MLPY/Lib/site-packages/sympy/matrices/expressions/matmul.py

from sympy.assumptions.ask import ask, Q
from sympy.assumptions.refine import handlers_dict
from sympy.core import Basic, sympify, S
from sympy.core.mul import mul, Mul
from sympy.core.numbers import Number, Integer
from sympy.core.symbol import Dummy
from sympy.functions import adjoint
from sympy.strategies import (rm_id, unpack, typed, flatten, exhaust,
        do_one, new)
from sympy.matrices.exceptions import NonInvertibleMatrixError
from sympy.matrices.matrixbase import MatrixBase
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.matrices.expressions._shape import validate_matmul_integer as validate

from .inverse import Inverse
from .matexpr import MatrixExpr
from .matpow import MatPow
from .transpose import transpose
from .permutation import PermutationMatrix
from .special import ZeroMatrix, Identity, GenericIdentity, OneMatrix


# XXX: MatMul should perhaps not subclass directly from Mul
class MatMul(MatrixExpr, Mul):
    """
    A product of matrix expressions

    Examples
    ========

    >>> from sympy import MatMul, MatrixSymbol
    >>> A = MatrixSymbol('A', 5, 4)
    >>> B = MatrixSymbol('B', 4, 3)
    >>> C = MatrixSymbol('C', 3, 6)
    >>> MatMul(A, B, C)
    A*B*C
    """
    is_MatMul = True

    identity = GenericIdentity()

    def __new__(cls, *args, evaluate=False, check=None, _sympify=True):
        if not args:
            return cls.identity

        # This must be removed aggressively in the constructor to avoid
        # TypeErrors from GenericIdentity().shape
        args = list(filter(lambda i: cls.identity != i, args))
        if _sympify:
            args = list(map(sympify, args))
        obj = Basic.__new__(cls, *args)
        factor, matrices = obj.as_coeff_matrices()

        if check is not None:
            sympy_deprecation_warning(
                "Passing check to MatMul is deprecated and the check argument will be removed in a future version.",
                deprecated_since_version="1.11",
                active_deprecations_target='remove-check-argument-from-matrix-operations')

        if check is not False:
            validate(*matrices)

        if not matrices:
            # Should it be
            #
            # return Basic.__neq__(cls, factor, GenericIdentity()) ?
            return factor

        if evaluate:
            return cls._evaluate(obj)

        return obj

    @classmethod
    def _evaluate(cls, expr):
        return canonicalize(expr)

    @property
    def shape(self):
        matrices = [arg for arg in self.args if arg.is_Matrix]
        return (matrices[0].rows, matrices[-1].cols)

    def _entry(self, i, j, expand=True, **kwargs):
        # Avoid cyclic imports
        from sympy.concrete.summations import Sum
        from sympy.matrices.immutable import ImmutableMatrix

        coeff, matrices = self.as_coeff_matrices()

        if len(matrices) == 1:  # situation like 2*X, matmul is just X
            return coeff * matrices[0][i, j]

        indices = [None]*(len(matrices) + 1)
        ind_ranges = [None]*(len(matrices) - 1)
        indices[0] = i
        indices[-1] = j

        def f():
            counter = 1
            while True:
                yield Dummy("i_%i" % counter)
                counter += 1

        dummy_generator = kwargs.get("dummy_generator", f())

        for i in range(1, len(matrices)):
            indices[i] = next(dummy_generator)

        for i, arg in enumerate(matrices[:-1]):
            ind_ranges[i] = arg.shape[1] - 1
        matrices = [arg._entry(indices[i], indices[i+1], dummy_generator=dummy_generator) for i, arg in enumerate(matrices)]
        expr_in_sum = Mul.fromiter(matrices)
        if any(v.has(ImmutableMatrix) for v in matrices):
            expand = True
        result = coeff*Sum(
                expr_in_sum,
                *zip(indices[1:-1], [0]*len(ind_ranges), ind_ranges)
            )

