Roots of equations 1

ESCUELA DE INGENIERÍA DE PETROLEOS

RUBEN DARIO ARISMENDI RUEDA


CHAPTER 4: ‘ROOTS OF EQUATIONS’


The roots of equations are the values of x that makes f(x)=0. There are many forms to obtain this values of x, but the most common is the quadratic formula. The other forms are mostly numerical methods and graphical methods that are used when is not to easy to find the root of the function.


There are some different kind of methods to find the roots of Equation:

GRAPHICSOPEN

METHODSCLOSED

METHODS

FIXED POINT NEWTON-

RAPHSONSECANT FALSE

POSITIONBISECTION


http://s4.hubimg.com/u/351_f520.jpg

f(x)=0


CLOSED METHODS.

1. Bisection


The objective of this Method consist in divide the interval to the half, looking forward for the change of sings.

If F(x) is Real and continous in the interval that goes from X(inf) to X(sup) and then there is at least 1 root

between the intervals

0)(.)( si xfxf)(),( si xx


2

sir

xxx

)(xf

ix sx HALFxr


0)(.)( ri xfxf

THE ROOT WILL BE IN THE Inf. SEGMENT SO:Xi= STILL THE SAMEXs= THE LAST Xr

0)(.)( ri xfxfTHE ROOT WILL BE IN THE Sup. SEGMENT SO:Xi= THE LAST XrXs= STILL THE SAME


Example

CALCULATE THE ROOT OF THE NEXT EQUATION.

%100actualr

anteriorr

actualr

a x

xxE

ERROR FOR THE NEW RESULT


In the table, we can see that the value in the 7th iteration is 0,42578125 which is approximate to the real value with an error of 0,00917431.


2. False Position.

The steps are the same that are used in the Bisection Method. The only difference is the Value of Xr.

This method consit in the intersection of a line-segment with the X axis, and using similar triangles the next expression is obtained.


ix rx

xRaíz Falsa

sxRaíz Verdadera

)(xf

)()(

)(

si

sissr xfxf

xxxfxx


Example



xi xs xr Fxi Fxs Fxr Fxi*Fxr error

0 1 0,53628944 1 -0,86466472 -0,19416436 -0,19416436

0 0,53628944 0,44909182 1 -0,19416436 -0,041783 -0,041783 0,19416436

0 0,44909182 0,43108 1 -0,041783 -0,00883096 -0,00883096 0,041783

0 0,43108 0,42730647 1 -0,00883096 -0,00185864 -0,00185864 0,00883096

0 0,42730647 0,42651374 1 -0,00185864 -0,00039083 -0,00039083 0,00185864

In conclusion with this method, we can see that the value that is looking for, is obtain faster than in the Bisection Method.


OPEN METHODS.

1. Fixed Point.

There are two different ways to find the root of a equation with this method.

a. We add X in both parts of the equation.b. We reflect the variable X from the equation. (depending on the reflect of

the variable, the method will converge in a different way).

)(xgx


xix

)(1 xf

)(2 xf

)(xf

Root


CONVERGENCE.

1. 1)('2 xf Is convergent

2. 1)('2 xf Is Divergent

)()(... 2 xgxfWHEN When the equation doesn’t converge, we wont get the root.


2. Newton-Raphson

This Method consist in take an initial value and start to make tangents from this value to the value of the root.


)(

)('1 xf

xfxx iii

%1001

1

i

ii

x

xxE

1ix WILL BE THE VALUE OF THE ROOT

ERROR


Example



xi Fxi F'xi xi+1 error

0 1 -3 0,33333333

1 0,33333333 0,18008379 -2,02683424 0,42218312 0,21045319

2 0,42218312 0,00764656 -1,85965936 0,42629493 0,00964546

3 0,42629493 1,4494E-05 -1,85261884 0,42630275 1,8353E-05

4 0,42630275 5,219E-11 -1,8526055 0,42630275 6,6082E-11

This method is faster, than the Closed methods.


3.Secant

This method consist in get the Root of the equation giving two intial values.


)()(

))((

1

11

ii

iiiii xfxf

xxxfxx1ix WILL BE THE VALUE OF

THE ROOT

%1001

1

i

ii

x

xxE

ERROR


Example



xi xi-1 Fxi Fxi-1 xi+1 error

0 0,2 1 0,47032005 0,3775865

1 0,3775865 0 0,09234281 1 0,41600122 0,09234281

2 0,41600122 0,3775865 0,01917577 0,09234281 0,42606903 0,02362953

3 0,42606903 0,41600122 0,00043303 0,01917577 0,42630164 0,00054564


Bibliography:

•Numerical Methods for Engineers . Steven C. Chapra•Prf. Eduardo Carrillo's presentation ''METODOS NUMERICOS EN INGENIERIA DE PETROLEOS''.•*PPT ''Metodos iterativos para la resolucion de ecuacioens de una variable'' (www.google.com)

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