Module

for

Muller's Method

 

Background

    
Muller's method is a generalization of the secant method, in the sense that it does not require the derivative of the function. It is an iterative method that requires three starting points  [Graphics:Images/MullersMethodMod_gr_1.gif], [Graphics:Images/MullersMethodMod_gr_2.gif], and [Graphics:Images/MullersMethodMod_gr_3.gif].   A parabola is constructed that passes through the three points; then the quadratic formula is used to find a root of the quadratic for the next approximation.  It has been proved that near a simple root Muller's method converges faster than the secant method and almost as fast as Newton's method.  The method can be used to find real or complex zeros of a function and can be programmed to use complex arithmetic.

Proof  Muller's Method  Muller's Method  

 

Computer Programs  Muller's Method  Muller's Method  

Mathematica Subroutine (Newton-Raphson Iteration).

[Graphics:Images/MullersMethodMod_gr_4.gif]

Mathematica Subroutine (Muller's Method).

[Graphics:Images/MullersMethodMod_gr_5.gif]

Example 1.  Use Newton's method and Muller's method to find numerical approximations to the multiple root  [Graphics:Images/MullersMethodMod_gr_6.gif]  of the function  [Graphics:Images/MullersMethodMod_gr_7.gif].  
Show details of the computations for the starting value  [Graphics:Images/MullersMethodMod_gr_8.gif].  Compare the number of iterations for the two methods.
Solution 1.

 

Example 2.  Use Newton's method and Muller's method to find numerical approximations to the multiple root  [Graphics:Images/MullersMethodMod_gr_74.gif]  of the function  [Graphics:Images/MullersMethodMod_gr_75.gif].  
Show details of the computations for the starting value  [Graphics:Images/MullersMethodMod_gr_76.gif].  Compare the number of iterations for the two methods.
Solution 2.

 

Example 3.  Use Newton's method and Muller's method to find numerical approximations to the multiple root near  x = 2  of the function  [Graphics:Images/MullersMethodMod_gr_143.gif].  
Show details of the computations for the starting value  [Graphics:Images/MullersMethodMod_gr_144.gif].  Compare the number of iterations for the two methods.
Solution 3.

 

Research Experience for Undergraduates

Muller's Method  Muller's Method  Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook Muller's Method

 

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(c) John H. Mathews 2004