Module

for

The Regula Falsi Method

     

Background. The Regula Falsi method is one of the bracketing methods for finding roots of equations.
Implementation.  Given a function f(x) and an interval which might contain a root, perform a predetermined number of iterations using the Regula Falsi method.
Limitations.  Investigate the result of applying the Regula Falsi method over an interval where there is a discontinuity.  Apply the Regula Falsi method for a function using an interval where there are distinct roots.  Apply the Regula Falsi method over a "large" interval.  

 

Theorem (Regula Falsi Theorem). Assume that  [Graphics:Images/RegulaFalsiMod_gr_1.gif] and that there exists a number [Graphics:Images/RegulaFalsiMod_gr_2.gif] such that [Graphics:Images/RegulaFalsiMod_gr_3.gif].  
If  [Graphics:Images/RegulaFalsiMod_gr_4.gif] have opposite signs, and

    [Graphics:Images/RegulaFalsiMod_gr_5.gif]  

represents the sequence of points generated by the Regula Falsi process, then the sequence [Graphics:Images/RegulaFalsiMod_gr_6.gif] converges to the zero  [Graphics:Images/RegulaFalsiMod_gr_7.gif].  

That is,      [Graphics:Images/RegulaFalsiMod_gr_8.gif].  

Proof  False Position or Regula Falsi Method  False Position or Regula Falsi Method  

 

Computer Programs  False Position or Regula Falsi Method  False Position or Regula Falsi Method  

 

Mathematica Subroutine (Regula Falsi Method).

[Graphics:Images/RegulaFalsiMod_gr_9.gif]

Example 1.  Find all the real solutions to the cubic equation  [Graphics:Images/RegulaFalsiMod_gr_10.gif].  
Solution 1.

 

Remember. The Regula Falsi method can only be used to find a real root in an interval [a,b] in which f[x] changes sign.

Example 2.  Use the cubic equation  [Graphics:Images/RegulaFalsiMod_gr_256.gif]  in Example 1 and perform the following call to the Regula Falsi subroutine.
Solution 2.

 

Reduce the volume of printout.

After you have debugged you program and it is working properly, delete the unnecessary print statements.

Concise Program for the Regula Falsi

[Graphics:Images/RegulaFalsiMod_gr_426.gif]

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.

[Graphics:Images/RegulaFalsiMod_gr_427.gif]

[Graphics:Images/RegulaFalsiMod_gr_428.gif]
[Graphics:Images/RegulaFalsiMod_gr_429.gif]

Reducing the Computational Load for the Regula Falsi Method

The following program uses fewer computations in the Regula Falsi method and is the traditional way to do it.  Can you determine how many fewer functional evaluations are used ?

[Graphics:Images/RegulaFalsiMod_gr_430.gif]

Various Scenarios and Animations for Regula Falsi Method.

[Graphics:Images/RegulaFalsiMod_gr_431.gif]

Example 3.  Convergence  Find the solution to the cubic equation  [Graphics:Images/RegulaFalsiMod_gr_432.gif].  Use the starting interval  [Graphics:Images/RegulaFalsiMod_gr_433.gif].  
Solution 3.

 

Animations (Regula Falsi  Method  Regula Falsi  Method).  Internet hyperlinks to animations.  

 

Research Experience for Undergraduates

Regula Falsi Method  Regula Falsi Method  Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook Regula Falsi Method

 

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(c) John H. Mathews 2004