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image-handler/image-handler10.0/venv/lib/site-packages/sympy/polys/matrices/domainmatrix.py

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"""
Module for the DomainMatrix class.
A DomainMatrix represents a matrix with elements that are in a particular
Domain. Each DomainMatrix internally wraps a DDM which is used for the
lower-level operations. The idea is that the DomainMatrix class provides the
convenience routines for converting between Expr and the poly domains as well
as unifying matrices with different domains.
"""
from functools import reduce
from typing import Union as tUnion, Tuple as tTuple
from sympy.core.sympify import _sympify
from ..domains import Domain
from ..constructor import construct_domain
from .exceptions import (DMNonSquareMatrixError, DMShapeError,
DMDomainError, DMFormatError, DMBadInputError,
DMNotAField)
from .ddm import DDM
from .sdm import SDM
from .domainscalar import DomainScalar
from sympy.polys.domains import ZZ, EXRAW, QQ
def DM(rows, domain):
"""Convenient alias for DomainMatrix.from_list
Examples
=======
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DM
>>> DM([[1, 2], [3, 4]], ZZ)
DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ)
See also
=======
DomainMatrix.from_list
"""
return DomainMatrix.from_list(rows, domain)
class DomainMatrix:
r"""
Associate Matrix with :py:class:`~.Domain`
Explanation
===========
DomainMatrix uses :py:class:`~.Domain` for its internal representation
which makes it faster than the SymPy Matrix class (currently) for many
common operations, but this advantage makes it not entirely compatible
with Matrix. DomainMatrix are analogous to numpy arrays with "dtype".
In the DomainMatrix, each element has a domain such as :ref:`ZZ`
or :ref:`QQ(a)`.
Examples
========
Creating a DomainMatrix from the existing Matrix class:
>>> from sympy import Matrix
>>> from sympy.polys.matrices import DomainMatrix
>>> Matrix1 = Matrix([
... [1, 2],
... [3, 4]])
>>> A = DomainMatrix.from_Matrix(Matrix1)
>>> A
DomainMatrix({0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}, (2, 2), ZZ)
Directly forming a DomainMatrix:
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A
DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ)
See Also
========
DDM
SDM
Domain
Poly
"""
rep: tUnion[SDM, DDM]
shape: tTuple[int, int]
domain: Domain
def __new__(cls, rows, shape, domain, *, fmt=None):
"""
Creates a :py:class:`~.DomainMatrix`.
Parameters
==========
rows : Represents elements of DomainMatrix as list of lists
shape : Represents dimension of DomainMatrix
domain : Represents :py:class:`~.Domain` of DomainMatrix
Raises
======
TypeError
If any of rows, shape and domain are not provided
"""
if isinstance(rows, (DDM, SDM)):
raise TypeError("Use from_rep to initialise from SDM/DDM")
elif isinstance(rows, list):
rep = DDM(rows, shape, domain)
elif isinstance(rows, dict):
rep = SDM(rows, shape, domain)
else:
msg = "Input should be list-of-lists or dict-of-dicts"
raise TypeError(msg)
if fmt is not None:
if fmt == 'sparse':
rep = rep.to_sdm()
elif fmt == 'dense':
rep = rep.to_ddm()
else:
raise ValueError("fmt should be 'sparse' or 'dense'")
return cls.from_rep(rep)
def __getnewargs__(self):
rep = self.rep
if isinstance(rep, DDM):
arg = list(rep)
elif isinstance(rep, SDM):
arg = dict(rep)
else:
raise RuntimeError # pragma: no cover
return arg, self.shape, self.domain
def __getitem__(self, key):
i, j = key
m, n = self.shape
if not (isinstance(i, slice) or isinstance(j, slice)):
return DomainScalar(self.rep.getitem(i, j), self.domain)
if not isinstance(i, slice):
if not -m <= i < m:
raise IndexError("Row index out of range")
i = i % m
i = slice(i, i+1)
if not isinstance(j, slice):
if not -n <= j < n:
raise IndexError("Column index out of range")
j = j % n
j = slice(j, j+1)
return self.from_rep(self.rep.extract_slice(i, j))
def getitem_sympy(self, i, j):
return self.domain.to_sympy(self.rep.getitem(i, j))
def extract(self, rowslist, colslist):
return self.from_rep(self.rep.extract(rowslist, colslist))
def __setitem__(self, key, value):
i, j = key
if not self.domain.of_type(value):
raise TypeError
if isinstance(i, int) and isinstance(j, int):
self.rep.setitem(i, j, value)
else:
raise NotImplementedError
@classmethod
def from_rep(cls, rep):
"""Create a new DomainMatrix efficiently from DDM/SDM.
