env-llmeval/lib/python3.10/site-packages/sympy/polys/matrices/sdm.py

"""

Module for the SDM class.

"""

from operator import add, neg, pos, sub, mul
from collections import defaultdict

from sympy.utilities.iterables import _strongly_connected_components

from .exceptions import DMBadInputError, DMDomainError, DMShapeError

from .ddm import DDM
from .lll import ddm_lll, ddm_lll_transform
from sympy.polys.domains import QQ


class SDM(dict):
    r"""Sparse matrix based on polys domain elements

    This is a dict subclass and is a wrapper for a dict of dicts that supports
    basic matrix arithmetic +, -, *, **.


    In order to create a new :py:class:`~.SDM`, a dict
    of dicts mapping non-zero elements to their
    corresponding row and column in the matrix is needed.

    We also need to specify the shape and :py:class:`~.Domain`
    of our :py:class:`~.SDM` object.

    We declare a 2x2 :py:class:`~.SDM` matrix belonging
    to QQ domain as shown below.
    The 2x2 Matrix in the example is

    .. math::
           A = \left[\begin{array}{ccc}
                0 & \frac{1}{2} \\
                0 & 0 \end{array} \right]


    >>> from sympy.polys.matrices.sdm import SDM
    >>> from sympy import QQ
    >>> elemsdict = {0:{1:QQ(1, 2)}}
    >>> A = SDM(elemsdict, (2, 2), QQ)
    >>> A
    {0: {1: 1/2}}

    We can manipulate :py:class:`~.SDM` the same way
    as a Matrix class

    >>> from sympy import ZZ
    >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
    >>> B  = SDM({0:{0: ZZ(3)}, 1:{1:ZZ(4)}}, (2, 2), ZZ)
    >>> A + B
    {0: {0: 3, 1: 2}, 1: {0: 1, 1: 4}}

    Multiplication

    >>> A*B
    {0: {1: 8}, 1: {0: 3}}
    >>> A*ZZ(2)
    {0: {1: 4}, 1: {0: 2}}

    """

    fmt = 'sparse'

    def __init__(self, elemsdict, shape, domain):
        super().__init__(elemsdict)
        self.shape = self.rows, self.cols = m, n = shape
        self.domain = domain

        if not all(0 <= r < m for r in self):
            raise DMBadInputError("Row out of range")
        if not all(0 <= c < n for row in self.values() for c in row):
            raise DMBadInputError("Column out of range")

    def getitem(self, i, j):
        try:
            return self[i][j]
        except KeyError:
            m, n = self.shape
            if -m <= i < m and -n <= j < n:
                try:
                    return self[i % m][j % n]
                except KeyError:
                    return self.domain.zero
            else:
                raise IndexError("index out of range")

    def setitem(self, i, j, value):
        m, n = self.shape
        if not (-m <= i < m and -n <= j < n):
            raise IndexError("index out of range")
        i, j = i % m, j % n
        if value:
            try:
                self[i][j] = value
            except KeyError:
                self[i] = {j: value}
        else:
            rowi = self.get(i, None)
            if rowi is not None:
                try:
                    del rowi[j]
                except KeyError:
                    pass
                else:
                    if not rowi:
                        del self[i]

    def extract_slice(self, slice1, slice2):
        m, n = self.shape
        ri = range(m)[slice1]
        ci = range(n)[slice2]

        sdm = {}
        for i, row in self.items():
            if i in ri:
                row = {ci.index(j): e for j, e in row.items() if j in ci}
                if row:
                    sdm[ri.index(i)] = row

        return self.new(sdm, (len(ri), len(ci)), self.domain)

    def extract(self, rows, cols):
        if not (self and rows and cols):
            return self.zeros((len(rows), len(cols)), self.domain)

        m, n = self.shape
        if not (-m <= min(rows) <= max(rows) < m):
            raise IndexError('Row index out of range')
        if not (-n <= min(cols) <= max(cols) < n):
            raise IndexError('Column index out of range')

        # rows and cols can contain duplicates e.g. M[[1, 2, 2], [0, 1]]
        # Build a map from row/col in self to list of rows/cols in output
        rowmap = defaultdict(list)
        colmap = defaultdict(list)
        for i2, i1 in enumerate(rows):
            rowmap[i1 % m].append(i2)
        for j2, j1 in enumerate(cols):
            colmap[j1 % n].append(j2)

