Sources_For_Win/ParaView/Lib/site-packages/sympy/matrices/expressions/blockmatrix.py

from sympy.assumptions.ask import (Q, ask)
from sympy.core import Basic, Add, Mul, S
from sympy.core.sympify import _sympify
from sympy.functions.elementary.complexes import re, im
from sympy.strategies import typed, exhaust, condition, do_one, unpack
from sympy.strategies.traverse import bottom_up
from sympy.utilities.iterables import is_sequence, sift
from sympy.utilities.misc import filldedent

from sympy.matrices import Matrix, ShapeError
from sympy.matrices.common import NonInvertibleMatrixError
from sympy.matrices.expressions.determinant import det, Determinant
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.matexpr import MatrixExpr, MatrixElement
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions.special import ZeroMatrix, Identity
from sympy.matrices.expressions.trace import trace
from sympy.matrices.expressions.transpose import Transpose, transpose


class BlockMatrix(MatrixExpr):
    """A BlockMatrix is a Matrix comprised of other matrices.

    The submatrices are stored in a SymPy Matrix object but accessed as part of
    a Matrix Expression

    >>> from sympy import (MatrixSymbol, BlockMatrix, symbols,
    ...     Identity, ZeroMatrix, block_collapse)
    >>> n,m,l = symbols('n m l')
    >>> X = MatrixSymbol('X', n, n)
    >>> Y = MatrixSymbol('Y', m, m)
    >>> Z = MatrixSymbol('Z', n, m)
    >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
    >>> print(B)
    Matrix([
    [X, Z],
    [0, Y]])

    >>> C = BlockMatrix([[Identity(n), Z]])
    >>> print(C)
    Matrix([[I, Z]])

    >>> print(block_collapse(C*B))
    Matrix([[X, Z + Z*Y]])

    Some matrices might be comprised of rows of blocks with
    the matrices in each row having the same height and the
    rows all having the same total number of columns but
    not having the same number of columns for each matrix
    in each row. In this case, the matrix is not a block
    matrix and should be instantiated by Matrix.

    >>> from sympy import ones, Matrix
    >>> dat = [
    ... [ones(3,2), ones(3,3)*2],
    ... [ones(2,3)*3, ones(2,2)*4]]
    ...
    >>> BlockMatrix(dat)
    Traceback (most recent call last):
    ...
    ValueError:
    Although this matrix is comprised of blocks, the blocks do not fill
    the matrix in a size-symmetric fashion. To create a full matrix from
    these arguments, pass them directly to Matrix.
    >>> Matrix(dat)
    Matrix([
    [1, 1, 2, 2, 2],
    [1, 1, 2, 2, 2],
    [1, 1, 2, 2, 2],
    [3, 3, 3, 4, 4],
    [3, 3, 3, 4, 4]])

    See Also
    ========
    sympy.matrices.matrices.MatrixBase.irregular
    """
    def __new__(cls, *args, **kwargs):
        from sympy.matrices.immutable import ImmutableDenseMatrix
        isMat = lambda i: getattr(i, 'is_Matrix', False)
        if len(args) != 1 or \
                not is_sequence(args[0]) or \
                len({isMat(r) for r in args[0]}) != 1:
            raise ValueError(filldedent('''
                expecting a sequence of 1 or more rows
                containing Matrices.'''))
        rows = args[0] if args else []
        if not isMat(rows):
            if rows and isMat(rows[0]):
                rows = [rows]  # rows is not list of lists or []
            # regularity check
            # same number of matrices in each row
            blocky = ok = len({len(r) for r in rows}) == 1
            if ok:
                # same number of rows for each matrix in a row
                for r in rows:
                    ok = len({i.rows for i in r}) == 1
                    if not ok:
                        break
                blocky = ok
                if ok:
                    # same number of cols for each matrix in each col
                    for c in range(len(rows[0])):
                        ok = len({rows[i][c].cols
                            for i in range(len(rows))}) == 1
                        if not ok:
                            break
            if not ok:
                # same total cols in each row
                ok = len({
                    sum([i.cols for i in r]) for r in rows}) == 1
                if blocky and ok:
                    raise ValueError(filldedent('''
                        Although this matrix is comprised of blocks,
                        the blocks do not fill the matrix in a
                        size-symmetric fashion. To create a full matrix
                        from these arguments, pass them directly to
                        Matrix.'''))
                raise ValueError(filldedent('''
                    When there are not the same number of rows in each
                    row's matrices or there are not the same number of
                    total columns in each row, the matrix is not a
                    block matrix. If this matrix is known to consist of
                    blocks fully filling a 2-D space then see
                    Matrix.irregular.'''))
        mat = ImmutableDenseMatrix(rows, evaluate=False)
        obj = Basic.__new__(cls, mat)
        return obj