Examples
========
Create a :py:class:`~.DomainMatrix` with an dense internal
representation as :py:class:`~.DDM`:
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy.polys.matrices.ddm import DDM
>>> drep = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> dM = DomainMatrix.from_rep(drep)
>>> dM
DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ)
Create a :py:class:`~.DomainMatrix` with a sparse internal
representation as :py:class:`~.SDM`:
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy.polys.matrices.sdm import SDM
>>> from sympy import ZZ
>>> drep = SDM({0:{1:ZZ(1)},1:{0:ZZ(2)}}, (2, 2), ZZ)
>>> dM = DomainMatrix.from_rep(drep)
>>> dM
DomainMatrix({0: {1: 1}, 1: {0: 2}}, (2, 2), ZZ)
Parameters
==========
rep: SDM or DDM
The internal sparse or dense representation of the matrix.
Returns
=======
DomainMatrix
A :py:class:`~.DomainMatrix` wrapping *rep*.
Notes
=====
This takes ownership of rep as its internal representation. If rep is
being mutated elsewhere then a copy should be provided to
``from_rep``. Only minimal verification or checking is done on *rep*
as this is supposed to be an efficient internal routine.
"""
if not isinstance(rep, (DDM, SDM)):
raise TypeError("rep should be of type DDM or SDM")
self = super().__new__(cls)
self.rep = rep
self.shape = rep.shape
self.domain = rep.domain
return self
@classmethod
def from_list(cls, rows, domain):
r"""
Convert a list of lists into a DomainMatrix
Parameters
==========
rows: list of lists
Each element of the inner lists should be either the single arg,
or tuple of args, that would be passed to the domain constructor
in order to form an element of the domain. See examples.
Returns
=======
DomainMatrix containing elements defined in rows
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import FF, QQ, ZZ
>>> A = DomainMatrix.from_list([[1, 0, 1], [0, 0, 1]], ZZ)
>>> A
DomainMatrix([[1, 0, 1], [0, 0, 1]], (2, 3), ZZ)
>>> B = DomainMatrix.from_list([[1, 0, 1], [0, 0, 1]], FF(7))
>>> B
DomainMatrix([[1 mod 7, 0 mod 7, 1 mod 7], [0 mod 7, 0 mod 7, 1 mod 7]], (2, 3), GF(7))
>>> C = DomainMatrix.from_list([[(1, 2), (3, 1)], [(1, 4), (5, 1)]], QQ)
>>> C
DomainMatrix([[1/2, 3], [1/4, 5]], (2, 2), QQ)
See Also
========
from_list_sympy
"""
nrows = len(rows)
ncols = 0 if not nrows else len(rows[0])
conv = lambda e: domain(*e) if isinstance(e, tuple) else domain(e)
domain_rows = [[conv(e) for e in row] for row in rows]
return DomainMatrix(domain_rows, (nrows, ncols), domain)
@classmethod
def from_list_sympy(cls, nrows, ncols, rows, **kwargs):
r"""
Convert a list of lists of Expr into a DomainMatrix using construct_domain
Parameters
==========
nrows: number of rows
ncols: number of columns
rows: list of lists
Returns
=======
DomainMatrix containing elements of rows
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy.abc import x, y, z
>>> A = DomainMatrix.from_list_sympy(1, 3, [[x, y, z]])
>>> A
DomainMatrix([[x, y, z]], (1, 3), ZZ[x,y,z])
See Also
========
sympy.polys.constructor.construct_domain, from_dict_sympy
"""
assert len(rows) == nrows
assert all(len(row) == ncols for row in rows)
items_sympy = [_sympify(item) for row in rows for item in row]
domain, items_domain = cls.get_domain(items_sympy, **kwargs)
domain_rows = [[items_domain[ncols*r + c] for c in range(ncols)] for r in range(nrows)]
return DomainMatrix(domain_rows, (nrows, ncols), domain)
@classmethod
def from_dict_sympy(cls, nrows, ncols, elemsdict, **kwargs):
"""
Parameters
==========
nrows: number of rows
ncols: number of cols
elemsdict: dict of dicts containing non-zero elements of the DomainMatrix
Returns
=======
DomainMatrix containing elements of elemsdict
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy.abc import x,y,z
>>> elemsdict = {0: {0:x}, 1:{1: y}, 2: {2: z}}
>>> A = DomainMatrix.from_dict_sympy(3, 3, elemsdict)
>>> A
DomainMatrix({0: {0: x}, 1: {1: y}, 2: {2: z}}, (3, 3), ZZ[x,y,z])
See Also
========
from_list_sympy
"""
if not all(0 <= r < nrows for r in elemsdict):
raise DMBadInputError("Row out of range")
if not all(0 <= c < ncols for row in elemsdict.values() for c in row):
raise DMBadInputError("Column out of range")
items_sympy = [_sympify(item) for row in elemsdict.values() for item in row.values()]
