Source code for sympy.matrices.dense

from __future__ import print_function, division

import random
from sympy import Derivative

from sympy.core.basic import Basic
from sympy.core.compatibility import is_sequence, as_int, range
from sympy.core.function import count_ops
from sympy.core.decorators import call_highest_priority
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.functions.elementary.miscellaneous import sqrt
from sympy.simplify import simplify as _simplify
from sympy.utilities.misc import filldedent
from sympy.utilities.decorator import doctest_depends_on

from sympy.matrices.matrices import (MatrixBase,
ShapeError, a2idx, classof)

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

class DenseMatrix(MatrixBase):

is_MatrixExpr = False

_op_priority = 10.01
_class_priority = 4

def __getitem__(self, key):
"""Return portion of self defined by key. If the key involves a slice
then a list will be returned (if key is a single slice) or a matrix
(if key was a tuple involving a slice).

Examples
========

>>> from sympy import Matrix, I
>>> m = Matrix([
... [1, 2 + I],
... [3, 4    ]])

If the key is a tuple that doesn't involve a slice then that element
is returned:

>>> m[1, 0]
3

When a tuple key involves a slice, a matrix is returned. Here, the
first column is selected (all rows, column 0):

>>> m[:, 0]
Matrix([
[1],
[3]])

If the slice is not a tuple then it selects from the underlying
list of elements that are arranged in row order and a list is
returned if a slice is involved:

>>> m[0]
1
>>> m[::2]
[1, 3]
"""
if isinstance(key, tuple):
i, j = key
try:
i, j = self.key2ij(key)
return self._mat[i*self.cols + j]
except (TypeError, IndexError):
if isinstance(i, slice):
# XXX remove list() when PY2 support is dropped
i = list(range(self.rows))[i]
elif is_sequence(i):
pass
else:
i = [i]
if isinstance(j, slice):
# XXX remove list() when PY2 support is dropped
j = list(range(self.cols))[j]
elif is_sequence(j):
pass
else:
j = [j]
return self.extract(i, j)
else:
# row-wise decomposition of matrix
if isinstance(key, slice):
return self._mat[key]
return self._mat[a2idx(key)]

def __setitem__(self, key, value):
raise NotImplementedError()

@property
def is_Identity(self):
if not self.is_square:
return False
if not all(self[i, i] == 1 for i in range(self.rows)):
return False
for i in range(self.rows):
for j in range(i + 1, self.cols):
if self[i, j] or self[j, i]:
return False
return True

def tolist(self):
"""Return the Matrix as a nested Python list.

Examples
========

>>> from sympy import Matrix, ones
>>> m = Matrix(3, 3, range(9))
>>> m
Matrix([
[0, 1, 2],
[3, 4, 5],
[6, 7, 8]])
>>> m.tolist()
[[0, 1, 2], [3, 4, 5], [6, 7, 8]]
>>> ones(3, 0).tolist()
[[], [], []]

When there are no rows then it will not be possible to tell how
many columns were in the original matrix:

>>> ones(0, 3).tolist()
[]

"""
if not self.rows:
return []
if not self.cols:
return [[] for i in range(self.rows)]
return [self._mat[i: i + self.cols]
for i in range(0, len(self), self.cols)]

def row(self, i):
"""Elementary row selector.

Examples
========

>>> from sympy import eye
>>> eye(2).row(0)
Matrix([[1, 0]])

========

col
row_op
row_swap
row_del
row_join
row_insert
"""
return self[i, :]

def col(self, j):
"""Elementary column selector.

Examples
========

>>> from sympy import eye
>>> eye(2).col(0)
Matrix([
[1],
[0]])

========

row
col_op
col_swap
col_del
col_join
col_insert
"""
return self[:, j]

def _eval_trace(self):
"""Calculate the trace of a square matrix.

Examples
========

>>> from sympy.matrices import eye
>>> eye(3).trace()
3

"""
trace = 0
for i in range(self.cols):
trace += self._mat[i*self.cols + i]
return trace

def _eval_determinant(self):
return self.det()

def _eval_transpose(self):
"""Matrix transposition.

