AEF.NEC Numerical Estimation Comparison ("The More, The Merrier")
In this subsection we compare the three previously discussed techniques for making numerical estimates for solutions to an equation using mapping diagrams to visualize these techniques. Each method starts with the assumption that $f$ is a continuous function on the interval $[a,b]$ and $0$ is between $f(a)$ and $f(b)$, which allows the use of the following theorem to guarantee a solution to the equation $f(x)=0$, i.e., a number $c \in [a,b]$ where $f(c) = 0$

Theorem: CCD.IVT.0: If $f$ is a continuous function on the interval $[a,b]$ and $0$ is between $f(a)$ and $f(b)$, then there is a number $c \in [a,b]$  where  $f(c) = 0$.

There are three main issues with numerical methods once there is a guarantee that there is a number to be estimated:
1. Is the method easy to follow? Is the method pactical? Of the three methods, only the Newton secant method can be impractical because it has the potential to fail by enountering a loop where values for estimated repeat or there is a possibility that $f(a_n) = f(b_n) \ne 0$.
2. Will the method assure better estimates for the desired solution? Is the method effective? Again- Of the three methods, only the Newton secant method can be ineffective because it has the potential to fail by enountering a loop where values for estimated repeat or there is a possibility that $f(a_n) = f(b_n) \ne 0$ or fall outside the inital interval where a solution is guaranteed.
3. How quickly will the method arrive at a desired level of accuracy? Is the method efficient? Despite the risks, in practice the Newton secant method is often the most efficient, with false position also better than the slow but steady bisection method

Comparison of Methods
This example allows you to change the method being applied at any stage for comparison of the effectiveness and efficiency of each method:
• bisection,
• false position, or
• Newton secant.
To change the method being applied, move the position of the slider labeled "Method."

Notice how the arrows on the mapping diagram are paired with the point on the graph of the function $f$.
You can move the point for $x$ on the mapping diagram to see how the function value changes both on the diagram and on the graph.

Sliders: The sliders labeled $a$ and $b$ control the end values of the initial interval $[a,b]$.
The slider labeled $n$ controls which the number of times the Newton secant method is executed.
Checking the box to "Show/Hide Secant Point Position & Data" will display $a_n, b_n, P_n$ and the values of $f$ for each of these numbers.
It will also display a second box to "Show/Hide  Smallest Exact Root". Checking this box will display this root on both the graph and the mapping Diagram.

The function $f$ can be changed by making an entry in the appropriately labeled input box.

Martin Flashman, August. 10, 2017. Created with GeoGebra