mattbrepo/SymPyTest

Test of SymPy package

SymPyTest

Test of SymPy package.

Language: Python

Start: 2024

Why

I wanted to try SymPy which is a Python package for symbolic mathematics.

A few tests

Check the equality of two expressions:

$$ \ln(1 + \exp(a) + \exp(b) + \exp(c) + 1) = \ln(\exp(b - a) + \exp(c - a) + \frac{1}{\exp(a)} + 1) + a $$

expr1 = sp.log(sp.exp(a) + sp.exp(b) + sp.exp(c) + 1)
expr2 = sp.log(sp.exp(b - a) + sp.exp(c - a) + 1/sp.exp(a) + 1) + a
print(sp.simplify(expr1-expr2) == 0)

True

Numeric evaluation:

expr1_num = expr1.subs({a: 700, b: 600, c: 10})
expr1_num.evalf()

700.0

Unit conversion:

expr_u = (5 * spu.cm + 3 * spu.m).simplify()
spu.convert_to(expr_u, spu.m).n()

3.05m

Partial derivatives

I wrote a code to calculate the partial derivatives for all the variables of a function. This code can be helpful to determine the gradient of a function.

Example, given the function:

$$ f = \left(- l + \sqrt{\left(x_{1} - x_{2}\right)^{2} + \left(y_{1} - y_{2}\right)^{2} + \left(z_{1} - z_{2}\right)^{2}}\right)^{2} $$

where l is a constant and x1, x2, y1, y2, z1, z2 are the variables, the corresponding gradient is:

$$ \nabla f = \left[ \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \frac{\partial f}{\partial y_1}, \frac{\partial f}{\partial y_2}, \frac{\partial f}{\partial z_1}, \frac{\partial f}{\partial z_2} \right] $$

and the first of the 6 partial derivatives (x1) is:

$$ \frac{\partial f}{\partial x_1} = \frac{2 \left(- l + \sqrt{\left(x_{1} - x_{2}\right)^{2} + \left(y_{1} - y_{2}\right)^{2} + \left(z_{1} - z_{2}\right)^{2}}\right) \left(x_{1} - x_{2}\right)}{\sqrt{\left(x_{1} - x_{2}\right)^{2} + \left(y_{1} - y_{2}\right)^{2} + \left(z_{1} - z_{2}\right)^{2}}} $$

Resources

A playlist of introductory videos.