**for**

**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, .

**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).**

**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**

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.

**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 ?

**Various Scenarios and
Animations for ****Regula
Falsi
Method****.**

**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.

**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**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004