Newton Raphson

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The Newton-Raphson Method for Root Finding

Unlike the Bisection and False Position methods, the Newton-Raphson method is an open method.  It does not bracket the root and as a result does not provide a guarantee of convergence.  However, by including information about the derivative of the function, it tends to converge much faster than either of the other two methods.  When the approximation is sufficiently close to a root, the Newton-Raphson method doubles the number of significant digits in every iteration!

The method requires only a single starting point (not a bracket): the initial guess, . Subsequent approximations for the root are obtained by finding the root of the first-order Taylor series approximation at the current guess.  The first-order Taylor series approximation corresponds to the tangent line. The approach is summarized in the Newton-Raphson Formula:

As for all root-finding methods, the relative approximate error can be used as a termination criterion:

Note: For practical reasons, the number of iterations in the interactive window below is limited to 8.

One can show that when approaching the true root, the true absolute error is approximately proportional to the square of the true absolute error in the previous iteration:

This corresponds to a doubling of the number of significant digits in every iteration. 

For additional details about the Newton Raphson Method, review "Chapter 6: Open Methods" in Numerical Methods for Engineers by Chapra and Canale. A succinct on-line overview can be found at Wikipedia

Start Exploring!

In the interactive window below you can perform the following operations:

• Move a control point: click-and-drag a control point to a different position
• Add a control point: ctrl-click in the location where you want to add it
• Remove a control point: ctrl-click on the control point to be removed
• Retrieve a pre-programmed curve: click on one of the buttons S0 through S3
• Move the initial guess: click-and-drag the purple arrow at the bottom of the graph.
• Print this entire page: click the printer button at the top right of this web-page

Learn by Exploring

1. Using curve S0, move the initial guess through the entire interval from -1 to 2.  Does the Newton-Raphson method always converge?  If not, describe the scenarios in which it does and does not.  How are they different?
2. Can you create an example for which the Newton-Raphson method gets stuck, that is, it neither converges to nor diverges away from a root?  (note: we call such a situation a limit cycle -- the algorithm cycles for ever through the same sequence of guesses).  Describe the characteristics of the scenarios in which such limit cycles occur. 
3. Construct two clearly distinct examples of a limit cycle.
4. Can you create an example for which the Newton-Raphson method finds the exact root in the first iteration?  Is it possible to find a graph for which this is the case no matter which initial guess is chosen?