        # Don't waste time in result.doit() if the sum bounds are symbolic
        if not any(isinstance(v, (Integer, int)) for v in ind_ranges):
            expand = False
        return result.doit() if expand else result

    def as_coeff_matrices(self):
        scalars = [x for x in self.args if not x.is_Matrix]
        matrices = [x for x in self.args if x.is_Matrix]
        coeff = Mul(*scalars)
        if coeff.is_commutative is False:
            raise NotImplementedError("noncommutative scalars in MatMul are not supported.")

        return coeff, matrices

    def as_coeff_mmul(self):
        coeff, matrices = self.as_coeff_matrices()
        return coeff, MatMul(*matrices)

    def expand(self, **kwargs):
        expanded = super(MatMul, self).expand(**kwargs)
        return self._evaluate(expanded)

    def _eval_transpose(self):
        """Transposition of matrix multiplication.

        Notes
        =====

        The following rules are applied.

        Transposition for matrix multiplied with another matrix:
        `\\left(A B\\right)^{T} = B^{T} A^{T}`

        Transposition for matrix multiplied with scalar:
        `\\left(c A\\right)^{T} = c A^{T}`

        References
        ==========

        .. [1] https://en.wikipedia.org/wiki/Transpose
        """
        coeff, matrices = self.as_coeff_matrices()
        return MatMul(
            coeff, *[transpose(arg) for arg in matrices[::-1]]).doit()

    def _eval_adjoint(self):
        return MatMul(*[adjoint(arg) for arg in self.args[::-1]]).doit()

    def _eval_trace(self):
        factor, mmul = self.as_coeff_mmul()
        if factor != 1:
            from .trace import trace
            return factor * trace(mmul.doit())

    def _eval_determinant(self):
        from sympy.matrices.expressions.determinant import Determinant
        factor, matrices = self.as_coeff_matrices()
        square_matrices = only_squares(*matrices)
        return factor**self.rows * Mul(*list(map(Determinant, square_matrices)))

    def _eval_inverse(self):
        if all(arg.is_square for arg in self.args if isinstance(arg, MatrixExpr)):
            return MatMul(*(
                arg.inverse() if isinstance(arg, MatrixExpr) else arg**-1
                    for arg in self.args[::-1]
                )
            ).doit()
        return Inverse(self)

    def doit(self, **hints):
        deep = hints.get('deep', True)
        if deep:
            args = tuple(arg.doit(**hints) for arg in self.args)
        else:
            args = self.args

        # treat scalar*MatrixSymbol or scalar*MatPow separately
        expr = canonicalize(MatMul(*args))
        return expr

    # Needed for partial compatibility with Mul
    def args_cnc(self, cset=False, warn=True, **kwargs):
        coeff_c = [x for x in self.args if x.is_commutative]
        coeff_nc = [x for x in self.args if not x.is_commutative]
        if cset:
            clen = len(coeff_c)
            coeff_c = set(coeff_c)
            if clen and warn and len(coeff_c) != clen:
                raise ValueError('repeated commutative arguments: %s' %
                                 [ci for ci in coeff_c if list(self.args).count(ci) > 1])
        return [coeff_c, coeff_nc]

    def _eval_derivative_matrix_lines(self, x):
        from .transpose import Transpose
        with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
        lines = []
        for ind in with_x_ind:
            left_args = self.args[:ind]
            right_args = self.args[ind+1:]

            if right_args:
                right_mat = MatMul.fromiter(right_args)
            else:
                right_mat = Identity(self.shape[1])
            if left_args:
                left_rev = MatMul.fromiter([Transpose(i).doit() if i.is_Matrix else i for i in reversed(left_args)])
            else:
                left_rev = Identity(self.shape[0])

            d = self.args[ind]._eval_derivative_matrix_lines(x)
            for i in d:
                i.append_first(left_rev)
                i.append_second(right_mat)
                lines.append(i)

        return lines

mul.register_handlerclass((Mul, MatMul), MatMul)


