ai-content-maker/.venv/Lib/site-packages/sympy/matrices/dense.py

import random

from sympy.core.basic import Basic
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.utilities.decorator import doctest_depends_on
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.iterables import is_sequence

from .common import ShapeError
from .decompositions import _cholesky, _LDLdecomposition
from .matrices import MatrixBase
from .repmatrix import MutableRepMatrix, RepMatrix
from .solvers import _lower_triangular_solve, _upper_triangular_solve


def _iszero(x):
    """Returns True if x is zero."""
    return x.is_zero


class DenseMatrix(RepMatrix):
    """Matrix implementation based on DomainMatrix as the internal representation"""

    #
    # DenseMatrix is a superclass for both MutableDenseMatrix and
    # ImmutableDenseMatrix. Methods shared by both classes but not for the
    # Sparse classes should be implemented here.
    #

    is_MatrixExpr = False  # type: bool

    _op_priority = 10.01
    _class_priority = 4

    @property
    def _mat(self):
        sympy_deprecation_warning(
            """
            The private _mat attribute of Matrix is deprecated. Use the
            .flat() method instead.
            """,
            deprecated_since_version="1.9",
            active_deprecations_target="deprecated-private-matrix-attributes"
        )

        return self.flat()

    def _eval_inverse(self, **kwargs):
        return self.inv(method=kwargs.get('method', 'GE'),
                        iszerofunc=kwargs.get('iszerofunc', _iszero),
                        try_block_diag=kwargs.get('try_block_diag', False))

    def as_immutable(self):
        """Returns an Immutable version of this Matrix
        """
        from .immutable import ImmutableDenseMatrix as cls
        return cls._fromrep(self._rep.copy())

    def as_mutable(self):
        """Returns a mutable version of this matrix

        Examples
        ========

        >>> from sympy import ImmutableMatrix
        >>> X = ImmutableMatrix([[1, 2], [3, 4]])
        >>> Y = X.as_mutable()
        >>> Y[1, 1] = 5 # Can set values in Y
        >>> Y
        Matrix([
        [1, 2],
        [3, 5]])
        """
        return Matrix(self)

    def cholesky(self, hermitian=True):
        return _cholesky(self, hermitian=hermitian)

    def LDLdecomposition(self, hermitian=True):
        return _LDLdecomposition(self, hermitian=hermitian)

    def lower_triangular_solve(self, rhs):
        return _lower_triangular_solve(self, rhs)

    def upper_triangular_solve(self, rhs):
        return _upper_triangular_solve(self, rhs)

    cholesky.__doc__               = _cholesky.__doc__
    LDLdecomposition.__doc__       = _LDLdecomposition.__doc__
    lower_triangular_solve.__doc__ = _lower_triangular_solve.__doc__
    upper_triangular_solve.__doc__ = _upper_triangular_solve.__doc__


def _force_mutable(x):
    """Return a matrix as a Matrix, otherwise return x."""
    if getattr(x, 'is_Matrix', False):
        return x.as_mutable()
    elif isinstance(x, Basic):
        return x
    elif hasattr(x, '__array__'):
        a = x.__array__()
        if len(a.shape) == 0:
            return sympify(a)
        return Matrix(x)
    return x


class MutableDenseMatrix(DenseMatrix, MutableRepMatrix):

    def simplify(self, **kwargs):
        """Applies simplify to the elements of a matrix in place.

        This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure))

        See Also
        ========

        sympy.simplify.simplify.simplify
        """
        from sympy.simplify.simplify import simplify as _simplify
        for (i, j), element in self.todok().items():
            self[i, j] = _simplify(element, **kwargs)


MutableMatrix = Matrix = MutableDenseMatrix

###########
# Numpy Utility Functions:
# list2numpy, matrix2numpy, symmarray
###########


def list2numpy(l, dtype=object):  # pragma: no cover
    """Converts Python list of SymPy expressions to a NumPy array.

    See Also
    ========

    matrix2numpy
    """
    from numpy import empty
    a = empty(len(l), dtype)
    for i, s in enumerate(l):
        a[i] = s
    return a


def matrix2numpy(m, dtype=object):  # pragma: no cover
    """Converts SymPy's matrix to a NumPy array.

