## Playing with SymPy

I have been playing with SymPy lately. SymPy is, basically, a Python library for Symbolic Mathematics. It’s precisely the kind of CAS I was looking for: simple, lightweight, easy to use, syntactically tasteful, open-source, and free. SymPy just makes sense. Simplicity über alles.

SymPy was started in 2005, by Ondřej Čertík, a Physics student from Czech Republic. Kudos to Ondřej for his initiative! And thanks to all of those who contributed to the SymPy project! Here are some slides on SymPy, by Ondřej:

You can download SymPy from here. If you don’t have Python installed in your computer, you can play with the SymPy online shell (which runs on the Google App Engine).

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An example

Let’s consider the system of polynomial equations in $\mathbb{R}[x_1, x_2, x_3]$

$\begin{array}{rl}x_1 + x_2 + x_3 = y_1 + y_2 + y_3 \\x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2\\x_1^3 + x_2^3 + x_3^3 = y_1^3 + y_2^3 + y_3^3\end{array}$,

where $y = (y_1, y_2, y_3) \in \mathbb{R}^3$ is a known vector (whose entries are distinct). We will denote $x = (x_1, x_2, x_3)$. Let function $g_k: \mathbb{R}^3 \rightarrow \mathbb{R}$ be defined as

$g_k(z) = \displaystyle\sum_{i=1}^3 z_i^k$.

The system of polynomial equations can thus be rewritten as

$\begin{array}{rl}p_1(x) = 0\\p_2(x) = 0\\p_3(x) = 0\end{array}$

where $p_k(x) = g_k(x) - g_k(y)$ for all $k \in \{1,2,3\}$. According to my conjecture, this system of equations should have exactly $3! = 6$ solutions. Solving the system is not very easy because the equations $p_k(x) = 0$ are nonlinear in variables $x_1, x_2, x_3$. A CAS would be useful.

For example, we could implement function $g_k: \mathbb{R}^3 \rightarrow \mathbb{R}$ with the following Python script:

def g(z,k):
"""Computes g_k(z), where z is a list of reals and k is a positive integer"""

# checks function arguments for errors
if len(z)==0:
return "ERROR: First argument must be a non-empty list of symbols!"
if (type(k) != int) or (k < 1):
return "ERROR: Second argument must be a positive integer!"

# computes g_k(z) and returns it
acc = 0
for i in range(0,len(z)):
acc += (z[i])**k
return acc



We could then compute the Gröbner basis of the set of polynomials $\{p_1(x), p_2(x), p_3(x)\}$, and the solutions of the system of polynomial equations for a given $y$, say, $y = (1,2,3)$:

from sympy import *

# number of equations and symbolic variables
n = 3

# defines set S = {1, 2,..., n}
S = range(1,n+1)

# declares symbolic variables
x = [Symbol('x%d' % i) for i in S]

# initializes y vector
#y = [Symbol('y%d' % i) for i in S]
y = S

# defines system of polynomial equations in variables [x1, x2, x3]
P = [g(x,k) - g(y,k) for k in S]

# computes Groebner basis and solutions of the system of polynomials
GB = groebner(P, order='lex')
Sols = solve_system(P)

# prints results
print "Groebner basis: %s" % GB
print "Solutions: %s" % Sols
print "There are %d solutions." % len(Sols)



The output is:

• Gröbner basis: the Gröbner basis is a set of three polynomials $\{b_1(x), b_2(x), b_3(x)\}$, where

$\begin{array}{l} b_1(x) = -6 + x_1 + x_2 + x_3\\ b_2(x) = 11 - 6 x_2 - 6 x_3 + x_2 x_3 + x_2^2 + x_3^2\\ b_3(x) = -6 + 11 x_3 - 6 x_3^2 + x_3^3\end{array}$.

• Solutions: there are $3! = 6$ solutions

$\{(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)\}$.

As expected, the solutions of the system of polynomial equations are the $3! = 6$ permutations of the elements of $y = (1,2,3)$. Of course, this does not prove the conjecture. All it proves is that my conjecture works for the particular case where $n=3$ and $y = (1,2,3)$.

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Related:

### 4 Responses to “Playing with SymPy”

1. Alex Says:

I’ve used SAGE for a total of 5 minutes, but I do have a strong opinion on it. It’s difficult for me to believe that you can sucessfully paste the illusion of a uniform interface over several CASes disparate in goals, philosophies, and abilities. Maybe SAGE has done that, but I also have the more personal issue that if I were to use it, I’d like to be proficient in using the various backends it calls. It just wouldn’t feel right otherwise :)

You could try JACAL or MockMMA for short nontrivial (but probably confusingly LISPy) CAS codes; I think you’d like the feel of SymPy, and I bet the code’s readable; Maxima for a more industrial sized codebase; or wait forever and maybe the Axiom people will get around to documenting their code (I personally can’t wait for that).

Either way, you’ve got to learn a lot of hardcore math (boring boring algebra) to understand their algorithms.

2. Alex Says:

Maple looks ugly, Mathematica is syntactically annoying … and those are the only two worth considering for general use. I settled on Mathematica– next fall I’ll actually be co-teaching a course on it, so over the summer I must become some kind of expert :)

My hope is reserved for Axiom. The core is written in LISP with most algorithms implemented in the system’s extension language. I’m not sure about the syntax of the system; I haven’t seriously used it ever, and my dabblings were long ago, but the idea of a category theoretical CAS seems very powerful. On top of that, I messed around with Aldor (the latest of its extension languages), and I found it fun. Hopefully they’ll get around to documenting the code so people can go in and rip off the algorithms one by one and incorporate them into their own CASes.

That’s probably the ideal situation: having a compendium of code that groups can use to roll their own CASes that fit their own design specifications.

3. Ondrej Says:

Nice to hear that SymPy is useful. I’ve added your post to:

http://docs.sympy.org/outreach.html#blogs-news-magazines

and also cited it here:

If you have any suggestions for improvements, please let us know on our mailinglist. We are also looking for new users and developers if anyone is interested. :)

4. Rod Carvalho Says:

If you know Python, you’ll probably think that in the piece of code that implements function $g_k(\cdot)$, the following lines

# computes g_k(z) and returns it
acc = 0
for i in range(0,len(z)):
acc += (z[i])**k
return acc


don’t look very “Pythonic” (at all). I know, I know. These lines of code look as though they were translated from C or C++ to Python (I even use the “+=” operator). This is kind of “plain vanilla”.

It would be much cooler to replace the lines above with a single line of code that does it all:

# computes g_k(z) and returns it
return sum([elem**k for elem in z])


This would work if the function’s first argument were a list of reals. However, if it’s a list of symbols, then function sum() does not work. I suppose the problem is that sum() uses an operator which does not work on symbols. Operator overloading would be necessary.