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 mostcommon is the quadratic formula. The other forms are mostlynumerical methods and graphical methods that are used when isnot 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

METHODSCLOSE

METHODS

FIXED POINT

NEWTON-RAPHSON

SECANT FAKE POSITION

BISECCION

ESCUELA DE INGENIERÍA DE PETROLEOS

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

f(x)=0

ESCUELA DE INGENIERÍA DE PETROLEOS

CLOSE METHODS.

1. Biseccion

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

ixsxHALFxr

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 xfxf

THE 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.

%100actual

r

anterior

r

actual

r

ax

xxE

ERROR FOR THE NEW RESULT

ESCUELA DE INGENIERÍA DE PETROLEOS

xi xs xr Fxi Fxs Fxr Fxi*Fxr error

0 1 0,5 1 -0,86466472 -0,13212056 -0,13212056

1 0 0,5 0,25 1 -0,13212056 0,35653066 0,35653066 1

2 0,25 0,5 0,375 0,35653066 -0,13212056 0,09736655 0,03471416 0,33333333

3 0,375 0,5 0,4375 0,09736655 -0,13212056 -0,02063798 -0,00200945 0,14285714

4 0,375 0,4375 0,40625 0,09736655 -0,02063798 0,03749731 0,00365098 0,07692308

5 0,40625 0,4375 0,421875 0,03749731 -0,02063798 0,00821964 0,00030821 0,03703704

6 0,421875 0,4375 0,4296875 0,00821964 -0,02063798 -0,00626086 -5,1462E-05 0,01818182

7 0,421875 0,4296875 0,42578125 0,00821964 -0,00626086 0,00096637 7,9432E-06 0,00917431

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