    See Also
    ========

    list2numpy
    """
    from numpy import empty
    a = empty(m.shape, dtype)
    for i in range(m.rows):
        for j in range(m.cols):
            a[i, j] = m[i, j]
    return a


###########
# Rotation matrices:
# rot_givens, rot_axis[123], rot_ccw_axis[123]
###########


def rot_givens(i, j, theta, dim=3):
    r"""Returns a a Givens rotation matrix, a a rotation in the
    plane spanned by two coordinates axes.

    Explanation
    ===========

    The Givens rotation corresponds to a generalization of rotation
    matrices to any number of dimensions, given by:

    .. math::
        G(i, j, \theta) =
            \begin{bmatrix}
                1   & \cdots &    0   & \cdots &    0   & \cdots &    0   \\
                \vdots & \ddots & \vdots &        & \vdots &        & \vdots \\
                0   & \cdots &    c   & \cdots &   -s   & \cdots &    0   \\
                \vdots &        & \vdots & \ddots & \vdots &        & \vdots \\
                0   & \cdots &    s   & \cdots &    c   & \cdots &    0   \\
                \vdots &        & \vdots &        & \vdots & \ddots & \vdots \\
                0   & \cdots &    0   & \cdots &    0   & \cdots &    1
            \end{bmatrix}

    Where $c = \cos(\theta)$ and $s = \sin(\theta)$ appear at the intersections
    ``i``\th and ``j``\th rows and columns.

    For fixed ``i > j``\, the non-zero elements of a Givens matrix are
    given by:

    - $g_{kk} = 1$ for $k \ne i,\,j$
    - $g_{kk} = c$ for $k = i,\,j$
    - $g_{ji} = -g_{ij} = -s$

    Parameters
    ==========

    i : int between ``0`` and ``dim - 1``
        Represents first axis
    j : int between ``0`` and ``dim - 1``
        Represents second axis
    dim : int bigger than 1
        Number of dimentions. Defaults to 3.

    Examples
    ========

    >>> from sympy import pi, rot_givens

    A counterclockwise rotation of pi/3 (60 degrees) around
    the third axis (z-axis):

    >>> rot_givens(1, 0, pi/3)
    Matrix([
    [      1/2, -sqrt(3)/2, 0],
    [sqrt(3)/2,        1/2, 0],
    [        0,          0, 1]])

    If we rotate by pi/2 (90 degrees):

    >>> rot_givens(1, 0, pi/2)
    Matrix([
    [0, -1, 0],
    [1,  0, 0],
    [0,  0, 1]])

    This can be generalized to any number
    of dimensions:

    >>> rot_givens(1, 0, pi/2, dim=4)
    Matrix([
    [0, -1, 0, 0],
    [1,  0, 0, 0],
    [0,  0, 1, 0],
    [0,  0, 0, 1]])

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Givens_rotation

    See Also
    ========

    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
        about the 1-axis (clockwise around the x axis)
    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
        about the 2-axis (clockwise around the y axis)
    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
        about the 3-axis (clockwise around the z axis)
    rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians)
        about the 1-axis (counterclockwise around the x axis)
    rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians)
        about the 2-axis (counterclockwise around the y axis)
    rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians)
        about the 3-axis (counterclockwise around the z axis)
    """
    if not isinstance(dim, int) or dim < 2:
        raise ValueError('dim must be an integer biggen than one, '
                         'got {}.'.format(dim))

    if i == j:
        raise ValueError('i and j must be different, '
                         'got ({}, {})'.format(i, j))

    for ij in [i, j]:
        if not isinstance(ij, int) or ij < 0 or ij > dim - 1:
            raise ValueError('i and j must be integers between 0 and '
                             '{}, got i={} and j={}.'.format(dim-1, i, j))

    theta = sympify(theta)
    c = cos(theta)
    s = sin(theta)
    M = eye(dim)
    M[i, i] = c
    M[j, j] = c
    M[i, j] = s
    M[j, i] = -s
    return M


def rot_axis3(theta):
    r"""Returns a rotation matrix for a rotation of theta (in radians)
    about the 3-axis.