        # Used to efficiently skip zero rows/cols
        rowset = set(rowmap)
        colset = set(colmap)

        sdm1 = self
        sdm2 = {}
        for i1 in rowset & set(sdm1):
            row1 = sdm1[i1]
            row2 = {}
            for j1 in colset & set(row1):
                row1_j1 = row1[j1]
                for j2 in colmap[j1]:
                    row2[j2] = row1_j1
            if row2:
                for i2 in rowmap[i1]:
                    sdm2[i2] = row2.copy()

        return self.new(sdm2, (len(rows), len(cols)), self.domain)

    def __str__(self):
        rowsstr = []
        for i, row in self.items():
            elemsstr = ', '.join('%s: %s' % (j, elem) for j, elem in row.items())
            rowsstr.append('%s: {%s}' % (i, elemsstr))
        return '{%s}' % ', '.join(rowsstr)

    def __repr__(self):
        cls = type(self).__name__
        rows = dict.__repr__(self)
        return '%s(%s, %s, %s)' % (cls, rows, self.shape, self.domain)

    @classmethod
    def new(cls, sdm, shape, domain):
        """

        Parameters
        ==========

        sdm: A dict of dicts for non-zero elements in SDM
        shape: tuple representing dimension of SDM
        domain: Represents :py:class:`~.Domain` of SDM

        Returns
        =======

        An :py:class:`~.SDM` object

        Examples
        ========

        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> elemsdict = {0:{1: QQ(2)}}
        >>> A = SDM.new(elemsdict, (2, 2), QQ)
        >>> A
        {0: {1: 2}}

        """
        return cls(sdm, shape, domain)

    def copy(A):
        """
        Returns the copy of a :py:class:`~.SDM` object

        Examples
        ========

        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> elemsdict = {0:{1:QQ(2)}, 1:{}}
        >>> A = SDM(elemsdict, (2, 2), QQ)
        >>> B = A.copy()
        >>> B
        {0: {1: 2}, 1: {}}

        """
        Ac = {i: Ai.copy() for i, Ai in A.items()}
        return A.new(Ac, A.shape, A.domain)

    @classmethod
    def from_list(cls, ddm, shape, domain):
        """

        Parameters
        ==========

        ddm:
            list of lists containing domain elements
        shape:
            Dimensions of :py:class:`~.SDM` matrix
        domain:
            Represents :py:class:`~.Domain` of :py:class:`~.SDM` object

        Returns
        =======

        :py:class:`~.SDM` containing elements of ddm

        Examples
        ========

        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> ddm = [[QQ(1, 2), QQ(0)], [QQ(0), QQ(3, 4)]]
        >>> A = SDM.from_list(ddm, (2, 2), QQ)
        >>> A
        {0: {0: 1/2}, 1: {1: 3/4}}

        """

        m, n = shape
        if not (len(ddm) == m and all(len(row) == n for row in ddm)):
            raise DMBadInputError("Inconsistent row-list/shape")
        getrow = lambda i: {j:ddm[i][j] for j in range(n) if ddm[i][j]}
        irows = ((i, getrow(i)) for i in range(m))
        sdm = {i: row for i, row in irows if row}
        return cls(sdm, shape, domain)

    @classmethod
    def from_ddm(cls, ddm):
        """
        converts object of :py:class:`~.DDM` to
        :py:class:`~.SDM`

        Examples
        ========

        >>> from sympy.polys.matrices.ddm import DDM
        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> ddm = DDM( [[QQ(1, 2), 0], [0, QQ(3, 4)]], (2, 2), QQ)
        >>> A = SDM.from_ddm(ddm)
        >>> A
        {0: {0: 1/2}, 1: {1: 3/4}}

        """
        return cls.from_list(ddm, ddm.shape, ddm.domain)

    def to_list(M):
        """

        Converts a :py:class:`~.SDM` object to a list

        Examples
        ========

        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> elemsdict = {0:{1:QQ(2)}, 1:{}}
        >>> A = SDM(elemsdict, (2, 2), QQ)
        >>> A.to_list()
        [[0, 2], [0, 0]]

        """
        m, n = M.shape
        zero = M.domain.zero
        ddm = [[zero] * n for _ in range(m)]
        for i, row in M.items():
            for j, e in row.items():
                ddm[i][j] = e
        return ddm

    def to_list_flat(M):
        m, n = M.shape
        zero = M.domain.zero
        flat = [zero] * (m * n)
        for i, row in M.items():
            for j, e in row.items():
                flat[i*n + j] = e
        return flat

    def to_dok(M):
        return {(i, j): e for i, row in M.items() for j, e in row.items()}

    def to_ddm(M):
        """
        Convert a :py:class:`~.SDM` object to a :py:class:`~.DDM` object