    @property
    def shape(self):
        numrows = numcols = 0
        M = self.blocks
        for i in range(M.shape[0]):
            numrows += M[i, 0].shape[0]
        for i in range(M.shape[1]):
            numcols += M[0, i].shape[1]
        return (numrows, numcols)

    @property
    def blockshape(self):
        return self.blocks.shape

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

    @property
    def rowblocksizes(self):
        return [self.blocks[i, 0].rows for i in range(self.blockshape[0])]

    @property
    def colblocksizes(self):
        return [self.blocks[0, i].cols for i in range(self.blockshape[1])]

    def structurally_equal(self, other):
        return (isinstance(other, BlockMatrix)
            and self.shape == other.shape
            and self.blockshape == other.blockshape
            and self.rowblocksizes == other.rowblocksizes
            and self.colblocksizes == other.colblocksizes)

    def _blockmul(self, other):
        if (isinstance(other, BlockMatrix) and
                self.colblocksizes == other.rowblocksizes):
            return BlockMatrix(self.blocks*other.blocks)

        return self * other

    def _blockadd(self, other):
        if (isinstance(other, BlockMatrix)
                and self.structurally_equal(other)):
            return BlockMatrix(self.blocks + other.blocks)

        return self + other

    def _eval_transpose(self):
        # Flip all the individual matrices
        matrices = [transpose(matrix) for matrix in self.blocks]
        # Make a copy
        M = Matrix(self.blockshape[0], self.blockshape[1], matrices)
        # Transpose the block structure
        M = M.transpose()
        return BlockMatrix(M)

    def _eval_trace(self):
        if self.rowblocksizes == self.colblocksizes:
            return Add(*[trace(self.blocks[i, i])
                        for i in range(self.blockshape[0])])
        raise NotImplementedError(
            "Can't perform trace of irregular blockshape")

    def _eval_determinant(self):
        if self.blockshape == (1, 1):
            return det(self.blocks[0, 0])
        if self.blockshape == (2, 2):
            [[A, B],
             [C, D]] = self.blocks.tolist()
            if ask(Q.invertible(A)):
                return det(A)*det(D - C*A.I*B)
            elif ask(Q.invertible(D)):
                return det(D)*det(A - B*D.I*C)
        return Determinant(self)

    def as_real_imag(self):
        real_matrices = [re(matrix) for matrix in self.blocks]
        real_matrices = Matrix(self.blockshape[0], self.blockshape[1], real_matrices)

        im_matrices = [im(matrix) for matrix in self.blocks]
        im_matrices = Matrix(self.blockshape[0], self.blockshape[1], im_matrices)

        return (real_matrices, im_matrices)

    def transpose(self):
        """Return transpose of matrix.

