Sources_For_Win/ParaView/Lib/site-packages/sympy/polys/matrices/ddm.py

"""

Module for the DDM class.

The DDM class is an internal representation used by DomainMatrix. The letters
DDM stand for Dense Domain Matrix. A DDM instance represents a matrix using
elements from a polynomial Domain (e.g. ZZ, QQ, ...) in a dense-matrix
representation.

Basic usage:

    >>> from sympy import ZZ, QQ
    >>> from sympy.polys.matrices.ddm import DDM
    >>> A = DDM([[ZZ(0), ZZ(1)], [ZZ(-1), ZZ(0)]], (2, 2), ZZ)
    >>> A.shape
    (2, 2)
    >>> A
    [[0, 1], [-1, 0]]
    >>> type(A)
    <class 'sympy.polys.matrices.ddm.DDM'>
    >>> A @ A
    [[-1, 0], [0, -1]]

The ddm_* functions are designed to operate on DDM as well as on an ordinary
list of lists:

    >>> from sympy.polys.matrices.dense import ddm_idet
    >>> ddm_idet(A, QQ)
    1
    >>> ddm_idet([[0, 1], [-1, 0]], QQ)
    1
    >>> A
    [[-1, 0], [0, -1]]

Note that ddm_idet modifies the input matrix in-place. It is recommended to
use the DDM.det method as a friendlier interface to this instead which takes
care of copying the matrix:

    >>> B = DDM([[ZZ(0), ZZ(1)], [ZZ(-1), ZZ(0)]], (2, 2), ZZ)
    >>> B.det()
    1

Normally DDM would not be used directly and is just part of the internal
representation of DomainMatrix which adds further functionality including e.g.
unifying domains.

The dense format used by DDM is a list of lists of elements e.g. the 2x2
identity matrix is like [[1, 0], [0, 1]]. The DDM class itself is a subclass
of list and its list items are plain lists. Elements are accessed as e.g.
ddm[i][j] where ddm[i] gives the ith row and ddm[i][j] gets the element in the
jth column of that row. Subclassing list makes e.g. iteration and indexing
very efficient. We do not override __getitem__ because it would lose that
benefit.

The core routines are implemented by the ddm_* functions defined in dense.py.
Those functions are intended to be able to operate on a raw list-of-lists
representation of matrices with most functions operating in-place. The DDM
class takes care of copying etc and also stores a Domain object associated
with its elements. This makes it possible to implement things like A + B with
domain checking and also shape checking so that the list of lists
representation is friendlier.

"""
from itertools import chain

from .exceptions import DMBadInputError, DMShapeError, DMDomainError

from .dense import (
        ddm_transpose,
        ddm_iadd,
        ddm_isub,
        ddm_ineg,
        ddm_imul,
        ddm_irmul,
        ddm_imatmul,
        ddm_irref,
        ddm_idet,
        ddm_iinv,
        ddm_ilu_split,
        ddm_ilu_solve,
        ddm_berk,
        )


class DDM(list):
    """Dense matrix based on polys domain elements

    This is a list subclass and is a wrapper for a list of lists that supports
    basic matrix arithmetic +, -, *, **.
    """

    fmt = 'dense'

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

        if not (len(self) == m and all(len(row) == n for row in self)):
            raise DMBadInputError("Inconsistent row-list/shape")

    def getitem(self, i, j):
        return self[i][j]

    def setitem(self, i, j, value):
        self[i][j] = value

    def extract_slice(self, slice1, slice2):
        ddm = [row[slice2] for row in self[slice1]]
        rows = len(ddm)
        cols = len(ddm[0]) if ddm else len(range(self.shape[1])[slice2])
        return DDM(ddm, (rows, cols), self.domain)

    def extract(self, rows, cols):
        ddm = []
        for i in rows:
            rowi = self[i]
            ddm.append([rowi[j] for j in cols])
        return DDM(ddm, (len(rows), len(cols)), self.domain)

    def to_list(self):
        return list(self)

    def to_list_flat(self):
        flat = []
        for row in self:
            flat.extend(row)
        return flat

    def flatiter(self):
        return chain.from_iterable(self)

    def flat(self):
        items = []
        for row in self:
            items.extend(row)
        return items

    def to_dok(self):
        return {(i, j): e for i, row in enumerate(self) for j, e in enumerate(row)}

    def to_ddm(self):
        return self

    def to_sdm(self):
        return SDM.from_list(self, self.shape, self.domain)

    def convert_to(self, K):
        Kold = self.domain
        if K == Kold:
            return self.copy()
        rows = ([K.convert_from(e, Kold) for e in row] for row in self)
        return DDM(rows, self.shape, K)

    def __str__(self):
        rowsstr = ['[%s]' % ', '.join(map(str, row)) for row in self]
        return '[%s]' % ', '.join(rowsstr)

