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 :doc:`custom types ` have no equivalent in SymPy. 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. .. code:: python >>> 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. Further examples ---------------- The `cart-pole `_ example uses SymPy to help find the Euler-Lagrange equations of a cart-pole system featuring a double-pendulum.