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from __future__ import print_function, division 

 

from sympy import ask, Q 

from sympy.core import Basic, Add, sympify 

from sympy.core.compatibility import range 

from sympy.strategies import typed, exhaust, condition, do_one, unpack 

from sympy.strategies.traverse import bottom_up 

from sympy.utilities import sift 

 

from sympy.matrices.expressions.matexpr import MatrixExpr, ZeroMatrix, Identity 

from sympy.matrices.expressions.matmul import MatMul 

from sympy.matrices.expressions.matadd import MatAdd 

from sympy.matrices.expressions.transpose import Transpose, transpose 

from sympy.matrices.expressions.trace import Trace 

from sympy.matrices.expressions.determinant import det, Determinant 

from sympy.matrices.expressions.slice import MatrixSlice 

from sympy.matrices.expressions.inverse import Inverse 

from sympy.matrices import Matrix, ShapeError 

 

 

class BlockMatrix(MatrixExpr): 

    """A BlockMatrix is a Matrix composed of other smaller, submatrices 

 

    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*Y + Z]]) 

 

    """ 

    def __new__(cls, *args): 

        from sympy.matrices.immutable import ImmutableMatrix 

        args = map(sympify, args) 

        mat = ImmutableMatrix(*args) 

 

        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 == (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 transpose(self): 

        """Return transpose of matrix. 

 

        Examples 

        ======== 

 

        >>> from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix 

        >>> from sympy.abc import l, 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',  0], 

        [Z', Y']]) 

        >>> _.transpose() 

        Matrix([ 

        [X, Z], 

        [0, Y]]) 

        """ 

        return self._eval_transpose() 

 

    def _entry(self, i, j): 

        # Find row entry 

        for row_block, numrows in enumerate(self.rowblocksizes): 

            if (i < numrows) != False: 

                break 

            else: 

                i -= numrows 

        for col_block, numcols in enumerate(self.colblocksizes): 

            if (j < numcols) != False: 

                break 

            else: 

                j -= numcols 

        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(BlockMatrix, self).equals(other) 

 

class BlockDiagMatrix(BlockMatrix): 

    """ 

    A BlockDiagMatrix is a BlockMatrix with matrices only along the diagonal 

 

    >>> from sympy import MatrixSymbol, BlockDiagMatrix, symbols, Identity 

    >>> 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]]) 

 

    """ 

    def __new__(cls, *mats): 

        return Basic.__new__(BlockDiagMatrix, *mats) 

 

    @property 

    def diag(self): 

        return self.args 

 

    @property 

    def blocks(self): 

        from sympy.matrices.immutable import ImmutableMatrix 

        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 ImmutableMatrix(data) 

 

    @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 _eval_inverse(self, expand='ignored'): 

        return BlockDiagMatrix(*[mat.inverse() 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 block_collapse(expr): 

    """Evaluates a block matrix expression 

 

    >>> from sympy import MatrixSymbol, BlockMatrix, symbols, \ 

                          Identity, Matrix, 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*Y + Z]]) 

    """ 

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

    rule = exhaust( 

        bottom_up(exhaust(condition(hasbm, typed( 

            {MatAdd: do_one(bc_matadd, bc_block_plus_ident), 

             MatMul: do_one(bc_matmul, bc_dist), 

             Transpose: bc_transpose, 

             Inverse: bc_inverse, 

             BlockMatrix: do_one(bc_unpack, deblock)}))))) 

    result = rule(expr) 

    try: 

        return result.doit() 

    except AttributeError: 

        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]) 

        return MatAdd(block_id * len(idents), *blocks).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 and isinstance(unpack(mat), BlockMatrix): 

        B = unpack(mat).blocks 

        return BlockMatrix([[factor * B[i, j] for j in range(B.cols)] 

                                              for i in range(B.rows)]) 

    return expr 

 

 

def bc_matmul(expr): 

    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): 

    return BlockMatrix(block_collapse(expr.arg).blocks.applyfunc(transpose).T) 

 

 

def bc_inverse(expr): 

    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): 

        # Cite: The Matrix Cookbook Section 9.1.3 

        [[A, B], 

         [C, D]] = expr.arg.blocks.tolist() 

 

        return BlockMatrix([[ (A - B*D.I*C).I,  (-A).I*B*(D - C*A.I*B).I], 

                            [-(D - C*A.I*B).I*C*A.I,     (D - C*A.I*B).I]]) 

    else: 

        return expr 

 

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 

 

    from sympy import Matrix 

    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(B): 

    """ Reblock a BlockMatrix so that it has 2x2 blocks of block matrices """ 

    if not isinstance(B, BlockMatrix) or not all(d > 2 for d in B.blocks.shape): 

        return B 

 

    BM = BlockMatrix  # for brevity's sake 

    return BM([[   B.blocks[0,  0],  BM(B.blocks[0,  1:])], 

               [BM(B.blocks[1:, 0]), BM(B.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])