        Examples
        ========

        >>> from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix
        >>> from sympy.abc import m, n
        >>> X = MatrixSymbol('X', n, n)
        >>> Y = MatrixSymbol('Y', m, m)
        >>> Z = MatrixSymbol('Z', n, m)
        >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
        >>> B.transpose()
        Matrix([
        [X.T,  0],
        [Z.T, Y.T]])
        >>> _.transpose()
        Matrix([
        [X, Z],
        [0, Y]])
        """
        return self._eval_transpose()

    def schur(self, mat = 'A', generalized = False):
        """Return the Schur Complement of the 2x2 BlockMatrix

        Parameters
        ==========

        mat : String, optional
            The matrix with respect to which the
            Schur Complement is calculated. 'A' is
            used by default

        generalized : bool, optional
            If True, returns the generalized Schur
            Component which uses Moore-Penrose Inverse

        Examples
        ========

        >>> from sympy import symbols, MatrixSymbol, BlockMatrix
        >>> m, n = symbols('m n')
        >>> A = MatrixSymbol('A', n, n)
        >>> B = MatrixSymbol('B', n, m)
        >>> C = MatrixSymbol('C', m, n)
        >>> D = MatrixSymbol('D', m, m)
        >>> X = BlockMatrix([[A, B], [C, D]])

        The default Schur Complement is evaluated with "A"

        >>> X.schur()
        -C*A**(-1)*B + D
        >>> X.schur('D')
        A - B*D**(-1)*C

        Schur complement with non-invertible matrices is not
        defined. Instead, the generalized Schur complement can
        be calculated which uses the Moore-Penrose Inverse. To
        achieve this, `generalized` must be set to `True`

        >>> X.schur('B', generalized=True)
        C - D*(B.T*B)**(-1)*B.T*A
        >>> X.schur('C', generalized=True)
        -A*(C.T*C)**(-1)*C.T*D + B

        Returns
        =======

        M : Matrix
            The Schur Complement Matrix

        Raises
        ======

        ShapeError
            If the block matrix is not a 2x2 matrix

        NonInvertibleMatrixError
            If given matrix is non-invertible

        References
        ==========

        .. [1] Wikipedia Article on Schur Component : https://en.wikipedia.org/wiki/Schur_complement

        See Also
        ========

        sympy.matrices.matrices.MatrixBase.pinv
        """

        if self.blockshape == (2, 2):
            [[A, B],
             [C, D]] = self.blocks.tolist()
            d={'A' : A, 'B' : B, 'C' : C, 'D' : D}
            try:
                inv = (d[mat].T*d[mat]).inv()*d[mat].T if generalized else d[mat].inv()
                if mat == 'A':
                    return D - C * inv * B
                elif mat == 'B':
                    return C - D * inv * A
                elif mat == 'C':
                    return B - A * inv * D
                elif mat == 'D':
                    return A - B * inv * C
                #For matrices where no sub-matrix is square
                return self
            except NonInvertibleMatrixError:
                raise NonInvertibleMatrixError('The given matrix is not invertible. Please set generalized=True \
            to compute the generalized Schur Complement which uses Moore-Penrose Inverse')
        else:
            raise ShapeError('Schur Complement can only be calculated for 2x2 block matrices')

    def LDUdecomposition(self):
        """Returns the Block LDU decomposition of
        a 2x2 Block Matrix

        Returns
        =======

        (L, D, U) : Matrices
            L : Lower Diagonal Matrix
            D : Diagonal Matrix
            U : Upper Diagonal Matrix

        Examples
        ========

        >>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
        >>> m, n = symbols('m n')
        >>> A = MatrixSymbol('A', n, n)
        >>> B = MatrixSymbol('B', n, m)
        >>> C = MatrixSymbol('C', m, n)
        >>> D = MatrixSymbol('D', m, m)
        >>> X = BlockMatrix([[A, B], [C, D]])
        >>> L, D, U = X.LDUdecomposition()
        >>> block_collapse(L*D*U)
        Matrix([
        [A, B],
        [C, D]])