    Explanation
    ===========

    For a right-handed coordinate system, this corresponds to a
    clockwise rotation around the `z`-axis, given by:

    .. math::

        R  = \begin{bmatrix}
                 \cos(\theta) & \sin(\theta) & 0 \\
                -\sin(\theta) & \cos(\theta) & 0 \\
                            0 &            0 & 1
            \end{bmatrix}

    Examples
    ========

    >>> from sympy import pi, rot_axis3

    A rotation of pi/3 (60 degrees):

    >>> theta = pi/3
    >>> rot_axis3(theta)
    Matrix([
    [       1/2, sqrt(3)/2, 0],
    [-sqrt(3)/2,       1/2, 0],
    [         0,         0, 1]])

    If we rotate by pi/2 (90 degrees):

    >>> rot_axis3(pi/2)
    Matrix([
    [ 0, 1, 0],
    [-1, 0, 0],
    [ 0, 0, 1]])

    See Also
    ========

    rot_givens: Returns a Givens rotation matrix (generalized rotation for
        any number of dimensions)
    rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians)
        about the 3-axis (counterclockwise around the z axis)
    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
        about the 1-axis (clockwise around the x axis)
    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
        about the 2-axis (clockwise around the y axis)
    """
    return rot_givens(0, 1, theta, dim=3)


def rot_axis2(theta):
    r"""Returns a rotation matrix for a rotation of theta (in radians)
    about the 2-axis.

    Explanation
    ===========

    For a right-handed coordinate system, this corresponds to a
    clockwise rotation around the `y`-axis, given by:

    .. math::

        R  = \begin{bmatrix}
                \cos(\theta) & 0 & -\sin(\theta) \\
                           0 & 1 &             0 \\
                \sin(\theta) & 0 &  \cos(\theta)
            \end{bmatrix}

    Examples
    ========

    >>> from sympy import pi, rot_axis2

    A rotation of pi/3 (60 degrees):

    >>> theta = pi/3
    >>> rot_axis2(theta)
    Matrix([
    [      1/2, 0, -sqrt(3)/2],
    [        0, 1,          0],
    [sqrt(3)/2, 0,        1/2]])

    If we rotate by pi/2 (90 degrees):

    >>> rot_axis2(pi/2)
    Matrix([
    [0, 0, -1],
    [0, 1,  0],
    [1, 0,  0]])

    See Also
    ========

    rot_givens: Returns a Givens rotation matrix (generalized rotation for
        any number of dimensions)
    rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians)
        about the 2-axis (clockwise around the y axis)
    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
        about the 1-axis (counterclockwise around the x axis)
    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
        about the 3-axis (counterclockwise around the z axis)
    """
    return rot_givens(2, 0, theta, dim=3)


def rot_axis1(theta):
    r"""Returns a rotation matrix for a rotation of theta (in radians)
    about the 1-axis.

    Explanation
    ===========

    For a right-handed coordinate system, this corresponds to a
    clockwise rotation around the `x`-axis, given by:

    .. math::

        R  = \begin{bmatrix}
                1 &             0 &            0 \\
                0 &  \cos(\theta) & \sin(\theta) \\
                0 & -\sin(\theta) & \cos(\theta)
            \end{bmatrix}

    Examples
    ========

    >>> from sympy import pi, rot_axis1

    A rotation of pi/3 (60 degrees):

    >>> theta = pi/3
    >>> rot_axis1(theta)
    Matrix([
    [1,          0,         0],
    [0,        1/2, sqrt(3)/2],
    [0, -sqrt(3)/2,       1/2]])

    If we rotate by pi/2 (90 degrees):

    >>> rot_axis1(pi/2)
    Matrix([
    [1,  0, 0],
    [0,  0, 1],
    [0, -1, 0]])

    See Also
    ========

    rot_givens: Returns a Givens rotation matrix (generalized rotation for
        any number of dimensions)
    rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians)
        about the 1-axis (counterclockwise around the x axis)
    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
        about the 2-axis (clockwise around the y axis)
    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
        about the 3-axis (clockwise around the z axis)
    """
    return rot_givens(1, 2, theta, dim=3)


def rot_ccw_axis3(theta):
    r"""Returns a rotation matrix for a rotation of theta (in radians)
    about the 3-axis.

