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METHODS TO FIND
ROOTS OF EQUATION
FIXED POINT, NEWTON – RAPHSON, SECANTE
JONATHAN PEREZ UIS
CLOSE METHODS
BISECCION
1. BISECCION
Raíz
xsf(xs) xi
)(xf
x
+
-
y=f(x)
Biseccion
Biseccion
Raíz
xsf(xs) xi
)(xf
x
+
-
y=f(x)
Biseccion
Biseccion
ERROR FOR THE NEW
RESULT
Biseccion
iter xi xs xm fxi fxs fxm fxi*fxm fxs*fxr Ea
0 12 16 14 6.06694996 -2.26875421 1.56870973 9.51728341 -3.55901679
1 14 16 15 1.56870973 -2.26875421 -0.42483189 -0.66643791 0.96383913 6.66666667
2 14 15 14.5 1.56870973 -0.42483189 0.55232821 0.86644263 -0.23464663 3.44827586
3 14.5 15 14.75 0.55232821 -0.42483189 0.05896283 0.03256684 -0.02504929 1.69491525
4 14.75 15 14.875 0.05896283 -0.42483189 -0.18411653 -0.01085603 0.07821857 0.84033613
5 14.75 14.875 14.8125 0.05896283 -0.18411653 -0.06287413 -0.00370724 0.01157617 0.42194093
6 14.75 14.8125 14.78125 0.05896283 -0.06287413 -0.00203019 -0.00011971 0.00012765 0.21141649
7 14.75 14.78125 14.765625 0.05896283 -0.00203019 0.02844766 0.00167735 -5.7754E-05 0.10582011
Biseccion-Example
In the table, we can see that the value in the 7th iteration is 14. 76, which is
approximate to the real values whith an error of 0.1058
CLOSE METHODS
FALSE POSITION
FALSE POSITION If we cosider this grafic:
sx
)( ixf
ix rx
xRaíz
Falsa
)( sxf
Raíz
Verdad
era
)(xf
if instead of considering the midpoint of the interval,
we take the point where this line crosses the axis, we come close
faster root-this is in itself, the central idea of the rule method
false and this is really the only difference with the method of bisection, as in all other respects
the two methods are practically identical.
FALSE POSITION
sr
s
ir
i
xx
xf
xx
xf )()(
)()(
)(
si
sissr
xfxf
xxxfxx
sx
)( ixf
ix rx
xRaíz
Falsa
)( sxf
Raíz
Verdad
era
)(xf
EXAMPLE
FALSE POSITION
iter xi xs xm fxi fxs fxm fxi*fxm fxs*fxm error
0 1 2 1.26315789 -5 14 -1.60227438 8.01137192 -22.4318414
1 1.26315789 2 1.33882784 -1.60227438 14 -0.43036475 0.68956241 -6.02510647 5.65195478
2 1.33882784 2 1.35854634 -0.43036475 14 -0.11000879 0.0473439 -1.54012304 1.45144132
3 1.35854634 2 1.36354744 -0.11000879 14 -0.02776209 0.00305407 -0.38866927 0.36677112
4 1.36354744 2 1.36480703 -0.02776209 14 -0.00698342 0.00019387 -0.09776782 0.09229083
5 1.36480703 2 1.36512372 -0.00698342 14 -0.00175521 1.2257E-05 -0.02457293 0.02319834
6 1.36512372 2 1.3652033 -0.00175521 14 -0.00044106 7.7416E-07 -0.00617488 0.00582959
In the table, we can see that the value in the 6th iteration is 1.3652, which is
approximate to the real values whith an error of 0.005829
FALSE POSITION
REFERENCES AND
BIBLIOGRAPHY
• http://noosfera.indivia.net/metodos.html
• METODOS NUMERICOS PhD EDUARDO CARRILLO –
UNIVERSIDAD INDUSTRIAL DE SANTANDER 2010
• CHAPRA, Steven C. “Métodos Numéricos para Ingenieros”. Edit.
McGraw Hil.
