An Improved Newton's Method

Example.

Use the valueand compare Methods A,B and C for finding the double rootof the equation.

Solution.

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 withm=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:

Sincewe rejectand setand 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.

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.

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 formulashould be considered the "true value."
So one should be suspect of the computationfor values ofxis nearx=1.