        Examples
        ========

        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> A = SDM({0:{1:QQ(2)}, 1:{}}, (2, 2), QQ)
        >>> A.to_ddm()
        [[0, 2], [0, 0]]

        """
        return DDM(M.to_list(), M.shape, M.domain)

    def to_sdm(M):
        return M

    @classmethod
    def zeros(cls, shape, domain):
        r"""

        Returns a :py:class:`~.SDM` of size shape,
        belonging to the specified domain

        In the example below we declare a matrix A where,

        .. math::
            A := \left[\begin{array}{ccc}
            0 & 0 & 0 \\
            0 & 0 & 0 \end{array} \right]

        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> A = SDM.zeros((2, 3), QQ)
        >>> A
        {}

        """
        return cls({}, shape, domain)

    @classmethod
    def ones(cls, shape, domain):
        one = domain.one
        m, n = shape
        row = dict(zip(range(n), [one]*n))
        sdm = {i: row.copy() for i in range(m)}
        return cls(sdm, shape, domain)

    @classmethod
    def eye(cls, shape, domain):
        """

        Returns a identity :py:class:`~.SDM` matrix of dimensions
        size x size, belonging to the specified domain

        Examples
        ========

        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> I = SDM.eye((2, 2), QQ)
        >>> I
        {0: {0: 1}, 1: {1: 1}}

        """
        rows, cols = shape
        one = domain.one
        sdm = {i: {i: one} for i in range(min(rows, cols))}
        return cls(sdm, shape, domain)

    @classmethod
    def diag(cls, diagonal, domain, shape):
        sdm = {i: {i: v} for i, v in enumerate(diagonal) if v}
        return cls(sdm, shape, domain)

    def transpose(M):
        """

        Returns the transpose of a :py:class:`~.SDM` matrix

        Examples
        ========

        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy import QQ
        >>> A = SDM({0:{1:QQ(2)}, 1:{}}, (2, 2), QQ)
        >>> A.transpose()
        {1: {0: 2}}

        """
        MT = sdm_transpose(M)
        return M.new(MT, M.shape[::-1], M.domain)

    def __add__(A, B):
        if not isinstance(B, SDM):
            return NotImplemented
        return A.add(B)

    def __sub__(A, B):
        if not isinstance(B, SDM):
            return NotImplemented
        return A.sub(B)

    def __neg__(A):
        return A.neg()

    def __mul__(A, B):
        """A * B"""
        if isinstance(B, SDM):
            return A.matmul(B)
        elif B in A.domain:
            return A.mul(B)
        else:
            return NotImplemented

    def __rmul__(a, b):
        if b in a.domain:
            return a.rmul(b)
        else:
            return NotImplemented

    def matmul(A, B):
        """
        Performs matrix multiplication of two SDM matrices

        Parameters
        ==========

        A, B: SDM to multiply

        Returns
        =======

        SDM
            SDM after multiplication

        Raises
        ======

        DomainError
            If domain of A does not match
            with that of B

        Examples
        ========

        >>> from sympy import ZZ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
        >>> B = SDM({0:{0:ZZ(2), 1:ZZ(3)}, 1:{0:ZZ(4)}}, (2, 2), ZZ)
        >>> A.matmul(B)
        {0: {0: 8}, 1: {0: 2, 1: 3}}

        """
        if A.domain != B.domain:
            raise DMDomainError
        m, n = A.shape
        n2, o = B.shape
        if n != n2:
            raise DMShapeError
        C = sdm_matmul(A, B, A.domain, m, o)
        return A.new(C, (m, o), A.domain)

    def mul(A, b):
        """
        Multiplies each element of A with a scalar b

        Examples
        ========

        >>> from sympy import ZZ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
        >>> A.mul(ZZ(3))
        {0: {1: 6}, 1: {0: 3}}

        """
        Csdm = unop_dict(A, lambda aij: aij*b)
        return A.new(Csdm, A.shape, A.domain)

    def rmul(A, b):
        Csdm = unop_dict(A, lambda aij: b*aij)
        return A.new(Csdm, A.shape, A.domain)

    def mul_elementwise(A, B):
        if A.domain != B.domain:
            raise DMDomainError
        if A.shape != B.shape:
            raise DMShapeError
        zero = A.domain.zero
        fzero = lambda e: zero
        Csdm = binop_dict(A, B, mul, fzero, fzero)
        return A.new(Csdm, A.shape, A.domain)

    def add(A, B):
        """

