AEF.NENS Numerical Estimation Newton Secant
In many cases representing the exact value of a solution to an equation  as a decimal  can be very difficult or even impossible. In such cases techniques that provide a numerical estimate of the solution to some precision is very useful. Mapping diagrams can be used often to visualize these techniques.
Since many examples involve elementary functions that are continuous for an interval of the form $[a,b]$, one basis for estimating solutions is the result:

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  $f(c) = 0$.

An immediate consequence of the IVT is that if $f(a)\cdot f(b) < 0$, then there is a number $c \in [a,b]$ where $f(c) = 0$.

The Newton Secant Method
The Newton secant method does not repeatedly apply the IVT to estimate a solution to an equation with an elementary function, $f$, of the form $f(x)=0$ when $f(a)\cdot f(b) < 0$ for the initial numbers $a$ and $b$.
Instead it replaces $a_n$ with $b_n$ and replaces $b_n$ with $s_n$, the root (zero) of the linear function determined by the function values at the numbers $a_n$ and $b_n$ . In other words, assignments are successively made, $a_{n+1}=b_n$ and $b_{n+1}=s_n$ where $s_n = a_n-\frac{f(a_n)}{(f(b_n)-f(a_n))/(b_n-a_n)}= \frac {a_n f(b_n)-b_n f(a_n)}{f(b_n)-f(a_n)}$.

The procedure is as follows: Start with $a_1=a$ and $b_1=b$.
Let $a_{n+1}=b_n$ and $b_{n+1}=s_n$ where $s_n = a_n-\frac{f(a_n)}{(f(b_n)-f(a_n))/(b_n-a_n)}= \frac {a_n f(b_n)-b_n f(a_n)}{f(b_n)-f(a_n)}$.
Unfortunately there is no guarantee that at each step the numbers $a_n$  and $b_n$ is closer to each other or contain a solution between them.
So the solution guaranteed by the IVT to exist in the first interval $[a,b]$ may not be in some or any of the following intervals.
When this method does improve the estimates, it is often more efficient than either bisection or false position, but one needs to be aware of the possibility that later values of $s_n$ need not be good estimates for a solution.
The mapping diagram below illustrates the example where $f(x) = x^2-2, a=0,$ and $b=2$.
The Newton Secant 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 $a_n, b_n, SecP_n$ 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