    Explanation
    ===========

    For a right-handed coordinate system, this corresponds to a
    counterclockwise rotation around the `z`-axis, given by:

    .. math::

        R  = \begin{bmatrix}
                \cos(\theta) & -\sin(\theta) & 0 \\
                \sin(\theta) &  \cos(\theta) & 0 \\
                           0 &             0 & 1
            \end{bmatrix}

    Examples
    ========

    >>> from sympy import pi, rot_ccw_axis3

    A rotation of pi/3 (60 degrees):

    >>> theta = pi/3
    >>> rot_ccw_axis3(theta)
    Matrix([
    [      1/2, -sqrt(3)/2, 0],
    [sqrt(3)/2,        1/2, 0],
    [        0,          0, 1]])

    If we rotate by pi/2 (90 degrees):

    >>> rot_ccw_axis3(pi/2)
    Matrix([
    [0, -1, 0],
    [1,  0, 0],
    [0,  0, 1]])

    See Also
    ========

    rot_givens: Returns a Givens rotation matrix (generalized rotation for
        any number of dimensions)
    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
        about the 3-axis (clockwise around the z axis)
    rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians)
        about the 1-axis (counterclockwise around the x axis)
    rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians)
        about the 2-axis (counterclockwise around the y axis)
    """
    return rot_givens(1, 0, theta, dim=3)


def rot_ccw_axis2(theta):
    r"""Returns a rotation matrix for a rotation of theta (in radians)
    about the 2-axis.

    Explanation
    ===========

    For a right-handed coordinate system, this corresponds to a
    counterclockwise rotation around the `y`-axis, given by:

    .. math::

        R  = \begin{bmatrix}
                 \cos(\theta) & 0 & \sin(\theta) \\
                            0 & 1 &            0 \\
                -\sin(\theta) & 0 & \cos(\theta)
            \end{bmatrix}

    Examples
    ========

    >>> from sympy import pi, rot_ccw_axis2

    A rotation of pi/3 (60 degrees):

    >>> theta = pi/3
    >>> rot_ccw_axis2(theta)
    Matrix([
    [       1/2, 0, sqrt(3)/2],
    [         0, 1,         0],
    [-sqrt(3)/2, 0,       1/2]])

    If we rotate by pi/2 (90 degrees):

    >>> rot_ccw_axis2(pi/2)
    Matrix([
    [ 0,  0,  1],
    [ 0,  1,  0],
    [-1,  0,  0]])

    See Also
    ========

    rot_givens: Returns a Givens rotation matrix (generalized rotation for
        any number of dimensions)
    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
        about the 2-axis (clockwise around the y axis)
    rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians)
        about the 1-axis (counterclockwise around the x axis)
    rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians)
        about the 3-axis (counterclockwise around the z axis)
    """
    return rot_givens(0, 2, theta, dim=3)


def rot_ccw_axis1(theta):
    r"""Returns a rotation matrix for a rotation of theta (in radians)
    about the 1-axis.

    Explanation
    ===========

    For a right-handed coordinate system, this corresponds to a
    counterclockwise rotation around the `x`-axis, given by:

    .. math::

        R  = \begin{bmatrix}
                1 &            0 &             0 \\
                0 & \cos(\theta) & -\sin(\theta) \\
                0 & \sin(\theta) &  \cos(\theta)
            \end{bmatrix}

    Examples
    ========

    >>> from sympy import pi, rot_ccw_axis1

    A rotation of pi/3 (60 degrees):

    >>> theta = pi/3
    >>> rot_ccw_axis1(theta)
    Matrix([
    [1,         0,          0],
    [0,       1/2, -sqrt(3)/2],
    [0, sqrt(3)/2,        1/2]])

    If we rotate by pi/2 (90 degrees):

    >>> rot_ccw_axis1(pi/2)
    Matrix([
    [1, 0,  0],
    [0, 0, -1],
    [0, 1,  0]])

    See Also
    ========

    rot_givens: Returns a Givens rotation matrix (generalized rotation for
        any number of dimensions)
    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
        about the 1-axis (clockwise around the x axis)
    rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians)
        about the 2-axis (counterclockwise around the y axis)
    rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians)
        about the 3-axis (counterclockwise around the z axis)
    """
    return rot_givens(2, 1, theta, dim=3)


@doctest_depends_on(modules=('numpy',))
def symarray(prefix, shape, **kwargs):  # pragma: no cover
    r"""Create a numpy ndarray of symbols (as an object array).

    The created symbols are named ``prefix_i1_i2_``...  You should thus provide a
    non-empty prefix if you want your symbols to be unique for different output
    arrays, as SymPy symbols with identical names are the same object.