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

    def __eq__(self, other):
        if not isinstance(other, DDM):
            return False
        return (super().__eq__(other) and self.domain == other.domain)

    def __ne__(self, other):
        return not self.__eq__(other)

    @classmethod
    def zeros(cls, shape, domain):
        z = domain.zero
        m, n = shape
        rowslist = ([z] * n for _ in range(m))
        return DDM(rowslist, shape, domain)

    @classmethod
    def ones(cls, shape, domain):
        one = domain.one
        m, n = shape
        rowlist = ([one] * n for _ in range(m))
        return DDM(rowlist, shape, domain)

    @classmethod
    def eye(cls, size, domain):
        one = domain.one
        ddm = cls.zeros((size, size), domain)
        for i in range(size):
            ddm[i][i] = one
        return ddm

    def copy(self):
        copyrows = (row[:] for row in self)
        return DDM(copyrows, self.shape, self.domain)

    def transpose(self):
        rows, cols = self.shape
        if rows:
            ddmT = ddm_transpose(self)
        else:
            ddmT = [[]] * cols
        return DDM(ddmT, (cols, rows), self.domain)

    def __add__(a, b):
        if not isinstance(b, DDM):
            return NotImplemented
        return a.add(b)

    def __sub__(a, b):
        if not isinstance(b, DDM):
            return NotImplemented
        return a.sub(b)

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

    def __mul__(a, b):
        if b in a.domain:
            return a.mul(b)
        else:
            return NotImplemented

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

    def __matmul__(a, b):
        if isinstance(b, DDM):
            return a.matmul(b)
        else:
            return NotImplemented

    @classmethod
    def _check(cls, a, op, b, ashape, bshape):
        if a.domain != b.domain:
            msg = "Domain mismatch: %s %s %s" % (a.domain, op, b.domain)
            raise DMDomainError(msg)
        if ashape != bshape:
            msg = "Shape mismatch: %s %s %s" % (a.shape, op, b.shape)
            raise DMShapeError(msg)

    def add(a, b):
        """a + b"""
        a._check(a, '+', b, a.shape, b.shape)
        c = a.copy()
        ddm_iadd(c, b)
        return c

    def sub(a, b):
        """a - b"""
        a._check(a, '-', b, a.shape, b.shape)
        c = a.copy()
        ddm_isub(c, b)
        return c

    def neg(a):
        """-a"""
        b = a.copy()
        ddm_ineg(b)
        return b

    def mul(a, b):
        c = a.copy()
        ddm_imul(c, b)
        return c

    def rmul(a, b):
        c = a.copy()
        ddm_irmul(c, b)
        return c

    def matmul(a, b):
        """a @ b (matrix product)"""
        m, o = a.shape
        o2, n = b.shape
        a._check(a, '*', b, o, o2)
        c = a.zeros((m, n), a.domain)
        ddm_imatmul(c, a, b)
        return c

    def mul_elementwise(a, b):
        assert a.shape == b.shape
        assert a.domain == b.domain
        c = [[aij * bij for aij, bij in zip(ai, bi)] for ai, bi in zip(a, b)]
        return DDM(c, a.shape, a.domain)

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

        Examples
        ========

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

        >>> A = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
        >>> B = DDM([[ZZ(5), ZZ(6)], [ZZ(7), ZZ(8)]], (2, 2), ZZ)
        >>> A.hstack(B)
        [[1, 2, 5, 6], [3, 4, 7, 8]]

        >>> C = DDM([[ZZ(9), ZZ(10)], [ZZ(11), ZZ(12)]], (2, 2), ZZ)
        >>> A.hstack(B, C)
        [[1, 2, 5, 6, 9, 10], [3, 4, 7, 8, 11, 12]]
        """
        Anew = list(A.copy())
        rows, cols = A.shape
        domain = A.domain

