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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   and that there exists a number such that .
If   have opposite signs, and

represents the sequence of points generated by the Regula Falsi process, then the sequence converges to the zero  .

That is,      .

Mathematica Subroutine (Regula Falsi Method).

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Example 1.  Find all the real solutions to the cubic equation  .
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    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

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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.

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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 ?

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Various Scenarios and Animations for Regula Falsi Method.

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Example 3.  Convergence  Find the solution to the cubic equation  .  Use the starting interval  .
Solution 3.

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

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

(c) John H. Mathews 2004