False Position

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The False Position Method for Root Finding

The False Position Method is a bracketing method.  A bracket is an interval  for which the function values at the bounds of the interval have opposite signs:

Given an initial bracket (provided by the user), the bisection method divides the bracket into two intervals and determines which of the two intervals is again a bracket, that is, has bounds with function values with opposite signs. Unlike the Bisection Method, however, the False Position Method does not divide the initial bracket into two intervals of equal size. Instead, it assumes that the function can be approximated by a straight line so that the root of the function can be well-approximated by the root of the straight line. This approximate root is the false position.  The false position, , is given by the following equation:

It is at the false position that the current bracket is divided into two sub-intervals. After determining which of the two intervals is again a bracket (i.e., has bounds with function values with opposite signs), the method repeats the same process in an iterative fashion until the size of the remaining bracket is sufficiently small (smaller than a desired absolute error).  Note: For practical reasons, the number of iterations in the interactive window below is limited to 8.

Unlike the Bisection method which reduces the error in half in every iteration, the False Position method has no guaranteed convergence rate. For some functions, it may converge faster than Bisection, for others it may converge significantly more slowly, as you will experience when experimenting with the applet below.

For additional details about the Bisection Method, review "Chapter 5: Bracketing 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
• Print this entire page: click the printer button at the top right of this web-page

Learn by Exploring

1. Create an example in which the false position algorithm converges very slowly to a root; preferably such that it does not even come near a root within the maximum of 8 iterations for the applet.
2. Does the false position algorithm always find a root when one exist?  If not, can you generate an example illustrating this?  If convergence is slow as you illustrated in the previous task, would the false position method find a root ultimately, if one would iterate long enough?
3. How does the false position algorithm behave when the curve has more than one root? 
4. Construct a curve with two roots.  Does the false position algorithm converge?  Explain.  Does the method behave in the same way for all curves with two roots?