Example 2.  Consider the function  , 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 (a).

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Form the Newton-Raphson iteration function  g(x).

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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[,100].  Next, a short program is written to compute the first seven terms in the iteration:

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Since the root is known to be exactly     we can have Mathematica list the error    at each step in the iteration:

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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 .

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Therefore, the Newton-Raphson iteration is converging quadratically.

(c) John H. Mathews 2004