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# sympy.ordered

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```Return an iterator of the seq where keys are used to break ties.
Two default keys will be applied after and provided unless ``default``
is False. The two keys are _nodes and default_sort_key which will
place smaller expressions before larger ones (in terms of Basic nodes)
and where there are ties, they will be broken by the default_sort_key.

If ``warn`` is True then an error will be raised if there were no
keys remaining to break ties. This can be used if it was expected that
there should be no ties.
(more...)
```

```        def ordered(seq, keys=None, default=True, warn=False):
"""Return an iterator of the seq where keys are used to break ties.
Two default keys will be applied after and provided unless ``default``
is False. The two keys are _nodes and default_sort_key which will
place smaller expressions before larger ones (in terms of Basic nodes)
and where there are ties, they will be broken by the default_sort_key.

If ``warn`` is True then an error will be raised if there were no
keys remaining to break ties. This can be used if it was expected that
there should be no ties.

Examples
========

>>> from sympy.utilities.iterables import ordered
>>> from sympy import count_ops
>>> from sympy.abc import x, y

The count_ops is not sufficient to break ties in this list and the first
two items appear in their original order (i.e. the sorting is stable):

>>> list(ordered([y + 2, x + 2, x**2 + y + 3],
...    count_ops, default=False, warn=False))
...
[y + 2, x + 2, x**2 + y + 3]

The default_sort_key allows the tie to be broken:

>>> list(ordered([y + 2, x + 2, x**2 + y + 3]))
...
[x + 2, y + 2, x**2 + y + 3]

Here, sequences are sorted by length, then sum:

>>> seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], , ], [
...    lambda x: len(x),
...    lambda x: sum(x)]]
...
>>> list(ordered(seq, keys, default=False, warn=False))
[, , [1, 2, 1], [0, 3, 1], [1, 1, 3]]

If ``warn`` is True, an error will be raised if there were not
enough keys to break ties:

>>> list(ordered(seq, keys, default=False, warn=True))
Traceback (most recent call last):
...
ValueError: not enough keys to break ties

Notes
=====

The decorated sort is one of the fastest ways to sort a sequence for
which special item comparison is desired: the sequence is decorated,
sorted on the basis of the decoration (e.g. making all letters lower
case) and then undecorated. If one wants to break ties for items that
have the same decorated value, a second key can be used. But if the
second key is expensive to compute then it is inefficient to decorate
all items with both keys: only those items having identical first key
values need to be decorated. This function applies keys successively
only when needed to break ties. By yielding an iterator, use of the
tie-breaker is delayed as long as possible.

This function is best used in cases when use of the first key is
expected to be a good hashing function; if there are no unique hashes
from application of a key then that key should not have been used. The
exception, however, is that even if there are many collisions, if the
first group is small and one does not need to process all items in the
list then time will not be wasted sorting what one was not interested
in. For example, if one were looking for the minimum in a list and
there were several criteria used to define the sort order, then this
function would be good at returning that quickly if the first group
of candidates is small relative to the number of items being processed.

"""
d = defaultdict(list)
if keys:
if not isinstance(keys, (list, tuple)):
keys = [keys]
keys = list(keys)

f = keys.pop(0)
for a in seq:
d[f(a)].append(a)
else:
if not default:
raise ValueError('if default=False then keys must be provided')
d[None].extend(seq)

for k in sorted(d.keys()):
if len(d[k]) > 1:
if keys:
d[k] = ordered(d[k], keys, default, warn)
elif default:
d[k] = ordered(d[k], (_nodes, default_sort_key,),
default=False, warn=warn)
elif warn:
raise ValueError('not enough keys to break ties')
for v in d[k]:
yield v
d.pop(k)
```

```"""Interfaces to SDP solvers"""

from sympy import Basic, Dummy, S, ordered, sympify, BlockDiagMatrix
from numpy import array
```
```        for expr in lm:
variables |= expr.free_symbols
return list(ordered(variables))

```

```from sympy import (
acos, Add, atan, besselsimp, binomial, collect, collect_const, combsimp,
cos, cosh, cot, coth, count_ops, Derivative, diff, Dummy, E, Eq, erf, exp,
exp_polar, expand, exptrigsimp, factor, factorial, FallingFactorial, Float,
fraction, Function, gamma, GoldenRatio, hyper, hyper, hypersimp, I,
```

```from sympy import (