    Parameters
    ----------

    prefix : string
      A prefix prepended to the name of every symbol.

    shape : int or tuple
      Shape of the created array.  If an int, the array is one-dimensional; for
      more than one dimension the shape must be a tuple.

    \*\*kwargs : dict
      keyword arguments passed on to Symbol

    Examples
    ========
    These doctests require numpy.

    >>> from sympy import symarray
    >>> symarray('', 3)
    [_0 _1 _2]

    If you want multiple symarrays to contain distinct symbols, you *must*
    provide unique prefixes:

    >>> a = symarray('', 3)
    >>> b = symarray('', 3)
    >>> a[0] == b[0]
    True
    >>> a = symarray('a', 3)
    >>> b = symarray('b', 3)
    >>> a[0] == b[0]
    False

    Creating symarrays with a prefix:

    >>> symarray('a', 3)
    [a_0 a_1 a_2]

    For more than one dimension, the shape must be given as a tuple:

    >>> symarray('a', (2, 3))
    [[a_0_0 a_0_1 a_0_2]
     [a_1_0 a_1_1 a_1_2]]
    >>> symarray('a', (2, 3, 2))
    [[[a_0_0_0 a_0_0_1]
      [a_0_1_0 a_0_1_1]
      [a_0_2_0 a_0_2_1]]
    <BLANKLINE>
     [[a_1_0_0 a_1_0_1]
      [a_1_1_0 a_1_1_1]
      [a_1_2_0 a_1_2_1]]]

    For setting assumptions of the underlying Symbols:

    >>> [s.is_real for s in symarray('a', 2, real=True)]
    [True, True]
    """
    from numpy import empty, ndindex
    arr = empty(shape, dtype=object)
    for index in ndindex(shape):
        arr[index] = Symbol('%s_%s' % (prefix, '_'.join(map(str, index))),
                            **kwargs)
    return arr


###############
# Functions
###############

def casoratian(seqs, n, zero=True):
    """Given linear difference operator L of order 'k' and homogeneous
       equation Ly = 0 we want to compute kernel of L, which is a set
       of 'k' sequences: a(n), b(n), ... z(n).

       Solutions of L are linearly independent iff their Casoratian,
       denoted as C(a, b, ..., z), do not vanish for n = 0.

       Casoratian is defined by k x k determinant::

                  +  a(n)     b(n)     . . . z(n)     +
                  |  a(n+1)   b(n+1)   . . . z(n+1)   |
                  |    .         .     .        .     |
                  |    .         .       .      .     |
                  |    .         .         .    .     |
                  +  a(n+k-1) b(n+k-1) . . . z(n+k-1) +

       It proves very useful in rsolve_hyper() where it is applied
       to a generating set of a recurrence to factor out linearly
       dependent solutions and return a basis:

       >>> from sympy import Symbol, casoratian, factorial
       >>> n = Symbol('n', integer=True)

       Exponential and factorial are linearly independent:

       >>> casoratian([2**n, factorial(n)], n) != 0
       True

    """

    seqs = list(map(sympify, seqs))

    if not zero:
        f = lambda i, j: seqs[j].subs(n, n + i)
    else:
        f = lambda i, j: seqs[j].subs(n, i)

    k = len(seqs)

    return Matrix(k, k, f).det()


def eye(*args, **kwargs):
    """Create square identity matrix n x n

    See Also
    ========

    diag
    zeros
    ones
    """

    return Matrix.eye(*args, **kwargs)


def diag(*values, strict=True, unpack=False, **kwargs):
    """Returns a matrix with the provided values placed on the
    diagonal. If non-square matrices are included, they will
    produce a block-diagonal matrix.

    Examples
    ========

    This version of diag is a thin wrapper to Matrix.diag that differs
    in that it treats all lists like matrices -- even when a single list
    is given. If this is not desired, either put a `*` before the list or
    set `unpack=True`.

    >>> from sympy import diag

    >>> diag([1, 2, 3], unpack=True)  # = diag(1,2,3) or diag(*[1,2,3])
    Matrix([
    [1, 0, 0],
    [0, 2, 0],
    [0, 0, 3]])

    >>> diag([1, 2, 3])  # a column vector
    Matrix([
    [1],
    [2],
    [3]])

    See Also
    ========
    .common.MatrixCommon.eye
    .common.MatrixCommon.diagonal
    .common.MatrixCommon.diag
    .expressions.blockmatrix.BlockMatrix
    """
    return Matrix.diag(*values, strict=strict, unpack=unpack, **kwargs)


def GramSchmidt(vlist, orthonormal=False):
    """Apply the Gram-Schmidt process to a set of vectors.

