02bracketingmethods-101108031223-phpapp02

Numerical Analysis: Bracketing Methods

Mohammad Tawfik

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Roots of Nonlinear Equations


Mohammad Tawfik

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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


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Root of Nonlinear Equations

• Solve 0xf


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Bracketing Methods


Mohammad Tawfik

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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.


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Example

• For the parachutist problem

mctec

mgtv /1

• Find ‘c’ such that smv /4010

• Where, kgmsmg 1.68,/8.9 2


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Example (cont’d)

• You get 1.68/1018.9*1.68

40 cec

• OR:

• Giving,

40138.667 147.0 ce

ccf

269.216&067.612 ff


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Example (cont’d)

• Graphically


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The Bisection Method


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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


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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


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Algorithm: Modified

• So, let’s modify the algorithm


f(a).f(b)<0

2. Calculate ‘c’ where c=0.5(a+b)

3. If |f(c)|<es; end




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False-Position Method


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The False-Position Method


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Evaluating ‘c’

• The slope of the line

joining the two point

maybe written as:

bc

yymor

ac

yym bcac

bc

yy

ac

yy bcac

bcac yyacyybc


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Evaluating ‘c’

ba yacybc 00

aybycycy baab

ab

ab

yy

byayc

afbf

bafabfc


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False Position Algorithm


f(a).f(b)<0

2. Calculate ‘c’ where

c=(af(b)-bf(a))/(f(b)-f(a))

3. If |f(c)|<es; end




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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!


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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!

02bracketingmethods-101108031223-phpapp02

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