Examples
========

>>> from sympy import Matrix, I
>>> m=Matrix(((1, 2+I), (3, 4)))
>>> m
Matrix([
[1, 2 + I],
[3,     4]])
>>> m.transpose()
Matrix([
[    1, 3],
[2 + I, 4]])
>>> m.T == m.transpose()
True

========

conjugate: By-element conjugation
"""
a = []
for i in range(self.cols):
a.extend(self._mat[i::self.cols])
return self._new(self.cols, self.rows, a)

def _eval_conjugate(self):
"""By-element conjugation.

========

transpose: Matrix transposition
H: Hermite conjugation
D: Dirac conjugation
"""
out = self._new(self.rows, self.cols,
lambda i, j: self[i, j].conjugate())
return out

return self.T.C

def _eval_inverse(self, **kwargs):
"""Return the matrix inverse using the method indicated (default
is Gauss elimination).

kwargs
======

method : ('GE', 'LU', or 'ADJ')
iszerofunc
try_block_diag

Notes
=====

According to the method keyword, it calls the appropriate method:

GE .... inverse_GE(); default
LU .... inverse_LU()

According to the try_block_diag keyword, it will try to form block
diagonal matrices using the method get_diag_blocks(), invert these
individually, and then reconstruct the full inverse matrix.

Note, the GE and LU methods may require the matrix to be simplified
before it is inverted in order to properly detect zeros during
pivoting. In difficult cases a custom zero detection function can
be provided by setting the iszerosfunc argument to a function that
should return True if its argument is zero. The ADJ routine computes
the determinant and uses that to detect singular matrices in addition
to testing for zeros on the diagonal.

========

inverse_LU
inverse_GE
"""
from sympy.matrices import diag

method = kwargs.get('method', 'GE')
iszerofunc = kwargs.get('iszerofunc', _iszero)
if kwargs.get('try_block_diag', False):
blocks = self.get_diag_blocks()
r = []
for block in blocks:
r.append(block.inv(method=method, iszerofunc=iszerofunc))
return diag(*r)

M = self.as_mutable()
if method == "GE":
rv = M.inverse_GE(iszerofunc=iszerofunc)
elif method == "LU":
rv = M.inverse_LU(iszerofunc=iszerofunc)
else:
# make sure to add an invertibility check (as in inverse_LU)
# if a new method is added.
raise ValueError("Inversion method unrecognized")
return self._new(rv)

def _eval_diff(self, *args, **kwargs):
if kwargs.pop("evaluate", True):
return self.diff(*args)
else:
return Derivative(self, *args, **kwargs)

def equals(self, other, failing_expression=False):
"""Applies equals to corresponding elements of the matrices,
trying to prove that the elements are equivalent, returning True
if they are, False if any pair is not, and None (or the first
failing expression if failing_expression is True) if it cannot
be decided if the expressions are equivalent or not. This is, in
general, an expensive operation.

Examples
========

>>> from sympy.matrices import Matrix
>>> from sympy.abc import x
>>> from sympy import cos
>>> A = Matrix([x*(x - 1), 0])
>>> B = Matrix([x**2 - x, 0])
>>> A == B
False
>>> A.simplify() == B.simplify()
True
>>> A.equals(B)
True
>>> A.equals(2)
False

========
sympy.core.expr.equals
"""
try:
if self.shape != other.shape:
return False
rv = True
for i in range(self.rows):
for j in range(self.cols):
ans = self[i, j].equals(other[i, j], failing_expression)
if ans is False:
return False
elif ans is not True and rv is True:
rv = ans
return rv
except AttributeError:
return False

def __eq__(self, other):
try:
if self.shape != other.shape:
return False
if isinstance(other, Matrix):
return self._mat == other._mat
elif isinstance(other, MatrixBase):
return self._mat == Matrix(other)._mat
except AttributeError:
return False

def __ne__(self, other):
return not self == other

def _cholesky(self):
"""Helper function of cholesky.
Without the error checks.
To be used privately. """
L = zeros(self.rows, self.rows)
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / L[j, j])*(self[i, j] -
sum(L[i, k]*L[j, k] for k in range(j)))
L[i, i] = sqrt(self[i, i] -
sum(L[i, k]**2 for k in range(i)))
return self._new(L)

