1. Bracketing Methods (Bisection Method)
Faculty of Applied Engineering and Urban Planning
Civil Engineering Department
ECGD3110
Numerical Analysis
Root Finding
Bracketing Methods (Bisection Method)
Lecture 1
1st Semester 2009/2010
UP Copyrights 2009

2. Content Solution Methods Graphical Methods Bracketing Methods
Bisection Method

3. General Cases

5. Roots Solution with Bracketing Methods
It is a simple technique for obtaining an estimate of root for the equation f(x)=0.
It gives a rough approximation of the root solution(s)

6. The Bisection Method
Choose lower xl and upper xu with: f(xl)f(xu)<0
Estimate of the root
Check:
If f(xl)f(xr)<0 the root between xl & xr
If f(xl)f(xr)>0 the root between xu & xr
If f(xl)f(xr)=0 the root is xr

7. Example
Solve:
Y=X3 – using the bisection method

9. X  1.71

10. Example
Solve:
Y=X3 - 5
Solution

11. Bisection Method

12. True value = approximation + Error Et = true value – approximation
When using the iterative approach to compute an answer and there is no true solution:

13. Error Estimates for Bisection
Estimated solution in each iteration

15. Find the positive root knowing that it is between (0.5) and (1.5)

16. ϵa < ϵs Define xL, and xu Calculate xr xL = xr xu = xr End
f(xL)f(xu) < 0 No
Yes
Calculate xr
f(xL)f(xr) < 0
Yes No
xL = xr
xu = xr
ϵa < ϵs No
Yes
End