        Adds two :py:class:`~.SDM` matrices

        Examples
        ========

        >>> from sympy import ZZ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
        >>> B = SDM({0:{0: ZZ(3)}, 1:{1:ZZ(4)}}, (2, 2), ZZ)
        >>> A.add(B)
        {0: {0: 3, 1: 2}, 1: {0: 1, 1: 4}}

        """

        Csdm = binop_dict(A, B, add, pos, pos)
        return A.new(Csdm, A.shape, A.domain)

    def sub(A, B):
        """

        Subtracts two :py:class:`~.SDM` matrices

        Examples
        ========

        >>> from sympy import ZZ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
        >>> B  = SDM({0:{0: ZZ(3)}, 1:{1:ZZ(4)}}, (2, 2), ZZ)
        >>> A.sub(B)
        {0: {0: -3, 1: 2}, 1: {0: 1, 1: -4}}

        """
        Csdm = binop_dict(A, B, sub, pos, neg)
        return A.new(Csdm, A.shape, A.domain)

    def neg(A):
        """

        Returns the negative of a :py:class:`~.SDM` matrix

        Examples
        ========

        >>> from sympy import ZZ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
        >>> A.neg()
        {0: {1: -2}, 1: {0: -1}}

        """
        Csdm = unop_dict(A, neg)
        return A.new(Csdm, A.shape, A.domain)

    def convert_to(A, K):
        """

        Converts the :py:class:`~.Domain` of a :py:class:`~.SDM` matrix to K

        Examples
        ========

        >>> from sympy import ZZ, QQ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
        >>> A.convert_to(QQ)
        {0: {1: 2}, 1: {0: 1}}

        """
        Kold = A.domain
        if K == Kold:
            return A.copy()
        Ak = unop_dict(A, lambda e: K.convert_from(e, Kold))
        return A.new(Ak, A.shape, K)

    def scc(A):
        """Strongly connected components of a square matrix *A*.

        Examples
        ========

        >>> from sympy import ZZ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{0: ZZ(2)}, 1:{1:ZZ(1)}}, (2, 2), ZZ)
        >>> A.scc()
        [[0], [1]]

        See also
        ========

        sympy.polys.matrices.domainmatrix.DomainMatrix.scc
        """
        rows, cols = A.shape
        assert rows == cols
        V = range(rows)
        Emap = {v: list(A.get(v, [])) for v in V}
        return _strongly_connected_components(V, Emap)

    def rref(A):
        """

        Returns reduced-row echelon form and list of pivots for the :py:class:`~.SDM`

        Examples
        ========

        >>> from sympy import QQ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(2), 1:QQ(4)}}, (2, 2), QQ)
        >>> A.rref()
        ({0: {0: 1, 1: 2}}, [0])

        """
        B, pivots, _ = sdm_irref(A)
        return A.new(B, A.shape, A.domain), pivots

    def inv(A):
        """

        Returns inverse of a matrix A

        Examples
        ========

        >>> from sympy import QQ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
        >>> A.inv()
        {0: {0: -2, 1: 1}, 1: {0: 3/2, 1: -1/2}}

        """
        return A.from_ddm(A.to_ddm().inv())

    def det(A):
        """
        Returns determinant of A

        Examples
        ========

        >>> from sympy import QQ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
        >>> A.det()
        -2

        """
        return A.to_ddm().det()

    def lu(A):
        """

        Returns LU decomposition for a matrix A

        Examples
        ========

        >>> from sympy import QQ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
        >>> A.lu()
        ({0: {0: 1}, 1: {0: 3, 1: 1}}, {0: {0: 1, 1: 2}, 1: {1: -2}}, [])

        """
        L, U, swaps = A.to_ddm().lu()
        return A.from_ddm(L), A.from_ddm(U), swaps

    def lu_solve(A, b):
        """

        Uses LU decomposition to solve Ax = b,

        Examples
        ========

        >>> from sympy import QQ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
        >>> b = SDM({0:{0:QQ(1)}, 1:{0:QQ(2)}}, (2, 1), QQ)
        >>> A.lu_solve(b)
        {1: {0: 1/2}}

        """
        return A.from_ddm(A.to_ddm().lu_solve(b.to_ddm()))

    def nullspace(A):
        """

        Returns nullspace for a :py:class:`~.SDM` matrix A

        Examples
        ========

        >>> from sympy import QQ
        >>> from sympy.polys.matrices.sdm import SDM
        >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0: QQ(2), 1: QQ(4)}}, (2, 2), QQ)
        >>> A.nullspace()
        ({0: {0: -2, 1: 1}}, [1])

