ESCUELA DE INGENIERÍA DE PETROLEOS RUBEN DARIO ARISMENDI RUEDA
Jun 19, 2015
ESCUELA DE INGENIERÍA DE PETROLEOS
RUBEN DARIO ARISMENDI RUEDA
ESCUELA DE INGENIERÍA DE PETROLEOS
CHAPTER 4: ‘ROOTS OF EQUATIONS’
ESCUELA DE INGENIERÍA DE PETROLEOS
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.
ESCUELA DE INGENIERÍA DE PETROLEOS
There are some different kind of methods to find the roots of Equation:
GRAPHICSOPEN
METHODSCLOSED
METHODS
FIXED POINT NEWTON-
RAPHSONSECANT FALSE
POSITIONBISECTION
ESCUELA DE INGENIERÍA DE PETROLEOS
http://s4.hubimg.com/u/351_f520.jpg
f(x)=0
ESCUELA DE INGENIERÍA DE PETROLEOS
CLOSED METHODS.
1. Bisection
ESCUELA DE INGENIERÍA DE PETROLEOS
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
ESCUELA DE INGENIERÍA DE PETROLEOS
2
sir
xxx
)(xf
ix sx HALFxr
ESCUELA DE INGENIERÍA DE PETROLEOS
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
ESCUELA DE INGENIERÍA DE PETROLEOS
Example
CALCULATE THE ROOT OF THE NEXT EQUATION.
%100actualr
anteriorr
actualr
a x
xxE
ERROR FOR THE NEW RESULT
ESCUELA DE INGENIERÍA DE PETROLEOS
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.
ESCUELA DE INGENIERÍA DE PETROLEOS
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.
ESCUELA DE INGENIERÍA DE PETROLEOS
ix rx
xRaíz Falsa
sxRaíz Verdadera
)(xf
)()(
)(
si
sissr xfxf
xxxfxx
ESCUELA DE INGENIERÍA DE PETROLEOS
Example
CALCULATE THE ROOT OF THE NEXT EQUATION.
ESCUELA DE INGENIERÍA DE PETROLEOS
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.
ESCUELA DE INGENIERÍA DE PETROLEOS
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
ESCUELA DE INGENIERÍA DE PETROLEOS
xix
)(1 xf
)(2 xf
)(xf
Root
ESCUELA DE INGENIERÍA DE PETROLEOS
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.
ESCUELA DE INGENIERÍA DE PETROLEOS
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.
ESCUELA DE INGENIERÍA DE PETROLEOS
)(
)('1 xf
xfxx iii
%1001
1
i
ii
x
xxE
1ix WILL BE THE VALUE OF THE ROOT
ERROR
ESCUELA DE INGENIERÍA DE PETROLEOS
Example
CALCULATE THE ROOT OF THE NEXT EQUATION.
ESCUELA DE INGENIERÍA DE PETROLEOS
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.
ESCUELA DE INGENIERÍA DE PETROLEOS
3.Secant
This method consist in get the Root of the equation giving two intial values.
ESCUELA DE INGENIERÍA DE PETROLEOS
)()(
))((
1
11
ii
iiiii xfxf
xxxfxx1ix WILL BE THE VALUE OF
THE ROOT
%1001
1
i
ii
x
xxE
ERROR
ESCUELA DE INGENIERÍA DE PETROLEOS
Example
CALCULATE THE ROOT OF THE NEXT EQUATION.
ESCUELA DE INGENIERÍA DE PETROLEOS
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
ESCUELA DE INGENIERÍA DE PETROLEOS
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)