def _LDLdecomposition(self):
"""Helper function of LDLdecomposition.
Without the error checks.
To be used privately.
"""
D = zeros(self.rows, self.rows)
L = eye(self.rows)
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / D[j, j])*(self[i, j] - sum(
L[i, k]*L[j, k]*D[k, k] for k in range(j)))
D[i, i] = self[i, i] - sum(L[i, k]**2*D[k, k]
for k in range(i))
return self._new(L), self._new(D)

def _lower_triangular_solve(self, rhs):
"""Helper function of function lower_triangular_solve.
Without the error checks.
To be used privately.
"""
X = zeros(self.rows, rhs.cols)
for j in range(rhs.cols):
for i in range(self.rows):
if self[i, i] == 0:
raise TypeError("Matrix must be non-singular.")
X[i, j] = (rhs[i, j] - sum(self[i, k]*X[k, j]
for k in range(i))) / self[i, i]
return self._new(X)

def _upper_triangular_solve(self, rhs):
"""Helper function of function upper_triangular_solve.
Without the error checks, to be used privately. """
X = zeros(self.rows, rhs.cols)
for j in range(rhs.cols):
for i in reversed(range(self.rows)):
if self[i, i] == 0:
raise ValueError("Matrix must be non-singular.")
X[i, j] = (rhs[i, j] - sum(self[i, k]*X[k, j]
for k in range(i + 1, self.rows))) / self[i, i]
return self._new(X)

def _diagonal_solve(self, rhs):
"""Helper function of function diagonal_solve,
without the error checks, to be used privately.
"""
return self._new(rhs.rows, rhs.cols, lambda i, j: rhs[i, j] / self[i, i])

def applyfunc(self, f):
"""Apply a function to each element of the matrix.

Examples
========

>>> from sympy import Matrix
>>> m = Matrix(2, 2, lambda i, j: i*2+j)
>>> m
Matrix([
[0, 1],
[2, 3]])
>>> m.applyfunc(lambda i: 2*i)
Matrix([
[0, 2],
[4, 6]])

"""
if not callable(f):
raise TypeError("f must be callable.")

out = self._new(self.rows, self.cols, list(map(f, self._mat)))
return out

def reshape(self, rows, cols):
"""Reshape the matrix. Total number of elements must remain the same.

Examples
========

>>> from sympy import Matrix
>>> m = Matrix(2, 3, lambda i, j: 1)
>>> m
Matrix([
[1, 1, 1],
[1, 1, 1]])
>>> m.reshape(1, 6)
Matrix([[1, 1, 1, 1, 1, 1]])
>>> m.reshape(3, 2)
Matrix([
[1, 1],
[1, 1],
[1, 1]])

"""
if len(self) != rows*cols:
raise ValueError("Invalid reshape parameters %d %d" % (rows, cols))
return self._new(rows, cols, lambda i, j: self._mat[i*cols + j])

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 as_immutable(self):
"""Returns an Immutable version of this Matrix
"""
from .immutable import ImmutableMatrix as cls
if self.rows and self.cols:
return cls._new(self.tolist())
return cls._new(self.rows, self.cols, [])

@classmethod
def zeros(cls, r, c=None):
"""Return an r x c matrix of zeros, square if c is omitted."""
c = r if c is None else c
r = as_int(r)
c = as_int(c)
return cls._new(r, c, [cls._sympify(0)]*r*c)

@classmethod
def eye(cls, n):
"""Return an n x n identity matrix."""
n = as_int(n)
mat = [cls._sympify(0)]*n*n
mat[::n + 1] = [cls._sympify(1)]*n
return cls._new(n, n, mat)

############################
# Mutable matrix operators #
############################

@call_highest_priority('__rsub__')
def __sub__(self, other):
return super(DenseMatrix, self).__sub__(_force_mutable(other))

