EQUATION ROOTS

The methods described here in employ different strategies to reduce the answer interval in some bracket. The objective is find a x valor of root that f(x)=0.

EQUATION ROOTS----EQUATION ROOTS----EQUATION ROOTS----EQUATION ROOTS----EQUATION ROOTS----EQUATION ROOTS

The methods described here in employ different strategies to reduce the answer interval in some bracket. The objective is find a x valor of root that f(x)=0.

BRACKETING METHODS

GRAPHICAL METHODS
To obtain a f(x)=0 with this method, you can draw a plot of the function and observe where it crosses x axis, this point represent an approximation of the root.

Example:
f(x)=(667,38/x)*(1-e^(-0,1468*x))-40 ; interval: (10, 20):

Possible cases in [a,b] interval

THE BISECTION METHOD

Initial guesses: 2
Convergence rate: slow
Stability: always

Incremental search methods capitalize on this observation by locating an interval where the function changes sign. Then the location sign change is identified more precisely by dividing the interval into a number of subintervals. The process is repeated and the root estimated refined by dividing the subintervals into finer increments*.
To identify the answered interval:

-If f(xi)*f(xr)<0>

-If f(xr)*f(xs)<0 the root is in lower interval

Example: f(x)= (e^ (-x))-x; interval (0,1)

THE FALSE POSITION METHOD

Initial guesses: 2
Convergence rate: slow/medium
Stability: always

An alternative method that exploits the graphical insight is to join f(xi) and f(xs) by a straight line. The intersection of this line with the x axis represents an improved estimated of the root. The fact that replacement of the curve by straight lines give a “false position” of the root.
This method is very similar to bisection method, just defer in:
Xr = Xs- ((fxs)*(xi-xs))/((f(xi)-f(xs)).

Example: f(x)= (x^3)+(4*(x^2))-10

OPEN METHODS

The open methods are based on formulas that required only a single starting value of x or two starting values that do not necessary bracket the root*.

SIMPLE FIXED-POINT METHOD

Initial guesses: 1
Convergence rate: slow
Stability: possibly divergent

The simple fixed-point iteration by rearranging the function f(x)=0 so that x is on the left-hand side of equation*.
X i= g(xo)
Xi+1 = g(xi)
The approximate error for this equation can be determined using the error estimator, if the approximate error is low, the root is nearly them valor.

Example: f(x)= (2400/(30-X))^(1/2)

Can report four situations at the moment to want a root:








When the solution converge from the root, and





Case when the iterations diverge from the root.




NEWTON-RAPHSON METHOD

Initial guesses: 1
Convergence rate: fast
Stability: possibly divergent

If the initial guess at the root is xi, a tangent can be extended from the point [xi, f(xi)]. The point where this tangent crosses the x axis usually represents an improved estimated of the root. The Newton-Raphson method can be derived on basis of geometrical interpretation where the first derivate from function at x is equivalent to slope:

f’(xi) = (f(xi) - 0)/(Xi - Xi+1)
Xi+1 = xi – (f(xi)/f’(xi))----Newton-Raphson formula.

Example:
f(x)=[1,27*10^(-4)*X^(2)]-[6,77*10^(-9)*X^(4)]-[2,67*10^(-6)*X^(3)]-0,0185
f’(x)= [2,55*10^(-4)*X]-[(2,71*10^(-8)*X^(3)]-[8*10^(-6)*X^(2)]

SECANT METHOD

Initial guesses: 2
Convergence rate: medium to fast
Stability: possibly divergent

In some cases is complicate meet the derivate of the function, through similar triangle associated at the union of two original points, obtained the next equation which describe the two triangle shaped between the two points mention.
Xi+1 = xi - [f(xi)*(Xi-1 – Xi)]/[f(xi-1)-f(xi)]

Example: f(x) = (e^(-X))-X

Summary

REFERENCES

 CHAPRA, Steven C. y CANALE, Raymond P.: “Numerical Methods for engineers”. McGraw Hill, fifth edition, 2006.*
 CARRILLO, Eduardo, “Métodos Abiertos”, UIS, 2010.