2. Consider the
which has a root at .
2 (a). Use the Newton-Raphson formula to find the root. Use the starting value
2 (b). Use Halley's formula to find the root. Use the starting value
Solution 2 (b).
Now we will investigate Halley's iteration for finding square roots.
Form the Halley iteration function h(x).
We start the iteration with and carry 100 digits in the computations, by telling Mathematica the precision of by issuing the command p = N[,100]. Next, a short program is written to compute the first five terms in the iteration:
Since the root is known to be exactly we can have Mathematica list the error at 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
We can conclude that Halley's method is faster than Newton's method.
Verify the convergence rate. At the simple root we can explore the ratio .
Therefore, the Halley iteration is converging cubically.
(c) John H. Mathews 2004