@call_highest_priority('__sub__')
def __rsub__(self, other):
return super(DenseMatrix, self).__rsub__(_force_mutable(other))

@call_highest_priority('__rmul__')
def __mul__(self, other):
return super(DenseMatrix, self).__mul__(_force_mutable(other))

@call_highest_priority('__rmul__')
def __matmul__(self, other):
return super(DenseMatrix, self).__mul__(_force_mutable(other))

@call_highest_priority('__mul__')
def __rmul__(self, other):
return super(DenseMatrix, self).__rmul__(_force_mutable(other))

@call_highest_priority('__mul__')
def __rmatmul__(self, other):
return super(DenseMatrix, self).__rmul__(_force_mutable(other))

@call_highest_priority('__div__')
def __div__(self, other):
return super(DenseMatrix, self).__div__(_force_mutable(other))

@call_highest_priority('__truediv__')
def __truediv__(self, other):
return super(DenseMatrix, self).__truediv__(_force_mutable(other))

@call_highest_priority('__rpow__')
def __pow__(self, other):
return super(DenseMatrix, self).__pow__(other)

@call_highest_priority('__pow__')
def __rpow__(self, other):
raise NotImplementedError("Matrix Power not defined")

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

[docs]class MutableDenseMatrix(DenseMatrix, MatrixBase):
@classmethod
def _new(cls, *args, **kwargs):
rows, cols, flat_list = cls._handle_creation_inputs(*args, **kwargs)
self = object.__new__(cls)
self.rows = rows
self.cols = cols
self._mat = list(flat_list)  # create a shallow copy
return self

def __new__(cls, *args, **kwargs):
return cls._new(*args, **kwargs)

def as_mutable(self):
return self.copy()

def __setitem__(self, key, value):
"""

Examples
========

>>> from sympy import Matrix, I, zeros, ones
>>> m = Matrix(((1, 2+I), (3, 4)))
>>> m
Matrix([
[1, 2 + I],
[3,     4]])
>>> m[1, 0] = 9
>>> m
Matrix([
[1, 2 + I],
[9,     4]])
>>> m[1, 0] = [[0, 1]]

To replace row r you assign to position r*m where m
is the number of columns:

>>> M = zeros(4)
>>> m = M.cols
>>> M[3*m] = ones(1, m)*2; M
Matrix([
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[2, 2, 2, 2]])

And to replace column c you can assign to position c:

>>> M[2] = ones(m, 1)*4; M
Matrix([
[0, 0, 4, 0],
[0, 0, 4, 0],
[0, 0, 4, 0],
[2, 2, 4, 2]])
"""
rv = self._setitem(key, value)
if rv is not None:
i, j, value = rv
self._mat[i*self.cols + j] = value

[docs]    def copyin_matrix(self, key, value):
"""Copy in values from a matrix into the given bounds.

Parameters
==========

key : slice
The section of this matrix to replace.
value : Matrix
The matrix to copy values from.

Examples
========

>>> from sympy.matrices import Matrix, eye
>>> M = Matrix([[0, 1], [2, 3], [4, 5]])
>>> I = eye(3)
>>> I[:3, :2] = M
>>> I
Matrix([
[0, 1, 0],
[2, 3, 0],
[4, 5, 1]])
>>> I[0, 1] = M
>>> I
Matrix([
[0, 0, 1],
[2, 2, 3],
[4, 4, 5]])

========

copyin_list
"""
rlo, rhi, clo, chi = self.key2bounds(key)
shape = value.shape
dr, dc = rhi - rlo, chi - clo
if shape != (dr, dc):
raise ShapeError(filldedent("The Matrix value doesn't have the "
"same dimensions "
"as the in sub-Matrix given by key."))

for i in range(value.rows):
for j in range(value.cols):
self[i + rlo, j + clo] = value[i, j]

[docs]    def copyin_list(self, key, value):
"""Copy in elements from a list.

Parameters
==========

key : slice
The section of this matrix to replace.
value : iterable
The iterable to copy values from.