        Raises
        ======

        ShapeError
            If the block matrix is not a 2x2 matrix

        NonInvertibleMatrixError
            If the matrix "A" is non-invertible

        See Also
        ========
        sympy.matrices.expressions.blockmatrix.BlockMatrix.UDLdecomposition
        sympy.matrices.expressions.blockmatrix.BlockMatrix.LUdecomposition
        """
        if self.blockshape == (2,2):
            [[A, B],
             [C, D]] = self.blocks.tolist()
            try:
                AI = A.I
            except NonInvertibleMatrixError:
                raise NonInvertibleMatrixError('Block LDU decomposition cannot be calculated when\
                    "A" is singular')
            Ip = Identity(B.shape[0])
            Iq = Identity(B.shape[1])
            Z = ZeroMatrix(*B.shape)
            L = BlockMatrix([[Ip, Z], [C*AI, Iq]])
            D = BlockDiagMatrix(A, self.schur())
            U = BlockMatrix([[Ip, AI*B],[Z.T, Iq]])
            return L, D, U
        else:
            raise ShapeError("Block LDU decomposition is supported only for 2x2 block matrices")

    def UDLdecomposition(self):
        """Returns the Block UDL decomposition of
        a 2x2 Block Matrix

        Returns
        =======

        (U, D, L) : Matrices
            U : Upper Diagonal Matrix
            D : Diagonal Matrix
            L : Lower Diagonal Matrix

        Examples
        ========

        >>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
        >>> m, n = symbols('m n')
        >>> A = MatrixSymbol('A', n, n)
        >>> B = MatrixSymbol('B', n, m)
        >>> C = MatrixSymbol('C', m, n)
        >>> D = MatrixSymbol('D', m, m)
        >>> X = BlockMatrix([[A, B], [C, D]])
        >>> U, D, L = X.UDLdecomposition()
        >>> block_collapse(U*D*L)
        Matrix([
        [A, B],
        [C, D]])

        Raises
        ======

        ShapeError
            If the block matrix is not a 2x2 matrix

        NonInvertibleMatrixError
            If the matrix "D" is non-invertible

        See Also
        ========
        sympy.matrices.expressions.blockmatrix.BlockMatrix.LDUdecomposition
        sympy.matrices.expressions.blockmatrix.BlockMatrix.LUdecomposition
        """
        if self.blockshape == (2,2):
            [[A, B],
             [C, D]] = self.blocks.tolist()
            try:
                DI = D.I
            except NonInvertibleMatrixError:
                raise NonInvertibleMatrixError('Block UDL decomposition cannot be calculated when\
                    "D" is singular')
            Ip = Identity(A.shape[0])
            Iq = Identity(B.shape[1])
            Z = ZeroMatrix(*B.shape)
            U = BlockMatrix([[Ip, B*DI], [Z.T, Iq]])
            D = BlockDiagMatrix(self.schur('D'), D)
            L = BlockMatrix([[Ip, Z],[DI*C, Iq]])
            return U, D, L
        else:
            raise ShapeError("Block UDL decomposition is supported only for 2x2 block matrices")

    def LUdecomposition(self):
        """Returns the Block LU decomposition of
        a 2x2 Block Matrix

        Returns
        =======

        (L, U) : Matrices
            L : Lower Diagonal Matrix
            U : Upper Diagonal Matrix

        Examples
        ========

        >>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
        >>> m, n = symbols('m n')
        >>> A = MatrixSymbol('A', n, n)
        >>> B = MatrixSymbol('B', n, m)
        >>> C = MatrixSymbol('C', m, n)
        >>> D = MatrixSymbol('D', m, m)
        >>> X = BlockMatrix([[A, B], [C, D]])
        >>> L, U = X.LUdecomposition()
        >>> block_collapse(L*U)
        Matrix([
        [A, B],
        [C, D]])