domain, items_domain = cls.get_domain(items_sympy, **kwargs)
idx = 0
items_dict = {}
for i, row in elemsdict.items():
items_dict[i] = {}
for j in row:
items_dict[i][j] = items_domain[idx]
idx += 1
return DomainMatrix(items_dict, (nrows, ncols), domain)
@classmethod
def from_Matrix(cls, M, fmt='sparse',**kwargs):
r"""
Convert Matrix to DomainMatrix
Parameters
==========
M: Matrix
Returns
=======
Returns DomainMatrix with identical elements as M
Examples
========
>>> from sympy import Matrix
>>> from sympy.polys.matrices import DomainMatrix
>>> M = Matrix([
... [1.0, 3.4],
... [2.4, 1]])
>>> A = DomainMatrix.from_Matrix(M)
>>> A
DomainMatrix({0: {0: 1.0, 1: 3.4}, 1: {0: 2.4, 1: 1.0}}, (2, 2), RR)
We can keep internal representation as ddm using fmt='dense'
>>> from sympy import Matrix, QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix.from_Matrix(Matrix([[QQ(1, 2), QQ(3, 4)], [QQ(0, 1), QQ(0, 1)]]), fmt='dense')
>>> A.rep
[[1/2, 3/4], [0, 0]]
See Also
========
Matrix
"""
if fmt == 'dense':
return cls.from_list_sympy(*M.shape, M.tolist(), **kwargs)
return cls.from_dict_sympy(*M.shape, M.todod(), **kwargs)
@classmethod
def get_domain(cls, items_sympy, **kwargs):
K, items_K = construct_domain(items_sympy, **kwargs)
return K, items_K
def copy(self):
return self.from_rep(self.rep.copy())
def convert_to(self, K):
r"""
Change the domain of DomainMatrix to desired domain or field
Parameters
==========
K : Represents the desired domain or field.
Alternatively, ``None`` may be passed, in which case this method
just returns a copy of this DomainMatrix.
Returns
=======
DomainMatrix
DomainMatrix with the desired domain or field
Examples
========
>>> from sympy import ZZ, ZZ_I
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.convert_to(ZZ_I)
DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ_I)
"""
if K is None:
return self.copy()
return self.from_rep(self.rep.convert_to(K))
def to_sympy(self):
return self.convert_to(EXRAW)
def to_field(self):
r"""
Returns a DomainMatrix with the appropriate field
Returns
=======
DomainMatrix
DomainMatrix with the appropriate field
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.to_field()
DomainMatrix([[1, 2], [3, 4]], (2, 2), QQ)
"""
K = self.domain.get_field()
return self.convert_to(K)
def to_sparse(self):
"""
Return a sparse DomainMatrix representation of *self*.
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> A = DomainMatrix([[1, 0],[0, 2]], (2, 2), QQ)
>>> A.rep
[[1, 0], [0, 2]]
>>> B = A.to_sparse()
>>> B.rep
{0: {0: 1}, 1: {1: 2}}
"""
if self.rep.fmt == 'sparse':
return self
return self.from_rep(SDM.from_ddm(self.rep))
def to_dense(self):
"""
Return a dense DomainMatrix representation of *self*.
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> A = DomainMatrix({0: {0: 1}, 1: {1: 2}}, (2, 2), QQ)
>>> A.rep
{0: {0: 1}, 1: {1: 2}}
>>> B = A.to_dense()
>>> B.rep
[[1, 0], [0, 2]]
"""
if self.rep.fmt == 'dense':
return self
return self.from_rep(SDM.to_ddm(self.rep))
@classmethod
def _unify_domain(cls, *matrices):
"""Convert matrices to a common domain"""
domains = {matrix.domain for matrix in matrices}
if len(domains) == 1:
return matrices
domain = reduce(lambda x, y: x.unify(y), domains)
return tuple(matrix.convert_to(domain) for matrix in matrices)
@classmethod
def _unify_fmt(cls, *matrices, fmt=None):
"""Convert matrices to the same format.
If all matrices have the same format, then return unmodified.
Otherwise convert both to the preferred format given as *fmt* which
should be 'dense' or 'sparse'.
"""
formats = {matrix.rep.fmt for matrix in matrices}
if len(formats) == 1:
return matrices
if fmt == 'sparse':
return tuple(matrix.to_sparse() for matrix in matrices)
elif fmt == 'dense':
return tuple(matrix.to_dense() for matrix in matrices)
else:
raise ValueError("fmt should be 'sparse' or 'dense'")
def unify(self, *others, fmt=None):
"""
Unifies the domains and the format of self and other
matrices.
Parameters
==========
others : DomainMatrix
fmt: string 'dense', 'sparse' or `None` (default)
The preferred format to convert to if self and other are not
already in the same format. If `None` or not specified then no
conversion if performed.
