Substitution in polynomials
christi...@googlemail.com
First thank you very much for your work on sympy it is a really
usefull program.
I want to substitute variables in a polynomial, something similar
to what is possible with matrices:
---
>>> M
x 0 0
0 x 0
0 0 x
>>> M.subs(x, 4)
4 0 0
0 4 0
0 0 4
---
Suppose you have a polynomial q(x,y,z) and i want to substitute the
x,y,z
with other expressions, namely other polynomials. Suppose all used
variables
are already declared as symbols.
q = xy + y**2 + zx
x = x1 + x2**2
y = y1 + y3
z = z2*z3
now i want q to become
q = (x1 + x2**2)*(y1 + y3) + (y1 + y3)**2 + (x1 + x2**2)*(z2*z3)
I know i can achive this by declairing x,y,z first but that is not
possible
in this case.
How can I do it? :)
Thank you very much and greetings
Christian Brumm
Ondrej Certik
> First thank you very much for your work on sympy it is a really
> usefull program.
Nice to hear that. :)
How about this:
In [1]: q=x*y+y**2+z*x
In [2]: q
Out[2]:
2
y + x*y + x*z
In [3]: x1, x2, y1, y3, z2, z3 = symbols("abcdef")
In [4]: x1, x2, y1, y3, z2, z3
Out[4]: (a, b, c, d, e, f)
In [5]: x0 = x1+x2**2
In [6]: y0 = y1+y3
In [7]: z0 = z2*z3
In [8]: q.subs(x, x0).subs(y, y0).subs(z, z0)
Out[8]:
2 ⎛ 2⎞ ⎛ 2⎞
(c + d) + ⎝a + b ⎠*(c + d) + e*f*⎝a + b ⎠
Does it solve the problem? Or are you looking for something else?
Ondrej
Fredrik Johansson
Hi Christian,
You have to be careful to distinguish between Python variables and
SymPy symbols.
I would suggest naming the expressions
x_ = x1 + x2**2
y_ = y1 + y3
z_ = z2*z3
and doing
q.subs(x, x_).subs(y, y_).subs(z, z_).
Alternatively, you could do it like this:
x = x1 + x2**2
y = y1 + y3
z = z2*z3
x_ = Symbol('x')
y_ = Symbol('y')
z_ = Symbol('z')
q.subs(x_, x).subs(y_, y).subs(z_, z)
but I think it's generally better to name the variable referring to a
symbol exactly the same thing as the symbol.
This is a rather subtle issue in SymPy and we probably need more
documentation about it.
Fredrik
christi...@googlemail.com
Fredrik was right, i was a bit confused with variables and symbols.
subs() worked fine for me with a new set of symbols.
However things start to get messy when you need hundreds of variables
and polynomials.
For example if you want to initalize variables x1 ... x100 (with
symbols),
i did it with the following rather "dirty" and ineffizient code
(i guess exec causes the process to fork?):
---
vars = ['x'+str(i) for i in range(1,101)]
for i, value in enumerate(vars):
exec("x%d = Symbol(value)" %(i+1))
---
I used code similar to the above to perform substitutions on hundreds
of
variables, but all becomes pretty messy and unreadable. (I want to
implement
a cryptosystem that relies on multivariate polynomials.)
Is there a better way to handle these larger scales? Or do I need
something
like Maple or Mathematica here? I would really love to do it in Python
Greeting,
Christian
On 4 Nov., 20:08, "Fredrik Johansson" <fredrik.johans...@gmail.com>
wrote:
> On 11/4/07, christian.br...@googlemail.com
basti
> For example if you want to initalize variables x1 ... x100 (with
> symbols),
> i did it with the following rather "dirty" and ineffizient code
> (i guess exec causes the process to fork?):
> ---
> vars = ['x'+str(i) for i in range(1,101)]
> for i, value in enumerate(vars):
> exec("x%d = Symbol(value)" %(i+1))
> ---
why you don't just use:
vars = (Symbol('x'+str(i) for i in range(1,101))
you then con access them easiely through vars[i]
basti
Fredrik Johansson
<christi...@googlemail.com> wrote:
>
> Hallo Ondrej & Fredrik, thanks a lot for your answers.
>
> Fredrik was right, i was a bit confused with variables and symbols.
> subs() worked fine for me with a new set of symbols.
>
> However things start to get messy when you need hundreds of variables
> and polynomials.
>
> For example if you want to initalize variables x1 ... x100 (with
> symbols),
> i did it with the following rather "dirty" and ineffizient code
> (i guess exec causes the process to fork?):
> ---
> vars = ['x'+str(i) for i in range(1,101)]
> for i, value in enumerate(vars):
> exec("x%d = Symbol(value)" %(i+1))
> ---
>
> I used code similar to the above to perform substitutions on hundreds
> of
> variables, but all becomes pretty messy and unreadable. (I want to
> implement
> a cryptosystem that relies on multivariate polynomials.)
>
> Is there a better way to handle these larger scales? Or do I need
> something
> like Maple or Mathematica here? I would really love to do it in Python
Why do you need separate named variables? It seems natural to index
the symbols in a list or dict:
x = dict([(i, Symbol('x' + str(i))) for i in range(1, 101)])
gives
x[1] --> Symbol('x1')
x[2] --> Symbol('x2')
etc.
By the way, the exec statement just compiles and interprets the given
code like an ordinary statement. As far as I know, it does not fork
the process.
Fredrik
christi...@googlemail.com
I do not know what I was thinking.
Greetings
Christian
On Nov 5, 12:42 pm, "Fredrik Johansson" <fredrik.johans...@gmail.com>
wrote:
> On 11/5/07, christian.br...@googlemail.com