        Raises
        ======

        ShapeError
            If the block matrix is not a 2x2 matrix

        NonInvertibleMatrixError
            If the matrix "A" is non-invertible

        See Also
        ========
        sympy.matrices.expressions.blockmatrix.BlockMatrix.UDLdecomposition
        sympy.matrices.expressions.blockmatrix.BlockMatrix.LDUdecomposition
        """
        if self.blockshape == (2,2):
            [[A, B],
             [C, D]] = self.blocks.tolist()
            try:
                A = A**0.5
                AI = A.I
            except NonInvertibleMatrixError:
                raise NonInvertibleMatrixError('Block LU decomposition cannot be calculated when\
                    "A" is singular')
            Z = ZeroMatrix(*B.shape)
            Q = self.schur()**0.5
            L = BlockMatrix([[A, Z], [C*AI, Q]])
            U = BlockMatrix([[A, AI*B],[Z.T, Q]])
            return L, U
        else:
            raise ShapeError("Block LU decomposition is supported only for 2x2 block matrices")

    def _entry(self, i, j, **kwargs):
        # Find row entry
        orig_i, orig_j = i, j
        for row_block, numrows in enumerate(self.rowblocksizes):
            cmp = i < numrows
            if cmp == True:
                break
            elif cmp == False:
                i -= numrows
            elif row_block < self.blockshape[0] - 1:
                # Can't tell which block and it's not the last one, return unevaluated
                return MatrixElement(self, orig_i, orig_j)
        for col_block, numcols in enumerate(self.colblocksizes):
            cmp = j < numcols
            if cmp == True:
                break
            elif cmp == False:
                j -= numcols
            elif col_block < self.blockshape[1] - 1:
                return MatrixElement(self, orig_i, orig_j)
        return self.blocks[row_block, col_block][i, j]

    @property
    def is_Identity(self):
        if self.blockshape[0] != self.blockshape[1]:
            return False
        for i in range(self.blockshape[0]):
            for j in range(self.blockshape[1]):
                if i==j and not self.blocks[i, j].is_Identity:
                    return False
                if i!=j and not self.blocks[i, j].is_ZeroMatrix:
                    return False
        return True

    @property
    def is_structurally_symmetric(self):
        return self.rowblocksizes == self.colblocksizes

    def equals(self, other):
        if self == other:
            return True
        if (isinstance(other, BlockMatrix) and self.blocks == other.blocks):
            return True
        return super().equals(other)


class BlockDiagMatrix(BlockMatrix):
    """A sparse matrix with block matrices along its diagonals

    Examples
    ========

    >>> from sympy import MatrixSymbol, BlockDiagMatrix, symbols
    >>> n, m, l = symbols('n m l')
    >>> X = MatrixSymbol('X', n, n)
    >>> Y = MatrixSymbol('Y', m, m)
    >>> BlockDiagMatrix(X, Y)
    Matrix([
    [X, 0],
    [0, Y]])

    Notes
    =====

    If you want to get the individual diagonal blocks, use
    :meth:`get_diag_blocks`.

    See Also
    ========

    sympy.matrices.dense.diag
    """
    def __new__(cls, *mats):
        return Basic.__new__(BlockDiagMatrix, *[_sympify(m) for m in mats])

    @property
    def diag(self):
        return self.args

    @property
    def blocks(self):
        from sympy.matrices.immutable import ImmutableDenseMatrix
        mats = self.args
        data = [[mats[i] if i == j else ZeroMatrix(mats[i].rows, mats[j].cols)
                        for j in range(len(mats))]
                        for i in range(len(mats))]
        return ImmutableDenseMatrix(data, evaluate=False)

    @property
    def shape(self):
        return (sum(block.rows for block in self.args),
                sum(block.cols for block in self.args))

    @property
    def blockshape(self):
        n = len(self.args)
        return (n, n)

    @property
    def rowblocksizes(self):
        return [block.rows for block in self.args]

    @property
    def colblocksizes(self):
        return [block.cols for block in self.args]

    def _all_square_blocks(self):
        """Returns true if all blocks are square"""
        return all(mat.is_square for mat in self.args)

    def _eval_determinant(self):
        if self._all_square_blocks():
            return Mul(*[det(mat) for mat in self.args])
        # At least one block is non-square.  Since the entire matrix must be square we know there must
        # be at least two blocks in this matrix, in which case the entire matrix is necessarily rank-deficient
        return S.Zero

    def _eval_inverse(self, expand='ignored'):
        if self._all_square_blocks():
            return BlockDiagMatrix(*[mat.inverse() for mat in self.args])
        # See comment in _eval_determinant()
        raise NonInvertibleMatrixError('Matrix det == 0; not invertible.')

