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Roots of equations 1

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  • 1. RUBEN DARIO ARISMENDI RUEDA
  • 2. CHAPTER 4: ‘ROOTS OF EQUATIONS’
  • 3. 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.
  • 4. There are some different kind of methods to find the roots of Equation: GRAPHICS OPEN METHODS CLOSED METHODS FIXED POINT NEWTON-RAPHSON SECANT FALSE POSITION BISECTION
  • 5. http://s4.hubimg.com/u/351_f520.jpg f(x)=0
  • 6. CLOSED METHODS. 1. Bisection
  • 7. 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
  • 8.  
  • 9. THE ROOT WILL BE IN THE Inf. SEGMENT SO: Xi= STILL THE SAME Xs= THE LAST Xr THE ROOT WILL BE IN THE Sup. SEGMENT SO: Xi= THE LAST Xr Xs= STILL THE SAME
  • 10. Example CALCULATE THE ROOT OF THE NEXT EQUATION. ERROR FOR THE NEW RESULT
  • 11. 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.
  • 12. 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.
  • 13. Raíz Falsa Raíz Verdadera
  • 14. Example CALCULATE THE ROOT OF THE NEXT EQUATION.
  • 15. In conclusion with this method, we can see that the value that is looking for, is obtain faster than in the Bisection Method. 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
  • 16. OPEN METHODS.
    • Fixed Point.
    • There are two different ways to find the root of a equation with this method.
    • We add X in both parts of the equation.
    • We reflect the variable X from the equation. (depending on the reflect of the variable, the method will converge in a different way).
  • 17. Root
  • 18. CONVERGENCE. Is convergent Is Divergent When the equation doesn’t converge, we wont get the root. 1. 2.
  • 19. 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.
  • 20. WILL BE THE VALUE OF THE ROOT ERROR
  • 21. Example CALCULATE THE ROOT OF THE NEXT EQUATION.
  • 22. This method is faster, than the Closed methods. 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
  • 23. 3.Secant This method consist in get the Root of the equation giving two intial values.
  • 24. WILL BE THE VALUE OF THE ROOT ERROR
  • 25. Example CALCULATE THE ROOT OF THE NEXT EQUATION.
  • 26. 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
  • 27.
    • 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 )