        """
        ncols = A.shape[1]
        one = A.domain.one
        B, pivots, nzcols = sdm_irref(A)
        K, nonpivots = sdm_nullspace_from_rref(B, one, ncols, pivots, nzcols)
        K = dict(enumerate(K))
        shape = (len(K), ncols)
        return A.new(K, shape, A.domain), nonpivots

    def particular(A):
        ncols = A.shape[1]
        B, pivots, nzcols = sdm_irref(A)
        P = sdm_particular_from_rref(B, ncols, pivots)
        rep = {0:P} if P else {}
        return A.new(rep, (1, ncols-1), A.domain)

    def hstack(A, *B):
        """Horizontally stacks :py:class:`~.SDM` matrices.

        Examples
        ========

        >>> from sympy import ZZ
        >>> from sympy.polys.matrices.sdm import SDM

        >>> A = SDM({0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(4)}}, (2, 2), ZZ)
        >>> B = SDM({0: {0: ZZ(5), 1: ZZ(6)}, 1: {0: ZZ(7), 1: ZZ(8)}}, (2, 2), ZZ)
        >>> A.hstack(B)
        {0: {0: 1, 1: 2, 2: 5, 3: 6}, 1: {0: 3, 1: 4, 2: 7, 3: 8}}

        >>> C = SDM({0: {0: ZZ(9), 1: ZZ(10)}, 1: {0: ZZ(11), 1: ZZ(12)}}, (2, 2), ZZ)
        >>> A.hstack(B, C)
        {0: {0: 1, 1: 2, 2: 5, 3: 6, 4: 9, 5: 10}, 1: {0: 3, 1: 4, 2: 7, 3: 8, 4: 11, 5: 12}}
        """
        Anew = dict(A.copy())
        rows, cols = A.shape
        domain = A.domain

        for Bk in B:
            Bkrows, Bkcols = Bk.shape
            assert Bkrows == rows
            assert Bk.domain == domain

            for i, Bki in Bk.items():
                Ai = Anew.get(i, None)
                if Ai is None:
                    Anew[i] = Ai = {}
                for j, Bkij in Bki.items():
                    Ai[j + cols] = Bkij
            cols += Bkcols

        return A.new(Anew, (rows, cols), A.domain)

    def vstack(A, *B):
        """Vertically stacks :py:class:`~.SDM` matrices.

        Examples
        ========

        >>> from sympy import ZZ
        >>> from sympy.polys.matrices.sdm import SDM

        >>> A = SDM({0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(4)}}, (2, 2), ZZ)
        >>> B = SDM({0: {0: ZZ(5), 1: ZZ(6)}, 1: {0: ZZ(7), 1: ZZ(8)}}, (2, 2), ZZ)
        >>> A.vstack(B)
        {0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}, 2: {0: 5, 1: 6}, 3: {0: 7, 1: 8}}

        >>> C = SDM({0: {0: ZZ(9), 1: ZZ(10)}, 1: {0: ZZ(11), 1: ZZ(12)}}, (2, 2), ZZ)
        >>> A.vstack(B, C)
        {0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}, 2: {0: 5, 1: 6}, 3: {0: 7, 1: 8}, 4: {0: 9, 1: 10}, 5: {0: 11, 1: 12}}
        """
        Anew = dict(A.copy())
        rows, cols = A.shape
        domain = A.domain

        for Bk in B:
            Bkrows, Bkcols = Bk.shape
            assert Bkcols == cols
            assert Bk.domain == domain

            for i, Bki in Bk.items():
                Anew[i + rows] = Bki
            rows += Bkrows

        return A.new(Anew, (rows, cols), A.domain)

    def applyfunc(self, func, domain):
        sdm = {i: {j: func(e) for j, e in row.items()} for i, row in self.items()}
        return self.new(sdm, self.shape, domain)

    def charpoly(A):
        """
        Returns the coefficients of the characteristic polynomial
        of the :py:class:`~.SDM` matrix. These elements will be domain elements.
        The domain of the elements will be same as domain of the :py:class:`~.SDM`.

