introduction to numerical analysis i math/cmpsc 455 bisection method
TRANSCRIPT
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