**Module
for Newton's Method**

**Check
out the new Numerical Analysis Projects page.**

If
are continuous near a root ,
then this extra information regarding the nature of
can be used to develop algorithms that will produce sequences
that converge faster to
than either the bisection or false position method. The
Newton-Raphson (or simply Newton's) method is one of the most useful
and best known algorithms that relies on the continuity of . The
method is attributed to Sir
Isaac Newton (1643-1727) and Joseph
Raphson (1648-1715).

**Theorem (****Newton-Raphson
Theorem****).** Assume
that
and there exists a number ,
where . If ,
then there exists a
such that the sequence
defined by the iteration

for

will converge to
for any initial approximation .

**Animations (****Newton's
Method** **Newton's
Method****).** Internet
hyperlinks to animations.

**Algorithm
(****Newton-Raphson
Iteration****).** To
find a root of given
an initial approximation using
the iteration

for .

**Mathematica Subroutine (Newton-Raphson
Iteration).**

**Example 1.** Use
Newton's method to find the three roots of the cubic
polynomial . Determine
the Newton-Raphson iteration formula that
is used. Show details of the computations for the starting
value .

**Reduce the volume of
printout.
**After you have debugged you program and it is working
properly, delete the unnecessary print statements

and

**Concise
Program for the Newton-Raphson Method**

Now test the example to see if it still works. Use the last case in Example 1 given above and compare with the previous results.

**Error
Checking in the Newton-Raphson Method**

Error checking can be added to the Newton-Raphson method. Here we have added a third parameter to the subroutine which estimate the accuracy of the numerical solution.

The following subroutine call uses a maximum of 20 iterations, just to make sure enough iterations are performed. However, it will terminate when the difference between consecutive iterations is less than . By interrogating k afterward we can see how many iterations were actually performed.

**Old Lab Project (****Newton's
Method** **Newton's
Method****).** Internet
hyperlinks to an old lab project.

**Research Experience for
Undergraduates**

**Newton-Raphson
Method** **Newton-Raphson
Method** Internet hyperlinks to web sites and
a bibliography of articles.

**Downloads (****Newton's
Method**** ** **Newton's
Method****).** Download
this Mathematica notebook.

(c) John H. Mathews 2003