# Rules
def newmul(*args):
    if args[0] == 1:
        args = args[1:]
    return new(MatMul, *args)

def any_zeros(mul):
    if any(arg.is_zero or (arg.is_Matrix and arg.is_ZeroMatrix)
                       for arg in mul.args):
        matrices = [arg for arg in mul.args if arg.is_Matrix]
        return ZeroMatrix(matrices[0].rows, matrices[-1].cols)
    return mul

def merge_explicit(matmul):
    """ Merge explicit MatrixBase arguments

    >>> from sympy import MatrixSymbol, Matrix, MatMul, pprint
    >>> from sympy.matrices.expressions.matmul import merge_explicit
    >>> A = MatrixSymbol('A', 2, 2)
    >>> B = Matrix([[1, 1], [1, 1]])
    >>> C = Matrix([[1, 2], [3, 4]])
    >>> X = MatMul(A, B, C)
    >>> pprint(X)
      [1  1] [1  2]
    A*[    ]*[    ]
      [1  1] [3  4]
    >>> pprint(merge_explicit(X))
      [4  6]
    A*[    ]
      [4  6]

    >>> X = MatMul(B, A, C)
    >>> pprint(X)
    [1  1]   [1  2]
    [    ]*A*[    ]
    [1  1]   [3  4]
    >>> pprint(merge_explicit(X))
    [1  1]   [1  2]
    [    ]*A*[    ]
    [1  1]   [3  4]
    """
    if not any(isinstance(arg, MatrixBase) for arg in matmul.args):
        return matmul
    newargs = []
    last = matmul.args[0]
    for arg in matmul.args[1:]:
        if isinstance(arg, (MatrixBase, Number)) and isinstance(last, (MatrixBase, Number)):
            last = last * arg
        else:
            newargs.append(last)
            last = arg
    newargs.append(last)

    return MatMul(*newargs)

def remove_ids(mul):
    """ Remove Identities from a MatMul

    This is a modified version of sympy.strategies.rm_id.
    This is necesssary because MatMul may contain both MatrixExprs and Exprs
    as args.

    See Also
    ========

    sympy.strategies.rm_id
    """
    # Separate Exprs from MatrixExprs in args
    factor, mmul = mul.as_coeff_mmul()
    # Apply standard rm_id for MatMuls
    result = rm_id(lambda x: x.is_Identity is True)(mmul)
    if result != mmul:
        return newmul(factor, *result.args)  # Recombine and return
    else:
        return mul

def factor_in_front(mul):
    factor, matrices = mul.as_coeff_matrices()
    if factor != 1:
        return newmul(factor, *matrices)
    return mul

def combine_powers(mul):
    r"""Combine consecutive powers with the same base into one, e.g.
    $$A \times A^2 \Rightarrow A^3$$

    This also cancels out the possible matrix inverses using the
    knowledgebase of :class:`~.Inverse`, e.g.,
    $$ Y \times X \times X^{-1} \Rightarrow Y $$
    """
    factor, args = mul.as_coeff_matrices()
    new_args = [args[0]]

    for i in range(1, len(args)):
        A = new_args[-1]
        B = args[i]

        if isinstance(B, Inverse) and isinstance(B.arg, MatMul):
            Bargs = B.arg.args
            l = len(Bargs)
            if list(Bargs) == new_args[-l:]:
                new_args = new_args[:-l] + [Identity(B.shape[0])]
                continue

        if isinstance(A, Inverse) and isinstance(A.arg, MatMul):
            Aargs = A.arg.args
            l = len(Aargs)
            if list(Aargs) == args[i:i+l]:
                identity = Identity(A.shape[0])
                new_args[-1] = identity
                for j in range(i, i+l):
                    args[j] = identity
                continue

        if A.is_square == False or B.is_square == False:
            new_args.append(B)
            continue

        if isinstance(A, MatPow):
            A_base, A_exp = A.args
        else:
            A_base, A_exp = A, S.One

