Symbolic Computation

You may also know computer algorithms as numerical algorithms, since the computer uses finite-precision floating-point numbers to approximate the result. Although the numerical result can be arbitrarily precise by using arbitrary-precision arithmetic, it is still an approximation.

As humans, we use symbols to carry out computations analytically. This can also be realized on computers using symbolic computation. There are many so-called computer algebra systems (CAS) out there and you might already be familiar with one from school if you have used a graphing calculator there like the ones shown below.

But of course CAS is not limited to external devices but also available as software for normal computers. The most commonly used computer algebra systems are Mathematica and SageMath/SymPy. We will use Sympy in this course because it is open-source and a Python library compared to Mathematica. SageMath is based on Sympy and offers a number of additional functionalities, which we do not need in this class.

To illustrate the difference between numerical and symbolic computations, we compare the output of the square root functions from numpy and sympy. The code

import numpy as np
print(np.sqrt(8))

produces 2.8284271247461903 as output, which is an approximation to the true value of (\sqrt{8}), while

import sympy as sp
print(sp.sqrt(8))

produces 2*sqrt(2), which is exact.