        Examples
        ========

        >>> from sympy import QQ, Symbol
        >>> from sympy.polys.matrices.sdm import SDM
        >>> from sympy.polys import Poly
        >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
        >>> A.charpoly()
        [1, -5, -2]

        We can create a polynomial using the
        coefficients using :py:class:`~.Poly`

        >>> x = Symbol('x')
        >>> p = Poly(A.charpoly(), x, domain=A.domain)
        >>> p
        Poly(x**2 - 5*x - 2, x, domain='QQ')

        """
        return A.to_ddm().charpoly()

    def is_zero_matrix(self):
        """
        Says whether this matrix has all zero entries.
        """
        return not self

    def is_upper(self):
        """
        Says whether this matrix is upper-triangular. True can be returned
        even if the matrix is not square.
        """
        return all(i <= j for i, row in self.items() for j in row)

    def is_lower(self):
        """
        Says whether this matrix is lower-triangular. True can be returned
        even if the matrix is not square.
        """
        return all(i >= j for i, row in self.items() for j in row)

    def lll(A, delta=QQ(3, 4)):
        return A.from_ddm(ddm_lll(A.to_ddm(), delta=delta))

    def lll_transform(A, delta=QQ(3, 4)):
        reduced, transform = ddm_lll_transform(A.to_ddm(), delta=delta)
        return A.from_ddm(reduced), A.from_ddm(transform)


def binop_dict(A, B, fab, fa, fb):
    Anz, Bnz = set(A), set(B)
    C = {}

    for i in Anz & Bnz:
        Ai, Bi = A[i], B[i]
        Ci = {}
        Anzi, Bnzi = set(Ai), set(Bi)
        for j in Anzi & Bnzi:
            Cij = fab(Ai[j], Bi[j])
            if Cij:
                Ci[j] = Cij
        for j in Anzi - Bnzi:
            Cij = fa(Ai[j])
            if Cij:
                Ci[j] = Cij
        for j in Bnzi - Anzi:
            Cij = fb(Bi[j])
            if Cij:
                Ci[j] = Cij
        if Ci:
            C[i] = Ci

    for i in Anz - Bnz:
        Ai = A[i]
        Ci = {}
        for j, Aij in Ai.items():
            Cij = fa(Aij)
            if Cij:
                Ci[j] = Cij
        if Ci:
            C[i] = Ci

    for i in Bnz - Anz:
        Bi = B[i]
        Ci = {}
        for j, Bij in Bi.items():
            Cij = fb(Bij)
            if Cij:
                Ci[j] = Cij
        if Ci:
            C[i] = Ci

    return C


def unop_dict(A, f):
    B = {}
    for i, Ai in A.items():
        Bi = {}
        for j, Aij in Ai.items():
            Bij = f(Aij)
            if Bij:
                Bi[j] = Bij
        if Bi:
            B[i] = Bi
    return B


def sdm_transpose(M):
    MT = {}
    for i, Mi in M.items():
        for j, Mij in Mi.items():
            try:
                MT[j][i] = Mij
            except KeyError:
                MT[j] = {i: Mij}
    return MT


def sdm_matmul(A, B, K, m, o):
    #
    # Should be fast if A and B are very sparse.
    # Consider e.g. A = B = eye(1000).
    #
    # The idea here is that we compute C = A*B in terms of the rows of C and
    # B since the dict of dicts representation naturally stores the matrix as
    # rows. The ith row of C (Ci) is equal to the sum of Aik * Bk where Bk is
    # the kth row of B. The algorithm below loops over each nonzero element
    # Aik of A and if the corresponding row Bj is nonzero then we do
    #    Ci += Aik * Bk.
    # To make this more efficient we don't need to loop over all elements Aik.
    # Instead for each row Ai we compute the intersection of the nonzero
    # columns in Ai with the nonzero rows in B. That gives the k such that
    # Aik and Bk are both nonzero. In Python the intersection of two sets
    # of int can be computed very efficiently.
    #
    if K.is_EXRAW:
        return sdm_matmul_exraw(A, B, K, m, o)

    C = {}
    B_knz = set(B)
    for i, Ai in A.items():
        Ci = {}
        Ai_knz = set(Ai)
        for k in Ai_knz & B_knz:
            Aik = Ai[k]
            for j, Bkj in B[k].items():
                Cij = Ci.get(j, None)
                if Cij is not None:
                    Cij = Cij + Aik * Bkj
                    if Cij:
                        Ci[j] = Cij
                    else:
                        Ci.pop(j)
                else:
                    Cij = Aik * Bkj
                    if Cij:
                        Ci[j] = Cij
        if Ci:
            C[i] = Ci
    return C


def sdm_matmul_exraw(A, B, K, m, o):
    #
    # Like sdm_matmul above except that:
    #
    # - Handles cases like 0*oo -> nan (sdm_matmul skips multipication by zero)
    # - Uses K.sum (Add(*items)) for efficient addition of Expr
    #
    zero = K.zero
    C = {}
    B_knz = set(B)
    for i, Ai in A.items():
        Ci_list = defaultdict(list)
        Ai_knz = set(Ai)