Returns
=======
Tuple[DomainMatrix]
Matrices with unified domain and format
Examples
========
Unify the domain of DomainMatrix that have different domains:
>>> from sympy import ZZ, QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
>>> B = DomainMatrix([[QQ(1, 2), QQ(2)]], (1, 2), QQ)
>>> Aq, Bq = A.unify(B)
>>> Aq
DomainMatrix([[1, 2]], (1, 2), QQ)
>>> Bq
DomainMatrix([[1/2, 2]], (1, 2), QQ)
Unify the format (dense or sparse):
>>> A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
>>> B = DomainMatrix({0:{0: ZZ(1)}}, (2, 2), ZZ)
>>> B.rep
{0: {0: 1}}
>>> A2, B2 = A.unify(B, fmt='dense')
>>> B2.rep
[[1, 0], [0, 0]]
See Also
========
convert_to, to_dense, to_sparse
"""
matrices = (self,) + others
matrices = DomainMatrix._unify_domain(*matrices)
if fmt is not None:
matrices = DomainMatrix._unify_fmt(*matrices, fmt=fmt)
return matrices
def to_Matrix(self):
r"""
Convert DomainMatrix to Matrix
Returns
=======
Matrix
MutableDenseMatrix for the DomainMatrix
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.to_Matrix()
Matrix([
[1, 2],
[3, 4]])
See Also
========
from_Matrix
"""
from sympy.matrices.dense import MutableDenseMatrix
elemlist = self.rep.to_list()
elements_sympy = [self.domain.to_sympy(e) for row in elemlist for e in row]
return MutableDenseMatrix(*self.shape, elements_sympy)
def to_list(self):
return self.rep.to_list()
def to_list_flat(self):
return self.rep.to_list_flat()
def to_dok(self):
return self.rep.to_dok()
def __repr__(self):
return 'DomainMatrix(%s, %r, %r)' % (str(self.rep), self.shape, self.domain)
def transpose(self):
"""Matrix transpose of ``self``"""
return self.from_rep(self.rep.transpose())
def flat(self):
rows, cols = self.shape
return [self[i,j].element for i in range(rows) for j in range(cols)]
@property
def is_zero_matrix(self):
return self.rep.is_zero_matrix()
@property
def is_upper(self):
"""
Says whether this matrix is upper-triangular. True can be returned
even if the matrix is not square.
"""
return self.rep.is_upper()
@property
def is_lower(self):
"""
Says whether this matrix is lower-triangular. True can be returned
even if the matrix is not square.
"""
return self.rep.is_lower()
@property
def is_square(self):
return self.shape[0] == self.shape[1]
def rank(self):
rref, pivots = self.rref()
return len(pivots)
def hstack(A, *B):
r"""Horizontally stack the given matrices.
Parameters
==========
B: DomainMatrix
Matrices to stack horizontally.
Returns
=======
DomainMatrix
DomainMatrix by stacking horizontally.
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([[ZZ(5), ZZ(6)], [ZZ(7), ZZ(8)]], (2, 2), ZZ)
>>> A.hstack(B)
DomainMatrix([[1, 2, 5, 6], [3, 4, 7, 8]], (2, 4), ZZ)
>>> C = DomainMatrix([[ZZ(9), ZZ(10)], [ZZ(11), ZZ(12)]], (2, 2), ZZ)
>>> A.hstack(B, C)
DomainMatrix([[1, 2, 5, 6, 9, 10], [3, 4, 7, 8, 11, 12]], (2, 6), ZZ)
See Also
========
unify
"""
A, *B = A.unify(*B, fmt='dense')
return DomainMatrix.from_rep(A.rep.hstack(*(Bk.rep for Bk in B)))
def vstack(A, *B):
r"""Vertically stack the given matrices.
Parameters
==========
B: DomainMatrix
Matrices to stack vertically.
Returns
=======
DomainMatrix
DomainMatrix by stacking vertically.
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([[ZZ(5), ZZ(6)], [ZZ(7), ZZ(8)]], (2, 2), ZZ)
>>> A.vstack(B)
DomainMatrix([[1, 2], [3, 4], [5, 6], [7, 8]], (4, 2), ZZ)
>>> C = DomainMatrix([[ZZ(9), ZZ(10)], [ZZ(11), ZZ(12)]], (2, 2), ZZ)
>>> A.vstack(B, C)
DomainMatrix([[1, 2], [3, 4], [5, 6], [7, 8], [9, 10], [11, 12]], (6, 2), ZZ)
See Also
========
unify
"""
A, *B = A.unify(*B, fmt='dense')
return DomainMatrix.from_rep(A.rep.vstack(*(Bk.rep for Bk in B)))
def applyfunc(self, func, domain=None):
if domain is None:
domain = self.domain
return self.from_rep(self.rep.applyfunc(func, domain))
def __add__(A, B):
if not isinstance(B, DomainMatrix):
return NotImplemented
A, B = A.unify(B, fmt='dense')
return A.add(B)
def __sub__(A, B):
if not isinstance(B, DomainMatrix):
return NotImplemented
A, B = A.unify(B, fmt='dense')
return A.sub(B)
def __neg__(A):
return A.neg()
def __mul__(A, B):
"""A * B"""
if isinstance(B, DomainMatrix):
A, B = A.unify(B, fmt='dense')
return A.matmul(B)
elif B in A.domain:
return A.scalarmul(B)
elif isinstance(B, DomainScalar):
A, B = A.unify(B)
return A.scalarmul(B.element)
else:
return NotImplemented
