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 0xf
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
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
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!