1. Bracketing Methods Chapter 5 The Islamic University of Gaza
Faculty of Engineering
Civil Engineering Department
Numerical Analysis
ECIV 3306
Chapter 5
Bracketing Methods

2. PART II ROOTS OF EQUATIONS
Bracketing Methods
Bisection method
False Position Method
Open Methods
Simple fixed point iteration
Newton Raphson
Secant
Modified Newton Raphson
System of Nonlinear Equations
Roots of polynomials
Muller Method

3. Study Objectives for Part Two

4. ROOTS OF EQUATIONS
Root of an equation: is the value of the equation variable which make the equations = 0.0
But

5. ROOTS OF EQUATIONS Non-computer methods:
- Closed form solution (not always available)
- Graphical solution (inaccurate)
Numerical systematic methods suitable for computers

6. The roots exist where f(x) crosses the x-axis.
Graphical Solution
Plot the function f(x)
f(x)
roots
The roots exist where f(x) crosses the x-axis. x
f(x)=0
f(x)=0

7. Graphical Solution: Example
The parachutist velocity is
What is the drag coefficient c needed to reach a velocity of 40 m/s if m=68.1 kg, t =10 s, g= 9.8 m/s2 c
f(c)
c=14.75
Check: F (14.75) = ~ 0.0
v (c=14.75) = ~ 40 m/s

8. Numerical Systematic Methods I. Bracketing Methods
f(x)
f(x)
No roots or even number of roots
Odd number of roots
f(xl)=+ve
f(xl)=+ve
roots
roots
f(xu)=+ve x x xl xu
f(xu)=-ve xl xu

9. Bracketing Methods (cont.)
Two initial guesses (xl and xu) are required for the root which bracket the root (s).
If one root of a real and continuous function, f(x)=0, is bounded by values xl , xu then f(xl).f(xu) <0.
(The function changes sign on opposite sides of the root)

12. Bracketing Methods 1. Bisection Method
Generally, if f(x) is real and continuous in the interval xl to xu and f (xl).f(xu)<0, then there is at least one real root between xl and xu to this function.
The interval at which the function changes sign is located. Then the interval is divided in half with the root lies in the midpoint of the subinterval. This process is repeated to obtained refined estimates.

13. Step 1: Choose lower xl and upper xu guesses for the root such that:
f(x)
Step 1: Choose lower xl and upper xu guesses for the root such that:
f(xl).f(xu)<0
Step 2: The root estimate is:
xr = ( xl + xu )/2
Step 3: Subdivide the interval according to:
If (f(xl).f(xr)<0) the root lies in the lower subinterval; xu = xr and go to step 2.
If (f(xl).f(xr)>0) the root lies in the upper subinterval; xl = xr and go to step 2.
If (f(xl).f(xr)=0) the root is xr and stop
xr = ( xl + xu )/2
f(xu) xl
xr1 xu x
f(xu)
f(xr1)
f(x)
(f(xl).f(xr)<0): xu = xr
xr = ( xl + xu )/2
f(xu)
f(xr2) xl xu x
xr2
f(xu)

14. Bisection Method - Termination Criteria
For the Bisection Method ea > et
The computation is terminated when ea becomes less than a certain criterion (ea < es)

15. Bisection method: Example
The parachutist velocity is
What is the drag coefficient c needed to reach a velocity of 40 m/s if m = 68.1 kg, t = 10 s, g= 9.8 m/s2
f(c) c

16. and so on…... f(x) Assume xl =12 and xu=16
f(xl)=6.067 and f(xu)=-2.269
The root: xr=(xl+xu)/2= 14
Check f(12).f(14) = 6.067•1.569=9.517 >0;
the root lies between 14 and 16.
Set xl = 14 and xu=16, thus the new root xr=(14+ 16)/2= 15
Check f(14).f(15) = 1.569•-0.425= <0;
the root lies bet. 14 and 15.
Set xl = 14 and xu=15, thus the new root xr=(14+ 15)/2= 14.5
and so on…...
6.067
1.569 x 12 14 16
-2.269
f(x)
(f(12).f(14)>0): xl = 14
1.569 15 x 14 16
-0.425
-2.269

17. Bisection method: Example
In the previous example, if the stopping criterion is et = 0.5%; what is the root?
Iter. Xl Xu Xr ea% et%

18. Bisection method

19. Flow Chart –Bisection Start False Stop Input: xl , xu , s, maxi
f(xl). f(xu)<0
i=0
a=1.1s
False
while
a> s &
i <maxi
Stop
Print: xr , f(xr ) ,a , i

20. True Test=0 Test<0 False xu+xl =0 Test=f(xl). f(xr) a=0.0 xu=xr
xl=xr
False

21. Bracketing Methods 2. False-position Method
The bisection method divides the interval xl to xu in half not accounting for the magnitudes of f(xl) and f(xu). For example if f(xl) is closer to zero than f(xu), then it is more likely that the root will be closer to f(xl).
False position method is an alternative approach where f(xl) and f(xu) are joined by a straight line; the intersection of which with the x-axis represents and improved estimate of the root.

22. 2. False-position Method
False position method is an alternative approach where f(xl) and f(xu) are joined by a straight line; the intersection of which with the x-axis represents and improved estimate of the root.

23. False-position Method -Procedure
f(x)
f(xu) xl xr xu x
f(xl)
f(xr)

24. False-position Method -Procedure
Step 1: Choose lower xl and upper xu guesses for the root such that: f(xl).f(xu)<0
Step 2: The root estimate is:
Step 3: Subdivide the interval according to:
If (f(xl).f(xr)<0) the root lies in the lower subinterval; xu = xr and go to step 2.
If (f(xl).f(xr)>0) the root lies in the upper subinterval; xl = xr and go to step 2.
If (f(xl).f(xr)=0) the root is xr and stop

25. False position method: Example
The parachutist velocity is
What is the drag coefficient c needed to reach a velocity of 40 m/s if m =68.1 kg, t =10 s, g= 9.8 m/s2
f(c) c

26. False position method: Example
Assume xl = 12 and xu=16
f(xl)= and f(xu)=
The root: xr=
f(12) . f( ) = < 0;
The root lies bet. 12 and
Assume xl = 12 and xu= , f(xl)=6.067 and f(xu)=
The new root xr=
This has an approximate error of 0.79%
f(x)
6.067
14.91 x 12 16
-2.269

27. False position method: Example

28. Flow Chart –False Position
Start
Input: xl , x0 , s, maxi
f(xl). f(xu)<0
i=0
a=1.1s
False
while
a> s &
i <maxi
Stop
Print: xr , f(xr ) ,a , i

29. True Test=0 Test<0 False i=1 or xr=0 Test=f(xl). f(xr) a=0.0 xu=xr
xr0=xr
Test<0
xl=xr
False

30. False Position Method-Example 2

31. False Position Method - Example 2

32. Roots of Polynomials: Using Software Packages
MS Excel:Goal seek
f(x)=x-cos x

33. MS Excel: Solver
u(x,y)= x2+xy-10 =0
v(x,y)=y+3xy2-57=0