With this topic we can begin to describe methods to locate polynomials roots. The

first step would be to investigate the viability of the bracketing and open

approaches that we viewed in roots of equations. The efficacy of these approaches

depends on whether the problem being solved involves complex roots. If only real

roots exist, any of the previously described methods could have utility. The

problem of finding good initial guesses complicates both the bracketing and the

open methods, whereas the open methods could be susceptible to divergence.

When complex roots are possible, the bracketing methods cannot be used

because of the obvious problem that the criterion for defining a bracket does not

translated to complex guesses.

Muller´s Method

In this figure we have a comparison

relate approaches for locating roots, the

first figure is the secant method and the

last one is Muller´s method.

The method consists of deriving the

coefficients of the parabola that goes

through the tree points .The coefficients

can be substituted into the quadratic

formula to obtain the point where the

parabola intercepts the x axis that is, the

root estimate. The approach is facilitated

by writing the parabolic equation in a

convenient form,

( ) ( ) ( )

We want this parabola to intersect the

three points

[ ( )] [ ( )] [ ( )]

The coefficients of the first equation can

be evaluated by substituting each of the

three points to give

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

We have three equations, now we can solve for three unknown coefficients, a,b

and c. in the last one equation we can view that two of the terms are zero and it

can be immediately solved for ( ) This result can be substituted into the

rest equations to yield equations with two unknowns

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

With algebraic manipulation we can solve the remaining coefficients, a and b, one

way to do this involves defining a number of differences.

( ) ( )

( ) ( )

This can be substituted into the last equations and the results can be summarized

as

( )

To find the root, we apply the quadratic formula

√

Example

Use Muller´s method with guesses of

and

respectively, to determine a root of the

equation

𝒇(𝒙) 𝒙𝟑 𝟏𝟑𝒙 𝟏𝟐

Solution:

First evaluate the function at the

guesses

𝒇(𝟒 𝟓) 𝟐𝟎 𝟔𝟐𝟓 𝒇(𝟓 𝟓) 𝟖𝟐 𝟖𝟕𝟓

𝒇(𝟓) 48

Which can be used to calculated

These values in turn can be substituted to the equation where we can find a, b and

c.

( )

The square root of the discriminant can be evaluated as

√ ( )

So we can view that positive sign is employed in the denominator, so we can

estimate the new root

( )

√ ( )

Bibliography

Numerical Methods for Engineers, Fifth Edition, Steven C. Chapra and Raymond

P. Canale

BY:

FRANCY GUERRERO ZABALA

NUMERICAL METHODS

UNIVERSIDAD INDUSTRIAL DE SANTANDER

iteration Xr

0 5 -----

1 3.9764 25.74

2 4.0010 0.6139

3 4 0.262

4 4 0.000011