        if isinstance(B, MatPow):
            B_base, B_exp = B.args
        else:
            B_base, B_exp = B, S.One

        if A_base == B_base:
            new_exp = A_exp + B_exp
            new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
            continue
        elif not isinstance(B_base, MatrixBase):
            try:
                B_base_inv = B_base.inverse()
            except NonInvertibleMatrixError:
                B_base_inv = None
            if B_base_inv is not None and A_base == B_base_inv:
                new_exp = A_exp - B_exp
                new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
                continue
        new_args.append(B)

    return newmul(factor, *new_args)

def combine_permutations(mul):
    """Refine products of permutation matrices as the products of cycles.
    """
    args = mul.args
    l = len(args)
    if l < 2:
        return mul

    result = [args[0]]
    for i in range(1, l):
        A = result[-1]
        B = args[i]
        if isinstance(A, PermutationMatrix) and \
            isinstance(B, PermutationMatrix):
            cycle_1 = A.args[0]
            cycle_2 = B.args[0]
            result[-1] = PermutationMatrix(cycle_1 * cycle_2)
        else:
            result.append(B)

    return MatMul(*result)

def combine_one_matrices(mul):
    """
    Combine products of OneMatrix

    e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4)
    """
    factor, args = mul.as_coeff_matrices()
    new_args = [args[0]]

    for B in args[1:]:
        A = new_args[-1]
        if not isinstance(A, OneMatrix) or not isinstance(B, OneMatrix):
            new_args.append(B)
            continue
        new_args.pop()
        new_args.append(OneMatrix(A.shape[0], B.shape[1]))
        factor *= A.shape[1]

    return newmul(factor, *new_args)

def distribute_monom(mul):
    """
    Simplify MatMul expressions but distributing
    rational term to MatMul.

    e.g. 2*(A+B) -> 2*A + 2*B
    """
    args = mul.args
    if len(args) == 2:
        from .matadd import MatAdd
        if args[0].is_MatAdd and args[1].is_Rational:
            return MatAdd(*[MatMul(mat, args[1]).doit() for mat in args[0].args])
        if args[1].is_MatAdd and args[0].is_Rational:
            return MatAdd(*[MatMul(args[0], mat).doit() for mat in args[1].args])
    return mul

rules = (
    distribute_monom, any_zeros, remove_ids, combine_one_matrices, combine_powers, unpack, rm_id(lambda x: x == 1),
    merge_explicit, factor_in_front, flatten, combine_permutations)

canonicalize = exhaust(typed({MatMul: do_one(*rules)}))

def only_squares(*matrices):
    """factor matrices only if they are square"""
    if matrices[0].rows != matrices[-1].cols:
        raise RuntimeError("Invalid matrices being multiplied")
    out = []
    start = 0
    for i, M in enumerate(matrices):
        if M.cols == matrices[start].rows:
            out.append(MatMul(*matrices[start:i+1]).doit())
            start = i+1
    return out


def refine_MatMul(expr, assumptions):
    """
    >>> from sympy import MatrixSymbol, Q, assuming, refine
    >>> X = MatrixSymbol('X', 2, 2)
    >>> expr = X * X.T
    >>> print(expr)
    X*X.T
    >>> with assuming(Q.orthogonal(X)):
    ...     print(refine(expr))
    I
    """
    newargs = []
    exprargs = []

    for args in expr.args:
        if args.is_Matrix:
            exprargs.append(args)
        else:
            newargs.append(args)

    last = exprargs[0]
    for arg in exprargs[1:]:
        if arg == last.T and ask(Q.orthogonal(arg), assumptions):
            last = Identity(arg.shape[0])
        elif arg == last.conjugate() and ask(Q.unitary(arg), assumptions):
            last = Identity(arg.shape[0])
        else:
            newargs.append(last)
            last = arg
    newargs.append(last)

    return MatMul(*newargs)


handlers_dict['MatMul'] = refine_MatMul