On 10/22/2014 01:22 PM, [email protected] wrote:
> Hello to every one,
> I am a *primitive *sage user and I want to produce a function of n
> variables (|x[1],x[2],...,x[n]|), in nested 'for' loops. I've searched
> in sage's documentation, but I have still 2 problems:
> 1. How to define a list of variables, such that I can use its items, as
> independent variables, in loops. The loops are needed because the
> function is made up of sums of multiplications of clauses of these
> variables?
This isn't built in, but you can use the attached file to do it. There
are detailed examples in the docstring, but basically you load the file
and then create a SymbolSequence object with the name you want:
sage: load('symbol_sequence.py')
sage: x = SymbolSequence('x')
sage: x[0]
x_0
sage: x[1]
x_1
sage: x[10]
x_10
Multiple subscripts and slices also work the way you'd want them to (I
hope).
> 2. How to simplify the result of loops according to some extra
> conditions, such as |x[i]^2==1| ?
We have the "assume" function, which can do things like,
sage: assume(x^2 == 1)
but I don't want to get your hopes up. Even with that assumption, the
expression x^2 won't simplify:
sage: (x^2).simplify_full()
x^2
For other equations, it might help though.
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from sage.all import *
class SymbolSequence:
"""
An iterable object which acts like a sequence of symbolic
expressions (variables).
INPUT:
- ``name`` -- The sequence name. For example, if you name the
sequence `x`, the variables will be called `x_0`, `x_1`,...
- ``latex_name`` -- An optional latex expression (string) to
use instead of `name` when converting the symbols to latex.
- ``domain`` -- A string representing the domain of the symbol,
either 'real', 'complex', or 'positive'.
OUTPUT:
An iterable object containing symbolic expressions.
EXAMPLES:
The simplest use case::
sage: a = SymbolSequence('a')
sage: a[0]
a_0
sage: a[1]
a_1
Create polynomials with symbolic coefficients of arbitrary
degree::
sage: a = SymbolSequence('a')
sage: p = sum([ a[i]*x^i for i in range(0,5)])
sage: p
a_4*x^4 + a_3*x^3 + a_2*x^2 + a_1*x + a_0
Using a different latex name since 'lambda' is reserved::
sage: l = SymbolSequence('l', '\lambda')
sage: l[0]
l_0
sage: latex(l[0])
\lambda_{0}
Using multiple indices::
sage: a = SymbolSequence('a')
sage: a[0,1,2]
a_0_1_2
sage: latex(a[0,1,2])
a_{0}_{1}_{2}
sage: [ a[i,j] for i in range(0,2) for j in range(0,2) ]
[a_0_0, a_0_1, a_1_0, a_1_1]
You can pass slices instead of integers to obtain a list of
symbols::
sage: a = SymbolSequence('a')
sage: a[5:7]
[a_5, a_6]
This even works for the second, third, etc. indices::
sage: a = SymbolSequence('a')
sage: a[0:2, 0:2]
[a_0_0, a_0_1, a_1_0, a_1_1]
TESTS:
We shouldn't overwrite variables in the global namespace::
sage: a = SymbolSequence('a')
sage: a_0 = 4
sage: a[0]
a_0
sage: a_0
4
The symbol at a given index should always be the same, even when
the symbols themselves are unnamed. We store the string
representation and compare because the output is unpredictable::
sage: a = SymbolSequence()
sage: a0str = str(a[0])
sage: str(a[0]) == a0str
True
Slices and single indices work when combined::
sage: a = SymbolSequence('a')
sage: a[3, 0:2]
[a_3_0, a_3_1]
sage: a[0:2, 3]
[a_0_3, a_1_3]
"""
def __init__(self, name=None, latex_name=None, domain=None):
# We store a dict of already-created symbols so that we don't
# recreate a symbol which already exists. This is especially
# helpful when using unnamed variables, if you want e.g. a[0]
# to return the same variable each time.
self._symbols = {}
self._name = name
self._latex_name = latex_name
self._domain = domain
def _create_symbol_(self, subscript):
"""
Return a symbol with the given subscript. Creates the
appropriate name and latex_name before delegating to
SR.symbol().
