Example.

Consider the function, which has a root at.
1 (a).Use the Newton-Raphson formula to find the root.Use the starting value

Solution (a).

Form the Newton-Raphson iteration functiong(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 seven 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 doubling at each step in the computations, hence convergence is proceeding quadratically.

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

k

Therefore, the Newton-Raphson iteration is converging quadratically.