    Parameters
    ==========

    vlist : List of Matrix
        Vectors to be orthogonalized for.

    orthonormal : Bool, optional
        If true, return an orthonormal basis.

    Returns
    =======

    vlist : List of Matrix
        Orthogonalized vectors

    Notes
    =====

    This routine is mostly duplicate from ``Matrix.orthogonalize``,
    except for some difference that this always raises error when
    linearly dependent vectors are found, and the keyword ``normalize``
    has been named as ``orthonormal`` in this function.

    See Also
    ========

    .matrices.MatrixSubspaces.orthogonalize

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
    """
    return MutableDenseMatrix.orthogonalize(
        *vlist, normalize=orthonormal, rankcheck=True
    )


def hessian(f, varlist, constraints=()):
    """Compute Hessian matrix for a function f wrt parameters in varlist
    which may be given as a sequence or a row/column vector. A list of
    constraints may optionally be given.

    Examples
    ========

    >>> from sympy import Function, hessian, pprint
    >>> from sympy.abc import x, y
    >>> f = Function('f')(x, y)
    >>> g1 = Function('g')(x, y)
    >>> g2 = x**2 + 3*y
    >>> pprint(hessian(f, (x, y), [g1, g2]))
    [                   d               d            ]
    [     0        0    --(g(x, y))     --(g(x, y))  ]
    [                   dx              dy           ]
    [                                                ]
    [     0        0        2*x              3       ]
    [                                                ]
    [                     2               2          ]
    [d                   d               d           ]
    [--(g(x, y))  2*x   ---(f(x, y))   -----(f(x, y))]
    [dx                   2            dy dx         ]
    [                   dx                           ]
    [                                                ]
    [                     2               2          ]
    [d                   d               d           ]
    [--(g(x, y))   3   -----(f(x, y))   ---(f(x, y)) ]
    [dy                dy dx              2          ]
    [                                   dy           ]

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hessian_matrix

    See Also
    ========

    sympy.matrices.matrices.MatrixCalculus.jacobian
    wronskian
    """
    # f is the expression representing a function f, return regular matrix
    if isinstance(varlist, MatrixBase):
        if 1 not in varlist.shape:
            raise ShapeError("`varlist` must be a column or row vector.")
        if varlist.cols == 1:
            varlist = varlist.T
        varlist = varlist.tolist()[0]
    if is_sequence(varlist):
        n = len(varlist)
        if not n:
            raise ShapeError("`len(varlist)` must not be zero.")
    else:
        raise ValueError("Improper variable list in hessian function")
    if not getattr(f, 'diff'):
        # check differentiability
        raise ValueError("Function `f` (%s) is not differentiable" % f)
    m = len(constraints)
    N = m + n
    out = zeros(N)
    for k, g in enumerate(constraints):
        if not getattr(g, 'diff'):
            # check differentiability
            raise ValueError("Function `f` (%s) is not differentiable" % f)
        for i in range(n):
            out[k, i + m] = g.diff(varlist[i])
    for i in range(n):
        for j in range(i, n):
            out[i + m, j + m] = f.diff(varlist[i]).diff(varlist[j])
    for i in range(N):
        for j in range(i + 1, N):
            out[j, i] = out[i, j]
    return out


def jordan_cell(eigenval, n):
    """
    Create a Jordan block:

    Examples
    ========

    >>> from sympy import jordan_cell
    >>> from sympy.abc import x
    >>> jordan_cell(x, 4)
    Matrix([
    [x, 1, 0, 0],
    [0, x, 1, 0],
    [0, 0, x, 1],
    [0, 0, 0, x]])
    """

    return Matrix.jordan_block(size=n, eigenvalue=eigenval)


def matrix_multiply_elementwise(A, B):
    """Return the Hadamard product (elementwise product) of A and B

    >>> from sympy import Matrix, matrix_multiply_elementwise
    >>> A = Matrix([[0, 1, 2], [3, 4, 5]])
    >>> B = Matrix([[1, 10, 100], [100, 10, 1]])
    >>> matrix_multiply_elementwise(A, B)
    Matrix([
    [  0, 10, 200],
    [300, 40,   5]])

    See Also
    ========

    sympy.matrices.common.MatrixCommon.__mul__
    """
    return A.multiply_elementwise(B)


def ones(*args, **kwargs):
    """Returns a matrix of ones with ``rows`` rows and ``cols`` columns;
    if ``cols`` is omitted a square matrix will be returned.