EXAMPLES::
sage: a = SymbolSequence('a', 'alpha', 'real')
sage: a_1 = a._create_symbol_(1)
sage: a_1
a_1
sage: latex(a_1)
alpha_{1}
"""
# Allow creating unnamed symbols, for consistency with
# SR.symbol().
name = None
if self._name is not None:
name = '%s_%d' % (self._name, subscript)
latex_name = None
if self._latex_name is not None:
latex_name = r'%s_{%d}' % (self._latex_name, subscript)
return SR.symbol(name, latex_name, self._domain)
def _flatten_list_(self, l):
"""
Recursively flatten the given list, allowing for some elements
to be non-iterable. This is slow, but also works, which is
more than can be said about some of the snappier solutions of
lore.
EXAMPLES::
sage: a = SymbolSequence('a')
sage: a._flatten_list_([1,2,3])
[1, 2, 3]
sage: a._flatten_list_([1,[2,3]])
[1, 2, 3]
sage: a._flatten_list_([1,[2,[3]]])
[1, 2, 3]
sage: a._flatten_list_([[[[[1,[2,[3]]]]]]])
[1, 2, 3]
"""
result = []
for item in l:
if isinstance(item, list):
result += self._flatten_list_(item)
else:
result += [item]
return result
def __getitem__(self, key):
"""
This handles individual integer arguments, slices, and
tuples. It just hands off the real work to
self._subscript_foo_().
EXAMPLES:
An integer argument::
sage: a = SymbolSequence('a')
sage: a.__getitem__(1)
a_1
A tuple argument::
sage: a = SymbolSequence('a')
sage: a.__getitem__((1,2))
a_1_2
A slice argument::
sage: a = SymbolSequence('a')
sage: a.__getitem__(slice(1,4))
[a_1, a_2, a_3]
"""
if isinstance(key, tuple):
return self._subscript_tuple_(key)
if isinstance(key, slice):
return self._subscript_slice_(key)
# This is the most common case so it would make sense to have
# this test first. But there are too many different "integer"
# classes that you have to check for.
return self._subscript_integer_(key)
def _subscript_integer_(self, n):
"""
The subscript is a single integer, or something that acts like
one.
EXAMPLES::
sage: a = SymbolSequence('a')
sage: a._subscript_integer_(123)
a_123
"""
if n < 0:
# Cowardly refuse to create a variable named "a-1".
raise IndexError('Indices must be nonnegative')
try:
return self._symbols[n]
except KeyError:
self._symbols[n] = self._create_symbol_(n)
return self._symbols[n]
def _subscript_slice_(self, s):
"""
We were given a slice. Clean up some of its properties
first. The start/step are default for lists. We make
copies of these because they're read-only.
EXAMPLES::
sage: a = SymbolSequence('a')
sage: a._subscript_slice_(slice(1,3))
[a_1, a_2]
"""
(start, step) = (s.start, s.step)
if start is None:
start = 0
if s.stop is None:
# Would otherwise loop forever since our "length" is
# undefined.
raise ValueError('You must supply an terminal index')
if step is None:
step = 1
# If the user asks for a slice, we'll be returning a list
# of symbols.
return [ self._subscript_integer_(idx)
for idx in range(start, s.stop, step) ]
def _subscript_tuple_(self, args):
"""
When we have more than one level of subscripts, we pick off
the first one and generate the rest recursively.
EXAMPLES:
A simple two-tuple::
sage: a = SymbolSequence('a')
sage: a._subscript_tuple_((1,8))
a_1_8
Nested tuples::
sage: a._subscript_tuple_(( (1,2), (3,(4,5,6)) ))
a_1_2_3_4_5_6
"""
# We never call this method without an argument.
key = args[0]
args = args[1:] # Peel off the first arg, which we've called 'key'
# We don't know the type of 'key', but __getitem__ will figure
# it out and dispatch properly.
v = self[key]
if len(args) == 0:
# There was only one element left in the tuple.
return v
# At this point, we know we were given at least a two-tuple.
# The symbols corresponding to the first entry are already
# computed, in 'v'. Here we recursively compute the symbols
# corresponding to the second coordinate, with the first
# coordinate(s) fixed.
if isinstance(key, slice):
ss = [ SymbolSequence(w._repr_(), w._latex_(), self._domain)
for w in v ]
# This might be nested...
maybe_nested_list = [ s._subscript_tuple_(args) for s in ss ]
return self._flatten_list_(maybe_nested_list)
else:
# If it's not a slice, it's an integer.
ss = SymbolSequence(v._repr_(), v._latex_(), self._domain)
return ss._subscript_tuple_(args)