    def _eval_transpose(self):
        return BlockDiagMatrix(*[mat.transpose() for mat in self.args])

    def _blockmul(self, other):
        if (isinstance(other, BlockDiagMatrix) and
                self.colblocksizes == other.rowblocksizes):
            return BlockDiagMatrix(*[a*b for a, b in zip(self.args, other.args)])
        else:
            return BlockMatrix._blockmul(self, other)

    def _blockadd(self, other):
        if (isinstance(other, BlockDiagMatrix) and
                self.blockshape == other.blockshape and
                self.rowblocksizes == other.rowblocksizes and
                self.colblocksizes == other.colblocksizes):
            return BlockDiagMatrix(*[a + b for a, b in zip(self.args, other.args)])
        else:
            return BlockMatrix._blockadd(self, other)

    def get_diag_blocks(self):
        """Return the list of diagonal blocks of the matrix.

        Examples
        ========

        >>> from sympy import BlockDiagMatrix, Matrix

        >>> A = Matrix([[1, 2], [3, 4]])
        >>> B = Matrix([[5, 6], [7, 8]])
        >>> M = BlockDiagMatrix(A, B)

        How to get diagonal blocks from the block diagonal matrix:

        >>> diag_blocks = M.get_diag_blocks()
        >>> diag_blocks[0]
        Matrix([
        [1, 2],
        [3, 4]])
        >>> diag_blocks[1]
        Matrix([
        [5, 6],
        [7, 8]])
        """
        return self.args


def block_collapse(expr):
    """Evaluates a block matrix expression

    >>> from sympy import MatrixSymbol, BlockMatrix, symbols, Identity, ZeroMatrix, block_collapse
    >>> n,m,l = symbols('n m l')
    >>> X = MatrixSymbol('X', n, n)
    >>> Y = MatrixSymbol('Y', m, m)
    >>> Z = MatrixSymbol('Z', n, m)
    >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]])
    >>> print(B)
    Matrix([
    [X, Z],
    [0, Y]])

    >>> C = BlockMatrix([[Identity(n), Z]])
    >>> print(C)
    Matrix([[I, Z]])

    >>> print(block_collapse(C*B))
    Matrix([[X, Z + Z*Y]])
    """
    from sympy.strategies.util import expr_fns

    hasbm = lambda expr: isinstance(expr, MatrixExpr) and expr.has(BlockMatrix)

    conditioned_rl = condition(
        hasbm,
        typed(
            {MatAdd: do_one(bc_matadd, bc_block_plus_ident),
             MatMul: do_one(bc_matmul, bc_dist),
             MatPow: bc_matmul,
             Transpose: bc_transpose,
             Inverse: bc_inverse,
             BlockMatrix: do_one(bc_unpack, deblock)}
        )
    )

    rule = exhaust(
        bottom_up(
            exhaust(conditioned_rl),
            fns=expr_fns
        )
    )

    result = rule(expr)
    doit = getattr(result, 'doit', None)
    if doit is not None:
        return doit()
    else:
        return result

def bc_unpack(expr):
    if expr.blockshape == (1, 1):
        return expr.blocks[0, 0]
    return expr

def bc_matadd(expr):
    args = sift(expr.args, lambda M: isinstance(M, BlockMatrix))
    blocks = args[True]
    if not blocks:
        return expr

    nonblocks = args[False]
    block = blocks[0]
    for b in blocks[1:]:
        block = block._blockadd(b)
    if nonblocks:
        return MatAdd(*nonblocks) + block
    else:
        return block

def bc_block_plus_ident(expr):
    idents = [arg for arg in expr.args if arg.is_Identity]
    if not idents:
        return expr