def __rmul__(A, B):
if B in A.domain:
return A.rscalarmul(B)
elif isinstance(B, DomainScalar):
A, B = A.unify(B)
return A.rscalarmul(B.element)
else:
return NotImplemented
def __pow__(A, n):
"""A ** n"""
if not isinstance(n, int):
return NotImplemented
return A.pow(n)
def _check(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)
if a.rep.fmt != b.rep.fmt:
msg = "Format mismatch: %s %s %s" % (a.rep.fmt, op, b.rep.fmt)
raise DMFormatError(msg)
def add(A, B):
r"""
Adds two DomainMatrix matrices of the same Domain
Parameters
==========
A, B: DomainMatrix
matrices to add
Returns
=======
DomainMatrix
DomainMatrix after Addition
Raises
======
DMShapeError
If the dimensions of the two DomainMatrix are not equal
ValueError
If the domain of the two DomainMatrix are not same
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(4), ZZ(3)],
... [ZZ(2), ZZ(1)]], (2, 2), ZZ)
>>> A.add(B)
DomainMatrix([[5, 5], [5, 5]], (2, 2), ZZ)
See Also
========
sub, matmul
"""
A._check('+', B, A.shape, B.shape)
return A.from_rep(A.rep.add(B.rep))
def sub(A, B):
r"""
Subtracts two DomainMatrix matrices of the same Domain
Parameters
==========
A, B: DomainMatrix
matrices to subtract
Returns
=======
DomainMatrix
DomainMatrix after Subtraction
Raises
======
DMShapeError
If the dimensions of the two DomainMatrix are not equal
ValueError
If the domain of the two DomainMatrix are not same
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(4), ZZ(3)],
... [ZZ(2), ZZ(1)]], (2, 2), ZZ)
>>> A.sub(B)
DomainMatrix([[-3, -1], [1, 3]], (2, 2), ZZ)
See Also
========
add, matmul
"""
A._check('-', B, A.shape, B.shape)
return A.from_rep(A.rep.sub(B.rep))
def neg(A):
r"""
Returns the negative of DomainMatrix
Parameters
==========
A : Represents a DomainMatrix
Returns
=======
DomainMatrix
DomainMatrix after Negation
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.neg()
DomainMatrix([[-1, -2], [-3, -4]], (2, 2), ZZ)
"""
return A.from_rep(A.rep.neg())
def mul(A, b):
r"""
Performs term by term multiplication for the second DomainMatrix
w.r.t first DomainMatrix. Returns a DomainMatrix whose rows are
list of DomainMatrix matrices created after term by term multiplication.
Parameters
==========
A, B: DomainMatrix
matrices to multiply term-wise
Returns
=======
DomainMatrix
DomainMatrix after term by term multiplication
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(1), ZZ(1)],
... [ZZ(0), ZZ(1)]], (2, 2), ZZ)
>>> A.mul(B)
DomainMatrix([[DomainMatrix([[1, 1], [0, 1]], (2, 2), ZZ),
DomainMatrix([[2, 2], [0, 2]], (2, 2), ZZ)],
[DomainMatrix([[3, 3], [0, 3]], (2, 2), ZZ),
DomainMatrix([[4, 4], [0, 4]], (2, 2), ZZ)]], (2, 2), ZZ)
See Also
========
matmul
"""
return A.from_rep(A.rep.mul(b))
def rmul(A, b):
return A.from_rep(A.rep.rmul(b))
def matmul(A, B):
r"""
Performs matrix multiplication of two DomainMatrix matrices
Parameters
==========
A, B: DomainMatrix
to multiply
Returns
=======
DomainMatrix
DomainMatrix after multiplication
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(1), ZZ(1)],
... [ZZ(0), ZZ(1)]], (2, 2), ZZ)
>>> A.matmul(B)
DomainMatrix([[1, 3], [3, 7]], (2, 2), ZZ)
See Also
========
mul, pow, add, sub
"""
A._check('*', B, A.shape[1], B.shape[0])
return A.from_rep(A.rep.matmul(B.rep))
def _scalarmul(A, lamda, reverse):
if lamda == A.domain.zero:
return DomainMatrix.zeros(A.shape, A.domain)
elif lamda == A.domain.one:
return A.copy()
elif reverse:
return A.rmul(lamda)
else:
return A.mul(lamda)
def scalarmul(A, lamda):
return A._scalarmul(lamda, reverse=False)
def rscalarmul(A, lamda):
return A._scalarmul(lamda, reverse=True)
def mul_elementwise(A, B):
assert A.domain == B.domain
return A.from_rep(A.rep.mul_elementwise(B.rep))
def __truediv__(A, lamda):
""" Method for Scalar Division"""
if isinstance(lamda, int) or ZZ.of_type(lamda):
lamda = DomainScalar(ZZ(lamda), ZZ)
if not isinstance(lamda, DomainScalar):
return NotImplemented
A, lamda = A.to_field().unify(lamda)
if lamda.element == lamda.domain.zero:
raise ZeroDivisionError
if lamda.element == lamda.domain.one:
return A.to_field()
return A.mul(1 / lamda.element)
def pow(A, n):
r"""
Computes A**n
Parameters
==========
A : DomainMatrix
n : exponent for A
Returns
=======
DomainMatrix
DomainMatrix on computing A**n
Raises
======
NotImplementedError
if n is negative.
