Ostroswki-based enhacement for the False Position root-seeking Method
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01-08-2017, 03:09 PM
(This post was last modified: 01-08-2017 06:49 PM by Namir.)
Post: #1
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Ostroswki-based enhacement for the False Position root-seeking Method
The False Position method is a root-bracketing method. It has faster convergence than the Bisection method. The False Position has a major weakness is that the root-bracketing range does not always shrink tightly around the root. An enhancement, called the Illinois Algorithm fixes that weakness by reducing the function value of either A or B that is not changed by the core False Position method. Here is the pseudo-code for the Illinois Algorithm
Code: Given function Fx(x)=0, initial root-bracketing range [A, B], root tolerance Toler, and function value tolerance FxToler: The Russian mathematician Ostrowski had proposed an enhancement to Newton's method by calculating two refinements for the root per iteration. I present an enhancement to the Illinois Algorithm based on Ostroski's approach: Code:
In the above code you can replace the following statements: Code: Z = B - Fb / Slope With the following If statement to make the calculations of Z more robust, if needed. Code: If Abs(Fb) < Abs(Fa) Then The Ostrowski variation of the Illinois Algorithm converges faster than the Illinois Algorithm. Of course if you use Ostrowski's enhancement to Newton's method, you get a rate of convergence matching that of Hailey's method which has a third order of convergence rate. Enjoy! Namir |
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