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BY: DUBAN CASTRO FLOREZ

NUMERICS METHODS IN

ENGINEERING

2010

MULLER

METHOD

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This is a method to find the roots of equations polynomials in the general way:

Where n is the order of the polynomial and they are they constant coefficients. Continuing with the polynomials, these they fulfill the following rules:

• " For the order equation n, is n real or complex roots. It should be noticed that those roots are not necessarily different.

• " If n is odd, there is a real root at least. • " If the complex roots exist, a conjugated couple exists.

nnn xaxaxaaxf .......)( 2

210

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DEFINITION

• It is the method secant, which obtains roots, estimating a projection of a direct line in the axis x, through two values of the function.

• The method consists on obtaining the coefficients of the three points, to substitute them in the quadratic formula and to obtain the point where the parable intercepts the axis x.

cxxbxxaxf )()()( 22

22

4

f(x)

XX1 X0

x

x

x

Línea recta

Raíz estimada

Raíz

5

cxxbxxaxf )()()( 202

200

cxxbxxaxf )()()( 212

211

cxxbxxaxf )()()( 222

222

Raíz estimada

f(x)

XX2 X0

0

0

0

X1

Raíz

Parábola

x x

This way, this parable is looked for intersectar the three points [x0, f(x0)], [x1, f(x1)] and [x2, f(x2)].

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• The last equation generates that , this way, one can have a system of two equations with two incognito:

• Defining this way:

• Substituting in the system:

cxf )( 2

)()()()( 202

2020 xxbxxaxfxf

)()()()( 212

2121 xxbxxaxfxf

010 xxh 121 xxh

01

210

)()(

xx

xfxf

12

121

)()(

xx

xfxf

11002

1010 )()( hhahhbhh

11211 hahbh

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• Having the coefficients as a result:

• Finding the root, you to implement the conventional solution, but due to the error of rounding potential, an alternative formulation will be used:

• Error

01

01

hha

11 ahb )( 2xfc

acbb

cxx

4

2223

acbb

cxx

4

2223

%1003

23

x

xxEa

8

1213)( 3 xxxf

625,20)5,4( f 875,82)5,5( f 48)5( f

15,45,50 h 5,05,551 h

25,625,45,5

625,20875,820

75,695,55

875,82481

1515,0

25,6275,69

a 25,6275,69)5,0(15 b 48c

544,314815425,62 2 9765,3544,3125,62

48253

x

%74,25%1000235,1

3

x

Ea

h = 0,1 x2 = 5 x1 = 5,5 x0 =4,5

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BIBLIOGRAGHY

• http://lc.fie.umich.mx/~calderon/programacion/Mnumericos/Muller.html

• http://aprendeenlinea.udea.edu.co/lms/moodle/mod/resource/view.php?id=24508