introduction to numerical analysis i math/cmpsc 455 bisection method

Introduction to Numerical Analysis I

MATH/CMPSC 455

Bisection Method

SOLVING NONLINEAR EQUATIONS

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

THE BISECTION METHODBasic 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

Yes

No

No

Example: Find a root of the function

on the interval

ERROR ANALYSIS

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 .

PRACTICAL CONSIDERATIONS