What are symbolic computations?

All computations we did so far in this book were so-called numeric computations. These were a sequence of operations mainly on floating-point numbers. It is the nature of numeric computations that the result is an approximation of the exact solution.

Symbolic computations operate on formulas or symbols by transforming them as taught in algebra or calculus into other formulas. The last step of these transformations might then require that numbers are inserted and a numeric evaluation is performed.

We illustrate the difference by computing this definite integral:

Symbolically this expression can be transformed by considering the primitive function of the integrand:

We now obtain a formula for the definite integral by inserting the integral bounds:

This is called a closed-form expression for the integral. Very few mathematical problems have a solution that can be given in a closed-form expression. It is the exact value of the integral without any approximation...