    blocks = [arg for arg in expr.args if isinstance(arg, BlockMatrix)]
    if (blocks and all(b.structurally_equal(blocks[0]) for b in blocks)
               and blocks[0].is_structurally_symmetric):
        block_id = BlockDiagMatrix(*[Identity(k)
                                        for k in blocks[0].rowblocksizes])
        rest = [arg for arg in expr.args if not arg.is_Identity and not isinstance(arg, BlockMatrix)]
        return MatAdd(block_id * len(idents), *blocks, *rest).doit()

    return expr

def bc_dist(expr):
    """ Turn  a*[X, Y] into [a*X, a*Y] """
    factor, mat = expr.as_coeff_mmul()
    if factor == 1:
        return expr

    unpacked = unpack(mat)

    if isinstance(unpacked, BlockDiagMatrix):
        B = unpacked.diag
        new_B = [factor * mat for mat in B]
        return BlockDiagMatrix(*new_B)
    elif isinstance(unpacked, BlockMatrix):
        B = unpacked.blocks
        new_B = [
            [factor * B[i, j] for j in range(B.cols)] for i in range(B.rows)]
        return BlockMatrix(new_B)
    return expr


def bc_matmul(expr):
    if isinstance(expr, MatPow):
        if expr.args[1].is_Integer:
            factor, matrices = (1, [expr.args[0]]*expr.args[1])
        else:
            return expr
    else:
        factor, matrices = expr.as_coeff_matrices()

    i = 0
    while (i+1 < len(matrices)):
        A, B = matrices[i:i+2]
        if isinstance(A, BlockMatrix) and isinstance(B, BlockMatrix):
            matrices[i] = A._blockmul(B)
            matrices.pop(i+1)
        elif isinstance(A, BlockMatrix):
            matrices[i] = A._blockmul(BlockMatrix([[B]]))
            matrices.pop(i+1)
        elif isinstance(B, BlockMatrix):
            matrices[i] = BlockMatrix([[A]])._blockmul(B)
            matrices.pop(i+1)
        else:
            i+=1
    return MatMul(factor, *matrices).doit()

def bc_transpose(expr):
    collapse = block_collapse(expr.arg)
    return collapse._eval_transpose()


def bc_inverse(expr):
    if isinstance(expr.arg, BlockDiagMatrix):
        return expr.inverse()

    expr2 = blockinverse_1x1(expr)
    if expr != expr2:
        return expr2
    return blockinverse_2x2(Inverse(reblock_2x2(expr.arg)))

def blockinverse_1x1(expr):
    if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (1, 1):
        mat = Matrix([[expr.arg.blocks[0].inverse()]])
        return BlockMatrix(mat)
    return expr


def blockinverse_2x2(expr):
    if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (2, 2):
        # See: Inverses of 2x2 Block Matrices, Tzon-Tzer Lu and Sheng-Hua Shiou
        [[A, B],
         [C, D]] = expr.arg.blocks.tolist()

        formula = _choose_2x2_inversion_formula(A, B, C, D)
        if formula != None:
            MI = expr.arg.schur(formula).I
        if formula == 'A':
            AI = A.I
            return BlockMatrix([[AI + AI * B * MI * C * AI, -AI * B * MI], [-MI * C * AI, MI]])
        if formula == 'B':
            BI = B.I
            return BlockMatrix([[-MI * D * BI, MI], [BI + BI * A * MI * D * BI, -BI * A * MI]])
        if formula == 'C':
            CI = C.I
            return BlockMatrix([[-CI * D * MI, CI + CI * D * MI * A * CI], [MI, -MI * A * CI]])
        if formula == 'D':
            DI = D.I
            return BlockMatrix([[MI, -MI * B * DI], [-DI * C * MI, DI + DI * C * MI * B * DI]])

    return expr


def _choose_2x2_inversion_formula(A, B, C, D):
    """
    Assuming [[A, B], [C, D]] would form a valid square block matrix, find
    which of the classical 2x2 block matrix inversion formulas would be
    best suited.