Examples
========

>>> from sympy.matrices import eye
>>> I = eye(3)
>>> I[:2, 0] = [1, 2] # col
>>> I
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
>>> I[1, :2] = [[3, 4]]
>>> I
Matrix([
[1, 0, 0],
[3, 4, 0],
[0, 0, 1]])

========

copyin_matrix
"""
if not is_sequence(value):
raise TypeError("value must be an ordered iterable, not %s." % type(value))
return self.copyin_matrix(key, Matrix(value))

[docs]    def zip_row_op(self, i, k, f):
"""In-place operation on row i using two-arg functor whose args are
interpreted as (self[i, j], self[k, j]).

Examples
========

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])

========
row
row_op
col_op

"""
i0 = i*self.cols
k0 = k*self.cols

ri = self._mat[i0: i0 + self.cols]
rk = self._mat[k0: k0 + self.cols]

self._mat[i0: i0 + self.cols] = [ f(x, y) for x, y in zip(ri, rk) ]

[docs]    def row_op(self, i, f):
"""In-place operation on row i using two-arg functor whose args are
interpreted as (self[i, j], j).

Examples
========

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.row_op(1, lambda v, j: v + 2*M[0, j]); M
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])

========
row
zip_row_op
col_op

"""
i0 = i*self.cols
ri = self._mat[i0: i0 + self.cols]
self._mat[i0: i0 + self.cols] = [ f(x, j) for x, j in zip(ri, list(range(self.cols))) ]

[docs]    def col_op(self, j, f):
"""In-place operation on col j using two-arg functor whose args are
interpreted as (self[i, j], i).

Examples
========

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.col_op(1, lambda v, i: v + 2*M[i, 0]); M
Matrix([
[1, 2, 0],
[0, 1, 0],
[0, 0, 1]])

========
col
row_op
"""
self._mat[j::self.cols] = [f(*t) for t in list(zip(self._mat[j::self.cols], list(range(self.rows))))]

[docs]    def row_swap(self, i, j):
"""Swap the two given rows of the matrix in-place.

Examples
========

>>> from sympy.matrices import Matrix
>>> M = Matrix([[0, 1], [1, 0]])
>>> M
Matrix([
[0, 1],
[1, 0]])
>>> M.row_swap(0, 1)
>>> M
Matrix([
[1, 0],
[0, 1]])

========

row
col_swap
"""
for k in range(0, self.cols):
self[i, k], self[j, k] = self[j, k], self[i, k]

[docs]    def col_swap(self, i, j):
"""Swap the two given columns of the matrix in-place.

Examples
========

>>> from sympy.matrices import Matrix
>>> M = Matrix([[1, 0], [1, 0]])
>>> M
Matrix([
[1, 0],
[1, 0]])
>>> M.col_swap(0, 1)
>>> M
Matrix([
[0, 1],
[0, 1]])

========

col
row_swap
"""
for k in range(0, self.rows):
self[k, i], self[k, j] = self[k, j], self[k, i]

[docs]    def row_del(self, i):
"""Delete the given row.

Examples
========

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.row_del(1)
>>> M
Matrix([
[1, 0, 0],
[0, 0, 1]])

========

row
col_del
"""
if i < -self.rows or i >= self.rows:
raise IndexError("Index out of range: 'i = %s', valid -%s <= i"
" < %s" % (i, self.rows, self.rows))
if i < 0:
i += self.rows
del self._mat[i*self.cols:(i+1)*self.cols]
self.rows -= 1

[docs]    def col_del(self, i):
"""Delete the given column.

Examples
========

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.col_del(1)
>>> M
Matrix([
[1, 0],
[0, 0],
[0, 1]])

========

col
row_del
"""
if i < -self.cols or i >= self.cols:
raise IndexError("Index out of range: 'i=%s', valid -%s <= i < %s"
% (i, self.cols, self.cols))
for j in range(self.rows - 1, -1, -1):
del self._mat[i + j*self.cols]
self.cols -= 1

# Utility functions
[docs]    def simplify(self, ratio=1.7, measure=count_ops):
"""Applies simplify to the elements of a matrix in place.

