Solve nonlinear equations using bracketing methods: Bisection and False Position #WikiCourses https://wikicourses.wikispaces.com/Topic+Roots+of+Nonlinear+Equations

1. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Roots of Nonlinear Equations

2. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Root of Nonlinear Equations
• Solve   0xf

4. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Bracketing Methods

5. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example (cont’d)
• Graphically

9. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The Bisection Method

10. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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

11. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Algorithm (cont’d)
• That algorithm will go on forever!
• We need to define a termination
criterion
• Examples of termination criteria:
1. |f(c)|<es
2. |b-a|<es
3. ea=|(cnew -cold)/cnew|<es
4. Number of iterations > N

12. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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)|<es; 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

13. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
False-Position Method

14. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The False-Position Method

15. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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 

16. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Evaluating ‘c’
     ba yacybc  00
aybycycy baab 
 ab
ab
yy
byay
c



   
   afbf
bafabf
c




17. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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)|<es; 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

18. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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!

19. Numerical Analysis: Bracketing Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Homework #1
• Chapter 5, page 139, numbers:
5.3,5.6,5.7,5.8,5.12
• You are not required to get the solution
graphically!
• Homework due Next week!