9. SymPy Interoperability

wrenfold is not a full computer algebra system. Instead, we aim to support a limited set of common operations required for applications like robotics, computer vision, numerical optimization, and machine learning. In order to help supplement missing functionality, wrenfold expressions can be converted to-and-from SymPy.

There are a some limitations:

• SymPy manipulations may come with a significant performance penalty, since they all occur in python.

• Not all expressions have an equivalent (in either direction of conversion). For example, expressions involving custom types have no equivalent in SymPy.

9.1. Example: Computing eigenvalues

As a motivating example, suppose we need an expression for the eigenvalues of a 3x3 matrix. There is no function for this in wrenfold, but SymPy features an eigenvals() method that can return a closed form expression.

>>> import sympy as sp
>>> from wrenfold import sym
>>> from wrenfold import sympy_conversion

>>> m = sym.matrix_of_symbols('m', 3, 3)
>>> m
[[m_0_0, m_0_1, m_0_2], [m_1_0, m_1_1, m_1_2], [m_2_0, m_2_1, m_2_2]]

>>> m_sp = sympy_conversion.to_sympy(m, sp=sp) # Convert matrix to SymPy.
>>> m_sp
Matrix([
[m_0_0, m_0_1, m_0_2],
[m_1_0, m_1_1, m_1_2],
[m_2_0, m_2_1, m_2_2]])

>>> ev_sp = m_sp.eigenvals(multiple=True)
>>> sympy_conversion.from_sympy(ev_sp[0], sp=sp) # Convert eigenvalues back to wrenfold.
m_0_0/3 + m_1_1/3 + m_2_2/3 - (3*m_0_1*m_1_0 - 3*m_0_0*m_1_1 + 3*m_0_2*m_2_0 + ...

Given the eigenvalue expressions, we can now substitute the variables in m with our choice of expressions.