:]:
        if A.shape != B.shape:
            raise ShapeError("Matrices %s and %s are not aligned" % (A, B))

rules = (unpack,
         flatten,
         sort(str))

canonicalize = exhaust(condition(lambda x: isinstance(x, HadamardProduct),
                                 do_one(*rules)))