Example 1.  Use the value    and compare Methods A,B and C for finding the double root    of the equation  .

Solution 1.

First, we will compute the iterations for each method, and afterward a table comparing the methods is given.  For the method C, all the iterations in the linear search are included.  

Using formula (2), the standard Newton-Raphson method.  

Using formula (9) Method A, the accelerated Newton-Raphson method with  m=2.  

Using formula (11) Method B, the modified Newton-Raphson method.

Using formula (13) Method C, the adaptive Newton-Raphson method.

The details for obtaining   are:

Since    we reject    and set    and continue the iteration using formula (13).  The subroutine makes all these choices automatically.
        

Summary of the above results.

    Observe in this example that the standard Newton-Raphson method converges linearly and methods A and B converge quadratically.  The reader can use formulas (12) to verify that is the order of the root .  The new method C is almost as fast as methods A and B.

Reminder.  The goal of this investigation is to show how the adaptive Newton-Raphson method is superior to the standard Newton-Raphson method, because of the limitations of Methods A and B.

Caveat.  Why is it difficult to locate a multiple root.  Because the function values themselves are essentially "noise" when you get close to a multiple root.  

One way to tell is to graph their difference.  

The formula    should be considered the "true value."
So one should be suspect of the computation    for values of  x  is near  x=1.

(c) John H. Mathews 2004