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(1)],
... [ZZ(0), ZZ(1)]], (2, 2), ZZ)
>>> A.pow(2)
DomainMatrix([[1, 2], [0, 1]], (2, 2), ZZ)
See Also
========
matmul
"""
nrows, ncols = A.shape
if nrows != ncols:
raise DMNonSquareMatrixError('Power of a nonsquare matrix')
if n < 0:
raise NotImplementedError('Negative powers')
elif n == 0:
return A.eye(nrows, A.domain)
elif n == 1:
return A
elif n % 2 == 1:
return A * A**(n - 1)
else:
sqrtAn = A ** (n // 2)
return sqrtAn * sqrtAn
def scc(self):
"""Compute the strongly connected components of a DomainMatrix
Explanation
===========
A square matrix can be considered as the adjacency matrix for a
directed graph where the row and column indices are the vertices. In
this graph if there is an edge from vertex ``i`` to vertex ``j`` if
``M[i, j]`` is nonzero. This routine computes the strongly connected
components of that graph which are subsets of the rows and columns that
are connected by some nonzero element of the matrix. The strongly
connected components are useful because many operations such as the
determinant can be computed by working with the submatrices
corresponding to each component.
Examples
========
Find the strongly connected components of a matrix:
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> M = DomainMatrix([[ZZ(1), ZZ(0), ZZ(2)],
... [ZZ(0), ZZ(3), ZZ(0)],
... [ZZ(4), ZZ(6), ZZ(5)]], (3, 3), ZZ)
>>> M.scc()
[[1], [0, 2]]
Compute the determinant from the components:
>>> MM = M.to_Matrix()
>>> MM
Matrix([
[1, 0, 2],
[0, 3, 0],
[4, 6, 5]])
>>> MM[[1], [1]]
Matrix([[3]])
>>> MM[[0, 2], [0, 2]]
Matrix([
[1, 2],
[4, 5]])
>>> MM.det()
-9
>>> MM[[1], [1]].det() * MM[[0, 2], [0, 2]].det()
-9
The components are given in reverse topological order and represent a
permutation of the rows and columns that will bring the matrix into
block lower-triangular form:
>>> MM[[1, 0, 2], [1, 0, 2]]
Matrix([
[3, 0, 0],
[0, 1, 2],
[6, 4, 5]])
Returns
=======
List of lists of integers
Each list represents a strongly connected component.
See also
========
sympy.matrices.matrices.MatrixBase.strongly_connected_components
sympy.utilities.iterables.strongly_connected_components
"""
rows, cols = self.shape
assert rows == cols
return self.rep.scc()
def rref(self):
r"""
Returns reduced-row echelon form and list of pivots for the DomainMatrix
Returns
=======
(DomainMatrix, list)
reduced-row echelon form and list of pivots for the DomainMatrix
Raises
======
ValueError
If the domain of DomainMatrix not a Field
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(2), QQ(-1), QQ(0)],
... [QQ(-1), QQ(2), QQ(-1)],
... [QQ(0), QQ(0), QQ(2)]], (3, 3), QQ)
>>> rref_matrix, rref_pivots = A.rref()
>>> rref_matrix
DomainMatrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], (3, 3), QQ)
>>> rref_pivots
(0, 1, 2)
See Also
========
convert_to, lu
"""
if not self.domain.is_Field:
raise DMNotAField('Not a field')
rref_ddm, pivots = self.rep.rref()
return self.from_rep(rref_ddm), tuple(pivots)
def columnspace(self):
r"""
Returns the columnspace for the DomainMatrix
Returns
=======
DomainMatrix
The columns of this matrix form a basis for the columnspace.
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(1), QQ(-1)],
... [QQ(2), QQ(-2)]], (2, 2), QQ)
>>> A.columnspace()
DomainMatrix([[1], [2]], (2, 1), QQ)
"""
if not self.domain.is_Field:
raise DMNotAField('Not a field')
rref, pivots = self.rref()
rows, cols = self.shape
return self.extract(range(rows), pivots)
def rowspace(self):
r"""
Returns the rowspace for the DomainMatrix
Returns
=======
DomainMatrix
The rows of this matrix form a basis for the rowspace.
