**for**

**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 ,
,
and . 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).**

**Mathematica Subroutine (Muller's
Method).**

**Example 1.** Use
Newton's method and Muller's method to find numerical approximations
to the multiple root of
the function .

Show details of the computations for the starting
value . 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 of
the function .

Show details of the computations for the starting
value . 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 .

Show details of the computations for the starting
value . 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**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004