        # Nonzero row/column pair
        for k in Ai_knz & B_knz:
            Aik = Ai[k]
            if zero * Aik == zero:
                # This is the main inner loop:
                for j, Bkj in B[k].items():
                    Ci_list[j].append(Aik * Bkj)
            else:
                for j in range(o):
                    Ci_list[j].append(Aik * B[k].get(j, zero))

        # Zero row in B, check for infinities in A
        for k in Ai_knz - B_knz:
            zAik = zero * Ai[k]
            if zAik != zero:
                for j in range(o):
                    Ci_list[j].append(zAik)

        # Add terms using K.sum (Add(*terms)) for efficiency
        Ci = {}
        for j, Cij_list in Ci_list.items():
            Cij = K.sum(Cij_list)
            if Cij:
                Ci[j] = Cij
        if Ci:
            C[i] = Ci

    # Find all infinities in B
    for k, Bk in B.items():
        for j, Bkj in Bk.items():
            if zero * Bkj != zero:
                for i in range(m):
                    Aik = A.get(i, {}).get(k, zero)
                    # If Aik is not zero then this was handled above
                    if Aik == zero:
                        Ci = C.get(i, {})
                        Cij = Ci.get(j, zero) + Aik * Bkj
                        if Cij != zero:
                            Ci[j] = Cij
                        else:  # pragma: no cover
                            # Not sure how we could get here but let's raise an
                            # exception just in case.
                            raise RuntimeError
                        C[i] = Ci

    return C


def sdm_irref(A):
    """RREF and pivots of a sparse matrix *A*.

    Compute the reduced row echelon form (RREF) of the matrix *A* and return a
    list of the pivot columns. This routine does not work in place and leaves
    the original matrix *A* unmodified.

    Examples
    ========

    This routine works with a dict of dicts sparse representation of a matrix:

    >>> from sympy import QQ
    >>> from sympy.polys.matrices.sdm import sdm_irref
    >>> A = {0: {0: QQ(1), 1: QQ(2)}, 1: {0: QQ(3), 1: QQ(4)}}
    >>> Arref, pivots, _ = sdm_irref(A)
    >>> Arref
    {0: {0: 1}, 1: {1: 1}}
    >>> pivots
    [0, 1]

   The analogous calculation with :py:class:`~.Matrix` would be

    >>> from sympy import Matrix
    >>> M = Matrix([[1, 2], [3, 4]])
    >>> Mrref, pivots = M.rref()
    >>> Mrref
    Matrix([
    [1, 0],
    [0, 1]])
    >>> pivots
    (0, 1)

    Notes
    =====

    The cost of this algorithm is determined purely by the nonzero elements of
    the matrix. No part of the cost of any step in this algorithm depends on
    the number of rows or columns in the matrix. No step depends even on the
    number of nonzero rows apart from the primary loop over those rows. The
    implementation is much faster than ddm_rref for sparse matrices. In fact
    at the time of writing it is also (slightly) faster than the dense
    implementation even if the input is a fully dense matrix so it seems to be
    faster in all cases.

    The elements of the matrix should support exact division with ``/``. For
    example elements of any domain that is a field (e.g. ``QQ``) should be
    fine. No attempt is made to handle inexact arithmetic.

    """
    #
    # Any zeros in the matrix are not stored at all so an element is zero if
    # its row dict has no index at that key. A row is entirely zero if its
    # row index is not in the outer dict. Since rref reorders the rows and
    # removes zero rows we can completely discard the row indices. The first
    # step then copies the row dicts into a list sorted by the index of the
    # first nonzero column in each row.
    #
    # The algorithm then processes each row Ai one at a time. Previously seen
    # rows are used to cancel their pivot columns from Ai. Then a pivot from
    # Ai is chosen and is cancelled from all previously seen rows. At this
    # point Ai joins the previously seen rows. Once all rows are seen all
    # elimination has occurred and the rows are sorted by pivot column index.
    #
    # The previously seen rows are stored in two separate groups. The reduced
    # group consists of all rows that have been reduced to a single nonzero
    # element (the pivot). There is no need to attempt any further reduction
    # with these. Rows that still have other nonzeros need to be considered
    # when Ai is cancelled from the previously seen rows.
    #
    # A dict nonzerocolumns is used to map from a column index to a set of
    # previously seen rows that still have a nonzero element in that column.
    # This means that we can cancel the pivot from Ai into the previously seen
    # rows without needing to loop over each row that might have a zero in
    # that column.
    #

