Module for Newton's Method
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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
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
will converge to for any initial approximation .
Animations (Newton's Method Newton's Method). Internet hyperlinks to animations.
find a root of given
an initial approximation using
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
After you have debugged you program and it is working properly, delete the unnecessary print statements
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
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