Bracketing Methods

Solve nonlinear equations using bracketing methods: Bisection and False Position

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https://wikicourses.wikispaces.com/Topic+Roots+of+Nonlinear+Equations

1. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Roots of Nonlinear Equations

2. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Objectives
•Understand the need for numerical solutions of
nonlinear equations
•Be able to use the bisection algorithm to find a
root of an equation
•Be able to use the false position method to find a
root of an equations
•Write down an algorithm to outline the method
being used
•Realize the need for termination criteria

3. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Root of Nonlinear Equations
•Solve()0=xf

4. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Bracketing Methods

5. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Intermediate Value Theorem
•For our specific interest
If f(x) is continuous in the interval [a,b], and
f(a).f(b)<0, then there exists ‘c’ such that
a<c<b and f(c)=0.

6. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
•For the parachutist problem
() [ ]
mct
e
c
mg
tv
/
1
-
-=
•Find ‘c’ such that() smv /4010=
•Where, kgmsmg 1.68,/8.9
2
==

7. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example (cont’d)
•You get [ ]
1.68/10
1
8.9*1.68
40
c
e
c
-
-=
•OR:
•Giving,
() [ ]401
38.667
147.0
--=
- c
e
c
cf
() () 269.216&067.612 -== ff

8. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example (cont’d)
•Graphically

9. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
The Bisection Method

10. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example

11. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Algorithm
1.Search for a & b such that
f(a).f(b)<0
2.Calculate ‘c’ where c=0.5(a+b)
3.If f(c)=0; end
4.If f(a).f(c)>0 then let a=c; goto step 2
5.If f(b).f(c)>0 then let b=c; goto step 2

12. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Algorithm (cont’d)
•That algorithm will go on forever!
•We need to define a termination
criterion
•Examples of termination criteria:
1.|f(c)|<e
s
2.|b-a|<e
s
3.e
a
=|(c
new
-c
old
)/c
new
|<e
s
4.
Number of iterations > N

13. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Algorithm: Modified
•So, let’s modify the algorithm
1.Search for a & b such that
f(a).f(b)<0
2.Calculate ‘c’ where c=0.5(a+b)
3.If |f(c)|<e
s
; end
4.If f(a).f(c)>0 then let a=c; goto step 2
5.If f(b).f(c)>0 then let b=c; goto step 2

14. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
False-Position Method

15. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
The False-Position Method

16. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Evaluating ‘c’
•The slope of the line
joining the two point
maybe written as:
bc
yy
mor
ac
yy
m
bcac
-
-
=
-
-
=
bc
yy
ac
yy
bcac
-
-
=
-
-
( )( )( )( )
bcac
yyacyybc --=--

17. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Evaluating ‘c’
( )( )( )( )
ba
yacybc --=-- 00
aybycycy
baab +=-
( )
ab
ab
yy
byay
c
-
-
=
() ()
() ()afbf
bafabf
c
-
-
=

18. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example

19. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
False Position Algorithm
1.Search for a & b such that
f(a).f(b)<0
2.Calculate ‘c’ where
c=(af(b)-bf(a))/(f(b)-f(a))
3.If |f(c)|<e
s; end
4.If f(a).f(c)>0 then let a=c; goto step 2
5.If f(b).f(c)>0 then let b=c; goto step 2

20. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Conclusion
•The need for numerical solution of nonlinear
equations led to the invention of approximate
techniques!
•The bracketing techniques ensure that you will
find a solution for a continuous function if the
solution exists
•A termination criterion should be embedded into
the numerical algorithm to ensure its
termination!

21. ENME602 Spring 2007
Dr. Eng. Mohammad Tawfik
Homework #1
•Chapter 5, page 131, numbers:
5.3,5.6,5.7,5.8,5.12
•You are not required to get the solution
graphically!
•Homework due Next week!