ROOTS OF EQUATIONS

By: Maria Fernanda Vergara M.Universidad Industrial de Santander

A root or solution of equation f(x)=0 are the values of x for which the equation holds true. Sometimes roots of equations are called the zeros of the equation.

Numerical methods for finding roots of equations can often be easily programmed.

One method to determinate the roots is plotting the function and determinate where it crosses with the x axis. This point, represents the x value for which f(x)=0, give us an approximation of the root.

For example: Find the real roots of the function: f(x)=-0.5x + 1.8x + 6.3 with the graphical method.

Solution: You have to build a chart with values of x and f(x), trying to get a crossing with the x-axis.

Root Root

So we can see that there are two roots, one is approximately -2 and the other is approximately 6.

The bisection method is one type of incremental search method where the interval is always divided in the half. The root is determined as lying at the midpoint of the subinterval within which the sign change occurs. The process is iterative.

“An alternative method that exploits tis graphical insight is to join f(xi) and f(xu) by a straight line. The intersection of this line with the x axis represents an improved estimate of the root. The fact that the replacement of the curve by a straight line gives a “false position”. Source: CHAPRA, numerical methods for engineers.

The intersection of the line with the x axis can be estimated as

Source: Internet

Another option to find this roots is to incorporate an incremental search at the beginning of the computer program. This method consist in taking one end of the region of interest and then evaluate the function at small increments across the region. The point is: When the function changes the sign this mean that in that point there is a root.

Open methods employ a formula to predict the root

To rearrange the function f(x)=0

x=g(x)

If the initial guess at the root is xi, a tangent can be extended from the point {xi,x(xi)}. The point where this tangent crosses the x axis usaually represents an improved estimate of the root.

The Newton-Raphson formula is:

In the Secant method the derivative is approximated by a backward finite divided difference.

This approximation can be substituted with the following iterative equation:

The difference between the secant method and the false position method is how one of the initial values is replaced by the new estimate

CHAPRA, Steven; Numerical methods for engineers. Mc Graw Hill.

http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/Rathish-kumar/ratish-1/f3node3.html

http://www.virtualum.edu.co/antiguo/metnum/raices/metgraf.htm