Halley's Method Example

Example(b).

Consider the function, which has a root at.
1 (b).Use Halley's formula to find the root.Use the starting value

Solution 1 (b).

Now we will investigate Halley's iteration for finding square roots.

Form the Halley iteration functionh(x).

We start the iteration withand carry 100 digits in the computations, by telling Mathematica the precision of by issuing the command p[0] = N[2,100].Next, a short program is written to compute the first five terms in the iteration:

Since the root is known to be exactlywe can have Mathematica list the errorat each step in the iteration:

Looking at the error, we see that the number of accurate digits is tripling at each step in the computations, hence convergence is proceeding cubically.
We can conclude that Halley's method is faster than Newton's method.

Verify the convergence rate.At the simple rootwe can explore the ratio .

k

Therefore, the Halley iteration is converging cubically.