These methods on roots of equations deals with methods that exploit the fact that a function typically changes sign in the vincity of a root. These methods are called Bracketing methods because two initial values for the root are required. This initial guess will bracket the root of equation.
Graphical method:
If f(x)=0 is for any value of x, that value is the root of the equation. A simple method to obtain the root is make a plot of the function and observe where it crosses the X-axis. That value of x will be the root of equation.
In general, if f(x) is real and continuous in the interval Xl and Xu and f(Xl) and f(Xu) have opposite signs, that is f(Xl)*f(Xu)<0 then there exist one real root between Xl and Xu.
The location of the sign change is identified more precisely by dividing the interval into number of sub intervals.
Bisection method is one type of incremental search method in which the interval is always divided in half. If a function changes changes sign over an interval, the fun ction value at the mid point is evaluated. The location of the root is then determined as lying at the midpoint of the sub interval within which the sign change occurs.
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