    # Row dicts sorted by index of first nonzero column
    # (Maybe sorting is not needed/useful.)
    Arows = sorted((Ai.copy() for Ai in A.values()), key=min)

    # Each processed row has an associated pivot column.
    # pivot_row_map maps from the pivot column index to the row dict.
    # This means that we can represent a set of rows purely as a set of their
    # pivot indices.
    pivot_row_map = {}

    # Set of pivot indices for rows that are fully reduced to a single nonzero.
    reduced_pivots = set()

    # Set of pivot indices for rows not fully reduced
    nonreduced_pivots = set()

    # Map from column index to a set of pivot indices representing the rows
    # that have a nonzero at that column.
    nonzero_columns = defaultdict(set)

    while Arows:
        # Select pivot element and row
        Ai = Arows.pop()

        # Nonzero columns from fully reduced pivot rows can be removed
        Ai = {j: Aij for j, Aij in Ai.items() if j not in reduced_pivots}

        # Others require full row cancellation
        for j in nonreduced_pivots & set(Ai):
            Aj = pivot_row_map[j]
            Aij = Ai[j]
            Ainz = set(Ai)
            Ajnz = set(Aj)
            for k in Ajnz - Ainz:
                Ai[k] = - Aij * Aj[k]
            Ai.pop(j)
            Ainz.remove(j)
            for k in Ajnz & Ainz:
                Aik = Ai[k] - Aij * Aj[k]
                if Aik:
                    Ai[k] = Aik
                else:
                    Ai.pop(k)

        # We have now cancelled previously seen pivots from Ai.
        # If it is zero then discard it.
        if not Ai:
            continue

        # Choose a pivot from Ai:
        j = min(Ai)
        Aij = Ai[j]
        pivot_row_map[j] = Ai
        Ainz = set(Ai)

        # Normalise the pivot row to make the pivot 1.
        #
        # This approach is slow for some domains. Cross cancellation might be
        # better for e.g. QQ(x) with division delayed to the final steps.
        Aijinv = Aij**-1
        for l in Ai:
            Ai[l] *= Aijinv

        # Use Aij to cancel column j from all previously seen rows
        for k in nonzero_columns.pop(j, ()):
            Ak = pivot_row_map[k]
            Akj = Ak[j]
            Aknz = set(Ak)
            for l in Ainz - Aknz:
                Ak[l] = - Akj * Ai[l]
                nonzero_columns[l].add(k)
            Ak.pop(j)
            Aknz.remove(j)
            for l in Ainz & Aknz:
                Akl = Ak[l] - Akj * Ai[l]
                if Akl:
                    Ak[l] = Akl
                else:
                    # Drop nonzero elements
                    Ak.pop(l)
                    if l != j:
                        nonzero_columns[l].remove(k)
            if len(Ak) == 1:
                reduced_pivots.add(k)
                nonreduced_pivots.remove(k)

        if len(Ai) == 1:
            reduced_pivots.add(j)
        else:
            nonreduced_pivots.add(j)
            for l in Ai:
                if l != j:
                    nonzero_columns[l].add(j)

    # All done!
    pivots = sorted(reduced_pivots | nonreduced_pivots)
    pivot2row = {p: n for n, p in enumerate(pivots)}
    nonzero_columns = {c: {pivot2row[p] for p in s} for c, s in nonzero_columns.items()}
    rows = [pivot_row_map[i] for i in pivots]
    rref = dict(enumerate(rows))
    return rref, pivots, nonzero_columns


def sdm_nullspace_from_rref(A, one, ncols, pivots, nonzero_cols):
    """Get nullspace from A which is in RREF"""
    nonpivots = sorted(set(range(ncols)) - set(pivots))

    K = []
    for j in nonpivots:
        Kj = {j:one}
        for i in nonzero_cols.get(j, ()):
            Kj[pivots[i]] = -A[i][j]
        K.append(Kj)

    return K, nonpivots


def sdm_particular_from_rref(A, ncols, pivots):
    """Get a particular solution from A which is in RREF"""
    P = {}
    for i, j in enumerate(pivots):
        Ain = A[i].get(ncols-1, None)
        if Ain is not None:
            P[j] = Ain / A[i][j]
    return P