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(1), QQ(-1)],
... [QQ(2), QQ(-2)]], (2, 2), QQ)
>>> A.rowspace()
DomainMatrix([[1, -1]], (1, 2), QQ)
"""
if not self.domain.is_Field:
raise DMNotAField('Not a field')
rref, pivots = self.rref()
rows, cols = self.shape
return self.extract(range(len(pivots)), range(cols))
def nullspace(self):
r"""
Returns the nullspace for the DomainMatrix
Returns
=======
DomainMatrix
The rows of this matrix form a basis for the nullspace.
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(1), QQ(-1)],
... [QQ(2), QQ(-2)]], (2, 2), QQ)
>>> A.nullspace()
DomainMatrix([[1, 1]], (1, 2), QQ)
"""
if not self.domain.is_Field:
raise DMNotAField('Not a field')
return self.from_rep(self.rep.nullspace()[0])
def inv(self):
r"""
Finds the inverse of the DomainMatrix if exists
Returns
=======
DomainMatrix
DomainMatrix after inverse
Raises
======
ValueError
If the domain of DomainMatrix not a Field
DMNonSquareMatrixError
If the DomainMatrix is not a not Square DomainMatrix
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(2), QQ(-1), QQ(0)],
... [QQ(-1), QQ(2), QQ(-1)],
... [QQ(0), QQ(0), QQ(2)]], (3, 3), QQ)
>>> A.inv()
DomainMatrix([[2/3, 1/3, 1/6], [1/3, 2/3, 1/3], [0, 0, 1/2]], (3, 3), QQ)
See Also
========
neg
"""
if not self.domain.is_Field:
raise DMNotAField('Not a field')
m, n = self.shape
if m != n:
raise DMNonSquareMatrixError
inv = self.rep.inv()
return self.from_rep(inv)
def det(self):
r"""
Returns the determinant of a Square DomainMatrix
Returns
=======
S.Complexes
determinant of Square DomainMatrix
Raises
======
ValueError
If the domain of DomainMatrix not a Field
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.det()
-2
"""
m, n = self.shape
if m != n:
raise DMNonSquareMatrixError
return self.rep.det()
def lu(self):
r"""
Returns Lower and Upper decomposition of the DomainMatrix
Returns
=======
(L, U, exchange)
L, U are Lower and Upper decomposition of the DomainMatrix,
exchange is the list of indices of rows exchanged in the decomposition.
Raises
======
ValueError
If the domain of DomainMatrix not a Field
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(1), QQ(-1)],
... [QQ(2), QQ(-2)]], (2, 2), QQ)
>>> A.lu()
(DomainMatrix([[1, 0], [2, 1]], (2, 2), QQ), DomainMatrix([[1, -1], [0, 0]], (2, 2), QQ), [])
See Also
========
lu_solve
"""
if not self.domain.is_Field:
raise DMNotAField('Not a field')
L, U, swaps = self.rep.lu()
return self.from_rep(L), self.from_rep(U), swaps
def lu_solve(self, rhs):
r"""
Solver for DomainMatrix x in the A*x = B
Parameters
==========
rhs : DomainMatrix B
Returns
=======
DomainMatrix
x in A*x = B
Raises
======
DMShapeError
If the DomainMatrix A and rhs have different number of rows
ValueError
If the domain of DomainMatrix A not a Field
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(1), QQ(2)],
... [QQ(3), QQ(4)]], (2, 2), QQ)
>>> B = DomainMatrix([
... [QQ(1), QQ(1)],
... [QQ(0), QQ(1)]], (2, 2), QQ)
>>> A.lu_solve(B)
DomainMatrix([[-2, -1], [3/2, 1]], (2, 2), QQ)
See Also
========
lu
"""
if self.shape[0] != rhs.shape[0]:
raise DMShapeError("Shape")
if not self.domain.is_Field:
raise DMNotAField('Not a field')
sol = self.rep.lu_solve(rhs.rep)
return self.from_rep(sol)
def _solve(A, b):
# XXX: Not sure about this method or its signature. It is just created
# because it is needed by the holonomic module.
if A.shape[0] != b.shape[0]:
raise DMShapeError("Shape")
if A.domain != b.domain or not A.domain.is_Field:
raise DMNotAField('Not a field')
Aaug = A.hstack(b)
Arref, pivots = Aaug.rref()
particular = Arref.from_rep(Arref.rep.particular())
nullspace_rep, nonpivots = Arref[:,:-1].rep.nullspace()
nullspace = Arref.from_rep(nullspace_rep)
return particular, nullspace
def charpoly(self):
r"""
Returns the coefficients of the characteristic polynomial
of the DomainMatrix. These elements will be domain elements.
The domain of the elements will be same as domain of the DomainMatrix.
Returns
=======
list
coefficients of the characteristic polynomial
Raises
======
DMNonSquareMatrixError
If the DomainMatrix is not a not Square DomainMatrix
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.charpoly()
[1, -5, -2]
"""
m, n = self.shape
if m != n:
raise DMNonSquareMatrixError("not square")
return self.rep.charpoly()
@classmethod
def eye(cls, shape, domain):
r"""
Return identity matrix of size n
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> DomainMatrix.eye(3, QQ)
DomainMatrix({0: {0: 1}, 1: {1: 1}, 2: {2: 1}}, (3, 3), QQ)
"""
if isinstance(shape, int):
shape = (shape, shape)
return cls.from_rep(SDM.eye(shape, domain))
@classmethod
def diag(cls, diagonal, domain, shape=None):
r"""
Return diagonal matrix with entries from ``diagonal``.