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

========

sympy.simplify.simplify.simplify
"""
for i in range(len(self._mat)):
self._mat[i] = _simplify(self._mat[i], ratio=ratio,
measure=measure)

[docs]    def fill(self, value):
"""Fill the matrix with the scalar value.

========

zeros
ones
"""
self._mat = [value]*len(self)

MutableMatrix = Matrix = MutableDenseMatrix

###########
# Numpy Utility Functions:
# list2numpy, matrix2numpy, symmarray, rot_axis[123]
###########

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

========

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

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

========

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

@doctest_depends_on(modules=('numpy',))
[docs]def symarray(prefix, shape, **kwargs):  # pragma: no cover
"""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

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

Examples
========

>>> from sympy import pi
>>> from sympy.matrices import 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]])

========

rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
"""
ct = cos(theta)
st = sin(theta)
lil = ((ct, st, 0),
(-st, ct, 0),
(0, 0, 1))
return Matrix(lil)

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

Examples
========

>>> from sympy import pi
>>> from sympy.matrices import 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]])

========

rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
"""
ct = cos(theta)
st = sin(theta)
lil = ((ct, 0, -st),
(0, 1, 0),
(st, 0, ct))
return Matrix(lil)

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

Examples
========

>>> from sympy import pi
>>> from sympy.matrices import 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]])

========

rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
"""
ct = cos(theta)
st = sin(theta)
lil = ((1, 0, 0),
(0, ct, st),
(0, -st, ct))
return Matrix(lil)

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

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

>>> from sympy.matrices import matrix_multiply_elementwise
>>> from sympy.matrices import Matrix
>>> 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]])

========

__mul__
"""
if A.shape != B.shape:
raise ShapeError()
shape = A.shape
return classof(A, B)._new(shape[0], shape[1],
lambda i, j: A[i, j]*B[i, j])

[docs]def ones(r, c=None):
"""Returns a matrix of ones with r rows and c columns;
if c is omitted a square matrix will be returned.

========

zeros
eye
diag
"""
from .dense import Matrix

c = r if c is None else c
r = as_int(r)
c = as_int(c)
return Matrix(r, c, [S.One]*r*c)

[docs]def zeros(r, c=None, cls=None):
"""Returns a matrix of zeros with r rows and c columns;
if c is omitted a square matrix will be returned.

========

ones
eye
diag
"""
if cls is None:
from .dense import Matrix as cls
return cls.zeros(r, c)

[docs]def eye(n, cls=None):
"""Create square identity matrix n x n

========

diag
zeros
ones
"""
if cls is None:
from sympy.matrices import Matrix as cls
return cls.eye(n)

[docs]def diag(*values, **kwargs):
"""Create a sparse, diagonal matrix from a list of diagonal values.

Notes
=====

When arguments are matrices they are fitted in resultant matrix.

The returned matrix is a mutable, dense matrix. To make it a different
type, send the desired class for keyword cls.

Examples
========

>>> from sympy.matrices import diag, Matrix, ones
>>> diag(1, 2, 3)
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
>>> diag(*[1, 2, 3])
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])

The diagonal elements can be matrices; diagonal filling will
continue on the diagonal from the last element of the matrix:

>>> from sympy.abc import x, y, z
>>> a = Matrix([x, y, z])
>>> b = Matrix([[1, 2], [3, 4]])
>>> c = Matrix([[5, 6]])
>>> diag(a, 7, b, c)
Matrix([
[x, 0, 0, 0, 0, 0],
[y, 0, 0, 0, 0, 0],
[z, 0, 0, 0, 0, 0],
[0, 7, 0, 0, 0, 0],
[0, 0, 1, 2, 0, 0],
[0, 0, 3, 4, 0, 0],
[0, 0, 0, 0, 5, 6]])

When diagonal elements are lists, they will be treated as arguments
to Matrix:

>>> diag([1, 2, 3], 4)
Matrix([
[1, 0],
[2, 0],
[3, 0],
[0, 4]])
>>> diag([[1, 2, 3]], 4)
Matrix([
[1, 2, 3, 0],
[0, 0, 0, 4]])

A given band off the diagonal can be made by padding with a
vertical or horizontal "kerning" vector:

