Example
1. Consider the
function ,
which has a root at .
1
(a). Use the
Newton-Raphson formula to find the
root. Use the starting
value
Solution 1 (a).
Form the Newton-Raphson iteration function g(x).
We start the iteration with and 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 seven 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 doubling at each step in the computations, hence convergence is proceeding quadratically.
Verify the convergence rate. At the simple root we can explore the ratio .
k |
|||
Therefore, the Newton-Raphson iteration is converging quadratically.
(c) John H. Mathews 2004