    Returns 'A', 'B', 'C', 'D' to represent the algorithm involving inversion
    of the given argument or None if the matrix cannot be inverted using
    any of those formulas.
    """
    # Try to find a known invertible matrix.  Note that the Schur complement
    # is currently not being considered for this
    A_inv = ask(Q.invertible(A))
    if A_inv == True:
        return 'A'
    B_inv = ask(Q.invertible(B))
    if B_inv == True:
        return 'B'
    C_inv = ask(Q.invertible(C))
    if C_inv == True:
        return 'C'
    D_inv = ask(Q.invertible(D))
    if D_inv == True:
        return 'D'
    # Otherwise try to find a matrix that isn't known to be non-invertible
    if A_inv != False:
        return 'A'
    if B_inv != False:
        return 'B'
    if C_inv != False:
        return 'C'
    if D_inv != False:
        return 'D'
    return None


def deblock(B):
    """ Flatten a BlockMatrix of BlockMatrices """
    if not isinstance(B, BlockMatrix) or not B.blocks.has(BlockMatrix):
        return B
    wrap = lambda x: x if isinstance(x, BlockMatrix) else BlockMatrix([[x]])
    bb = B.blocks.applyfunc(wrap)  # everything is a block

    try:
        MM = Matrix(0, sum(bb[0, i].blocks.shape[1] for i in range(bb.shape[1])), [])
        for row in range(0, bb.shape[0]):
            M = Matrix(bb[row, 0].blocks)
            for col in range(1, bb.shape[1]):
                M = M.row_join(bb[row, col].blocks)
            MM = MM.col_join(M)

        return BlockMatrix(MM)
    except ShapeError:
        return B


def reblock_2x2(expr):
    """
    Reblock a BlockMatrix so that it has 2x2 blocks of block matrices.  If
    possible in such a way that the matrix continues to be invertible using the
    classical 2x2 block inversion formulas.
    """
    if not isinstance(expr, BlockMatrix) or not all(d > 2 for d in expr.blockshape):
        return expr

    BM = BlockMatrix  # for brevity's sake
    rowblocks, colblocks = expr.blockshape
    blocks = expr.blocks
    for i in range(1, rowblocks):
        for j in range(1, colblocks):
            # try to split rows at i and cols at j
            A = bc_unpack(BM(blocks[:i, :j]))
            B = bc_unpack(BM(blocks[:i, j:]))
            C = bc_unpack(BM(blocks[i:, :j]))
            D = bc_unpack(BM(blocks[i:, j:]))

            formula = _choose_2x2_inversion_formula(A, B, C, D)
            if formula is not None:
                return BlockMatrix([[A, B], [C, D]])

    # else: nothing worked, just split upper left corner
    return BM([[blocks[0, 0], BM(blocks[0, 1:])],
               [BM(blocks[1:, 0]), BM(blocks[1:, 1:])]])


def bounds(sizes):
    """ Convert sequence of numbers into pairs of low-high pairs

    >>> from sympy.matrices.expressions.blockmatrix import bounds
    >>> bounds((1, 10, 50))
    [(0, 1), (1, 11), (11, 61)]
    """
    low = 0
    rv = []
    for size in sizes:
        rv.append((low, low + size))
        low += size
    return rv

def blockcut(expr, rowsizes, colsizes):
    """ Cut a matrix expression into Blocks

    >>> from sympy import ImmutableMatrix, blockcut
    >>> M = ImmutableMatrix(4, 4, range(16))
    >>> B = blockcut(M, (1, 3), (1, 3))
    >>> type(B).__name__
    'BlockMatrix'
    >>> ImmutableMatrix(B.blocks[0, 1])
    Matrix([[1, 2, 3]])
    """

    rowbounds = bounds(rowsizes)
    colbounds = bounds(colsizes)
    return BlockMatrix([[MatrixSlice(expr, rowbound, colbound)
                         for colbound in colbounds]
                         for rowbound in rowbounds])