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

            cols += Bkcols

            for i, Bki in enumerate(Bk):
                Anew[i].extend(Bki)

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

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

        Examples
        ========

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

        >>> A = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
        >>> B = DDM([[ZZ(5), ZZ(6)], [ZZ(7), ZZ(8)]], (2, 2), ZZ)
        >>> A.vstack(B)
        [[1, 2], [3, 4], [5, 6], [7, 8]]

        >>> C = DDM([[ZZ(9), ZZ(10)], [ZZ(11), ZZ(12)]], (2, 2), ZZ)
        >>> A.vstack(B, C)
        [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10], [11, 12]]
        """
        Anew = list(A.copy())
        rows, cols = A.shape
        domain = A.domain

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

            rows += Bkrows

            Anew.extend(Bk.copy())

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

    def applyfunc(self, func, domain):
        elements = (list(map(func, row)) for row in self)
        return DDM(elements, self.shape, domain)

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

        Examples
        ========

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

        See also
        ========

        sympy.polys.matrices.domainmatrix.DomainMatrix.scc

        """
        return a.to_sdm().scc()

    def rref(a):
        """Reduced-row echelon form of a and list of pivots"""
        b = a.copy()
        K = a.domain
        partial_pivot = K.is_RealField or K.is_ComplexField
        pivots = ddm_irref(b, _partial_pivot=partial_pivot)
        return b, pivots

    def nullspace(a):
        rref, pivots = a.rref()
        rows, cols = a.shape
        domain = a.domain

        basis = []
        nonpivots = []
        for i in range(cols):
            if i in pivots:
                continue
            nonpivots.append(i)
            vec = [domain.one if i == j else domain.zero for j in range(cols)]
            for ii, jj in enumerate(pivots):
                vec[jj] -= rref[ii][i]
            basis.append(vec)

        return DDM(basis, (len(basis), cols), domain), nonpivots

    def particular(a):
        return a.to_sdm().particular().to_ddm()

    def det(a):
        """Determinant of a"""
        m, n = a.shape
        if m != n:
            raise DMShapeError("Determinant of non-square matrix")
        b = a.copy()
        K = b.domain
        deta = ddm_idet(b, K)
        return deta

    def inv(a):
        """Inverse of a"""
        m, n = a.shape
        if m != n:
            raise DMShapeError("Determinant of non-square matrix")
        ainv = a.copy()
        K = a.domain
        ddm_iinv(ainv, a, K)
        return ainv

    def lu(a):
        """L, U decomposition of a"""
        m, n = a.shape
        K = a.domain

        U = a.copy()
        L = a.eye(m, K)
        swaps = ddm_ilu_split(L, U, K)

        return L, U, swaps

    def lu_solve(a, b):
        """x where a*x = b"""
        m, n = a.shape
        m2, o = b.shape
        a._check(a, 'lu_solve', b, m, m2)

        L, U, swaps = a.lu()
        x = a.zeros((n, o), a.domain)
        ddm_ilu_solve(x, L, U, swaps, b)
        return x

    def charpoly(a):
        """Coefficients of characteristic polynomial of a"""
        K = a.domain
        m, n = a.shape
        if m != n:
            raise DMShapeError("Charpoly of non-square matrix")
        vec = ddm_berk(a, K)
        coeffs = [vec[i][0] for i in range(n+1)]
        return coeffs

    def is_zero_matrix(self):
        """
        Says whether this matrix has all zero entries.
        """
        zero = self.domain.zero
        return all(Mij == zero for Mij in self.flatiter())

    def is_upper(self):
        """
        Says whether this matrix is upper-triangular. True can be returned
        even if the matrix is not square.
        """
        zero = self.domain.zero
        return all(Mij == zero for i, Mi in enumerate(self) for Mij in Mi[:i])

    def is_lower(self):
        """
        Says whether this matrix is lower-triangular. True can be returned
        even if the matrix is not square.
        """
        zero = self.domain.zero
        return all(Mij == zero for i, Mi in enumerate(self) for Mij in Mi[i+1:])


from .sdm import SDM