Numerical Methods

## Numerical solution of equations

Let us consider a real-valued function $F(x)$. We seek the real values of $x$ for which:

$$\label{gl1} F(x)=0.$$

The values of $x$ that obey this relation are called solutions, zeroes, or roots of (\ref{gl1}).

In literature, one can find many specialized methods such as the method of Lobatschewski and Graeffe [Poloshi,1963] to determine numerically the zeroes of algebraic equations (polynomials) equations of the type:

$F(x)\equiv P_{m}(x)=\sum_{j=1}^{m} \alpha_{j} x^{j-1} = 0.$

These specialized methods will not be treated in this lecture.

Here we will consider general methods that solve any function of the form $F(x)=0$ including transcendental equations. In this lecture we will discuss the following:

• Graphical solutions
• Iterative methods (Newton-Raphson method; Regula Falsi)
• The bissection method
• The numerical treatment of non-linear systems of equations

### Graphical solutions

An equation of the form $F(x)=0$ can be solved graphically simply by plotting it. The solutions are the points where the plot intersects the horizontal $y=0$ axis. By zooming in to the intersection, the solutions can be determined with reasonable accuracy.

 $F(x)$ $x$
 $F(x)=$

from $x=$  to $x=$