    See Also
    ========

    zeros
    eye
    diag
    """

    if 'c' in kwargs:
        kwargs['cols'] = kwargs.pop('c')

    return Matrix.ones(*args, **kwargs)


def randMatrix(r, c=None, min=0, max=99, seed=None, symmetric=False,
               percent=100, prng=None):
    """Create random matrix with dimensions ``r`` x ``c``. If ``c`` is omitted
    the matrix will be square. If ``symmetric`` is True the matrix must be
    square. If ``percent`` is less than 100 then only approximately the given
    percentage of elements will be non-zero.

    The pseudo-random number generator used to generate matrix is chosen in the
    following way.

    * If ``prng`` is supplied, it will be used as random number generator.
      It should be an instance of ``random.Random``, or at least have
      ``randint`` and ``shuffle`` methods with same signatures.
    * if ``prng`` is not supplied but ``seed`` is supplied, then new
      ``random.Random`` with given ``seed`` will be created;
    * otherwise, a new ``random.Random`` with default seed will be used.

    Examples
    ========

    >>> from sympy import randMatrix
    >>> randMatrix(3) # doctest:+SKIP
    [25, 45, 27]
    [44, 54,  9]
    [23, 96, 46]
    >>> randMatrix(3, 2) # doctest:+SKIP
    [87, 29]
    [23, 37]
    [90, 26]
    >>> randMatrix(3, 3, 0, 2) # doctest:+SKIP
    [0, 2, 0]
    [2, 0, 1]
    [0, 0, 1]
    >>> randMatrix(3, symmetric=True) # doctest:+SKIP
    [85, 26, 29]
    [26, 71, 43]
    [29, 43, 57]
    >>> A = randMatrix(3, seed=1)
    >>> B = randMatrix(3, seed=2)
    >>> A == B
    False
    >>> A == randMatrix(3, seed=1)
    True
    >>> randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP
    [77, 70,  0],
    [70,  0,  0],
    [ 0,  0, 88]
    """
    # Note that ``Random()`` is equivalent to ``Random(None)``
    prng = prng or random.Random(seed)

    if c is None:
        c = r

    if symmetric and r != c:
        raise ValueError('For symmetric matrices, r must equal c, but %i != %i' % (r, c))

    ij = range(r * c)
    if percent != 100:
        ij = prng.sample(ij, int(len(ij)*percent // 100))

    m = zeros(r, c)

    if not symmetric:
        for ijk in ij:
            i, j = divmod(ijk, c)
            m[i, j] = prng.randint(min, max)
    else:
        for ijk in ij:
            i, j = divmod(ijk, c)
            if i <= j:
                m[i, j] = m[j, i] = prng.randint(min, max)

    return m


def wronskian(functions, var, method='bareiss'):
    """
    Compute Wronskian for [] of functions

    ::

                         | f1       f2        ...   fn      |
                         | f1'      f2'       ...   fn'     |
                         |  .        .        .      .      |
        W(f1, ..., fn) = |  .        .         .     .      |
                         |  .        .          .    .      |
                         |  (n)      (n)            (n)     |
                         | D   (f1) D   (f2)  ...  D   (fn) |

    see: https://en.wikipedia.org/wiki/Wronskian

    See Also
    ========

    sympy.matrices.matrices.MatrixCalculus.jacobian
    hessian
    """

    functions = [sympify(f) for f in functions]
    n = len(functions)
    if n == 0:
        return S.One
    W = Matrix(n, n, lambda i, j: functions[i].diff(var, j))
    return W.det(method)


def zeros(*args, **kwargs):
    """Returns a matrix of zeros with ``rows`` rows and ``cols`` columns;
    if ``cols`` is omitted a square matrix will be returned.

    See Also
    ========

    ones
    eye
    diag
    """

    if 'c' in kwargs:
        kwargs['cols'] = kwargs.pop('c')

    return Matrix.zeros(*args, **kwargs)