Matrix([
[0, 0, 4, 0, 0],
[0, 0, 0, 5, 0],
[1, 0, 0, 0, 6],
[0, 2, 0, 0, 0],
[0, 0, 3, 0, 0]])

The type is mutable by default but can be made immutable by setting
the mutable flag to False:

>>> type(diag(1))
<class 'sympy.matrices.dense.MutableDenseMatrix'>
>>> from sympy.matrices import ImmutableMatrix
>>> type(diag(1, cls=ImmutableMatrix))
<class 'sympy.matrices.immutable.ImmutableMatrix'>

========

eye
"""
from .sparse import MutableSparseMatrix

cls = kwargs.pop('cls', None)
if cls is None:
from .dense import Matrix as cls

if kwargs:
raise ValueError('unrecognized keyword%s: %s' % (
's' if len(kwargs) > 1 else '',
', '.join(kwargs.keys())))
rows = 0
cols = 0
values = list(values)
for i in range(len(values)):
m = values[i]
if isinstance(m, MatrixBase):
rows += m.rows
cols += m.cols
elif is_sequence(m):
m = values[i] = Matrix(m)
rows += m.rows
cols += m.cols
else:
rows += 1
cols += 1
res = MutableSparseMatrix.zeros(rows, cols)
i_row = 0
i_col = 0
for m in values:
if isinstance(m, MatrixBase):
res[i_row:i_row + m.rows, i_col:i_col + m.cols] = m
i_row += m.rows
i_col += m.cols
else:
res[i_row, i_col] = m
i_row += 1
i_col += 1
return cls._new(res)

[docs]def jordan_cell(eigenval, n):
"""
Create matrix of Jordan cell kind:

Examples
========

>>> from sympy.matrices 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]])
"""
n = as_int(n)
out = zeros(n)
for i in range(n - 1):
out[i, i] = eigenval
out[i, i + 1] = S.One
out[n - 1, n - 1] = eigenval
return out

[docs]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
==========

http://en.wikipedia.org/wiki/Hessian_matrix

========

sympy.matrices.mutable.Matrix.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

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

see: http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
"""
out = []
m = len(vlist)
for i in range(m):
tmp = vlist[i]
for j in range(i):
tmp -= vlist[i].project(out[j])
if not tmp.values():
raise ValueError(
"GramSchmidt: vector set not linearly independent")
out.append(tmp)
if orthonormal:
for i in range(len(out)):
out[i] = out[i].normalized()
return out

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

::

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

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

========

sympy.matrices.mutable.Matrix.jacobian
hessian
"""
from .dense import Matrix

for index in range(0, len(functions)):
functions[index] = sympify(functions[index])
n = len(functions)
if n == 0:
return 1
W = Matrix(n, n, lambda i, j: functions[i].diff(var, j))
return W.det(method)

[docs]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

"""
from .dense import Matrix

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()

[docs]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 :class:random.Random, or at least have
randint and shuffle methods with same signatures.
* if prng is not supplied but seed is supplied, then new
:class:random.Random with given seed will be created;
* otherwise, a new :class:random.Random with default seed will be used.

Examples
========

>>> from sympy.matrices 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 # doctest:+SKIP
False
>>> A == randMatrix(3, seed=1)
True
>>> randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP
[0, 68, 43]
[0, 68,  0]
[0, 91, 34]
"""
if c is None:
c = r
# Note that Random() is equivalent to Random(None)
prng = prng or random.Random(seed)
if symmetric and r != c:
raise ValueError(
'For symmetric matrices, r must equal c, but %i != %i' % (r, c))
if not symmetric:
m = Matrix._new(r, c, lambda i, j: prng.randint(min, max))
else:
m = zeros(r)
for i in range(r):
for j in range(i, r):
m[i, j] = prng.randint(min, max)
for i in range(r):
for j in range(i):
m[i, j] = m[j, i]
if percent == 100:
return m
else:
z = int(r*c*percent // 100)
m._mat[:z] = [S.Zero]*z
prng.shuffle(m._mat)
return m