Introduction to Numerical Analysis I MATH/CMPSC 455 Bisection Method

S OLVING N ONLINEAR E QUATIONS General Mathematical Problem: Given a function, find the values of for which Existence of root: Let be a continuous function on, satisfying. Then has a root between and, that is, there exists a number satisfying and

T HE B ISECTION M ETHOD Basic Idea: Narrow down the interval by halving Start f(a)f(b)<0 c=(a+b)/2 b=c f(a)f(c)<0 |a-b|<tol output (a+b)/2 and stop a=c Yes No

Example: Find a root of the function on the interval

E RROR A NALYSIS Theorem (Error Analysis of Bisection Method): If denote the intervals obtained by the Bisection method, then the limits and exist, equal and represent a zero (root) of. If and, then

Example: How many steps is needed, if we use Bisection Method to find a root of in the interval, and require the solution is correct with 6 decimal places? Example: Suppose that the bisection method is started with the interval, how many steps should be taken to compute a root that the relative error is less than.

P RACTICAL C ONSIDERATIONS