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import ZZ
>>> DomainMatrix.diag([ZZ(5), ZZ(6)], ZZ)
DomainMatrix({0: {0: 5}, 1: {1: 6}}, (2, 2), ZZ)
"""
if shape is None:
N = len(diagonal)
shape = (N, N)
return cls.from_rep(SDM.diag(diagonal, domain, shape))
@classmethod
def zeros(cls, shape, domain, *, fmt='sparse'):
"""Returns a zero DomainMatrix of size shape, belonging to the specified domain
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> DomainMatrix.zeros((2, 3), QQ)
DomainMatrix({}, (2, 3), QQ)
"""
return cls.from_rep(SDM.zeros(shape, domain))
@classmethod
def ones(cls, shape, domain):
"""Returns a DomainMatrix of 1s, of size shape, belonging to the specified domain
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> DomainMatrix.ones((2,3), QQ)
DomainMatrix([[1, 1, 1], [1, 1, 1]], (2, 3), QQ)
"""
return cls.from_rep(DDM.ones(shape, domain))
def __eq__(A, B):
r"""
Checks for two DomainMatrix matrices to be equal or not
Parameters
==========
A, B: DomainMatrix
to check equality
Returns
=======
Boolean
True for equal, else False
Raises
======
NotImplementedError
If B is not a DomainMatrix
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(1), ZZ(1)],
... [ZZ(0), ZZ(1)]], (2, 2), ZZ)
>>> A.__eq__(A)
True
>>> A.__eq__(B)
False
"""
if not isinstance(A, type(B)):
return NotImplemented
return A.domain == B.domain and A.rep == B.rep
def unify_eq(A, B):
if A.shape != B.shape:
return False
if A.domain != B.domain:
A, B = A.unify(B)
return A == B
def lll(A, delta=QQ(3, 4)):
"""
Performs the Lenstra–Lenstra–Lovász (LLL) basis reduction algorithm.
See [1]_ and [2]_.
Parameters
==========
delta : QQ, optional
The Lovász parameter. Must be in the interval (0.25, 1), with larger
values producing a more reduced basis. The default is 0.75 for
historical reasons.
Returns
=======
The reduced basis as a DomainMatrix over ZZ.
Throws
======
DMValueError: if delta is not in the range (0.25, 1)
DMShapeError: if the matrix is not of shape (m, n) with m <= n
DMDomainError: if the matrix domain is not ZZ
DMRankError: if the matrix contains linearly dependent rows
Examples
========
>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.matrices import DM
>>> x = DM([[1, 0, 0, 0, -20160],
... [0, 1, 0, 0, 33768],
... [0, 0, 1, 0, 39578],
... [0, 0, 0, 1, 47757]], ZZ)
>>> y = DM([[10, -3, -2, 8, -4],
... [3, -9, 8, 1, -11],
... [-3, 13, -9, -3, -9],
... [-12, -7, -11, 9, -1]], ZZ)
>>> assert x.lll(delta=QQ(5, 6)) == y
Notes
=====
The implementation is derived from the Maple code given in Figures 4.3
and 4.4 of [3]_ (pp.68-69). It uses the efficient method of only calculating
state updates as they are required.
See also
========
lll_transform
References
==========
.. [1] https://en.wikipedia.org/wiki/Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
.. [2] https://web.archive.org/web/20221029115428/https://web.cs.elte.hu/~lovasz/scans/lll.pdf
.. [3] Murray R. Bremner, "Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications"
"""
return DomainMatrix.from_rep(A.rep.lll(delta=delta))
def lll_transform(A, delta=QQ(3, 4)):
"""
Performs the Lenstra–Lenstra–Lovász (LLL) basis reduction algorithm
and returns the reduced basis and transformation matrix.
Explanation
===========
Parameters, algorithm and basis are the same as for :meth:`lll` except that
the return value is a tuple `(B, T)` with `B` the reduced basis and
`T` a transformation matrix. The original basis `A` is transformed to
`B` with `T*A == B`. If only `B` is needed then :meth:`lll` should be
used as it is a little faster.
Examples
========
>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.matrices import DM
>>> X = DM([[1, 0, 0, 0, -20160],
... [0, 1, 0, 0, 33768],
... [0, 0, 1, 0, 39578],
... [0, 0, 0, 1, 47757]], ZZ)
>>> B, T = X.lll_transform(delta=QQ(5, 6))
>>> T * X == B
True
See also
========
lll
"""
reduced, transform = A.rep.lll_transform(delta=delta)
return DomainMatrix.from_rep(reduced), DomainMatrix.from_rep(transform)