Root Finding Methods

There are not many general tools available to determine in advance whether a root-finding problem can be solved. For our purposes, the main goal is to obtain information about the existence of a root, and if a root does exist, it is important to estimate an interval in which it lies.

To locate such a root, two main methods can be used:

1. Graphical Method

One of the first attempts is to plot the graph of the function. Indeed, if the goal is to solve \( f(x) = 0 \) and the graph of \( f \) can be drawn, then the intersection of this graph with the x-axis (axis of \( x \)) gives an approximate location of a root.

There is nothing wrong with using this method, but it can be difficult to apply when the graph is complex. Sometimes, it is possible to write \( f = f_1 - f_2 \), where \( f_1 \) and \( f_2 \) are simple functions. Then, we look for the intersection points of their graphs, the x-coordinates of which are exactly the roots of \( f \).

Example 3

Consider the function defined by:

\[ f(x) = \frac{\log x - 1}{x}, \quad x \in \left] 0, +\infty \right[ \]

This function has a unique root in the interval \( \left] \frac{3}{2}, 2 \right[ \). Let \( f_1(x) = \log x \) and \( f_2(x) = 1 \), then solving \( f(x) = 0 \) is equivalent to \( f_1(x) = f_2(x) \).

The intersection point of the graphs of \( f_1 \) and \( f_2 \) corresponds to the desired root.

2. Intermediate Value Theorem Method

Another method is to verify that the function \( f \) is continuous on an interval \( [a, b] \). If two points \( a \) and \( b \) can be found such that:

\[ f(a) > 0 \quad \text{and} \quad f(b) < 0 \]

Then, by the Intermediate Value Theorem, there exists a point \( c \in (a, b) \) such that \( f(c) = 0 \).

Finding such points \( a \) and \( b \) may require intuition, trial and error, reasoning, or graphical tools.

Example

Consider the following polynomial:

\[ p(x) = x^4 - x^3 - x^2 + 7x - 10 \]

Here are the values of the polynomial for various \( x \):

The table shows that the polynomial \( p \) changes sign between:

• \( x = -2 \) and \( x = -1 \) → a root in \( ]-2, -1[ \)
• \( x = 1 \) and \( x = 2 \) → a root in \( ]1, 2[ \)

Numerical Methods

In the following sections, we will present several numerical solution methods. All of these methods are iterative, meaning they start from an initial estimate and produce a sequence of approximations that is supposed to converge to the solution.

In some cases, this sequence converges to a limit. It must then be verified whether this limit is indeed a solution of the equation \( f(x) = 0 \). Another important question is the speed of convergence of the method.