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BAIRSTOW BAIRSTOW METHODMETHOD

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Sam Matahari [2311100134]

Bahang Surya [2312106005]

Maghvirah Januarty [2312106016]

Nurdina Bestari [2312106018]

Reka Erdanto [2311100075]

MEMBERS OF GROUPMEMBERS OF GROUP

I Made Pendi Adi Merta [2311100033]

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LOGODefinitionsDefinitions

Bairstow method is a method to find all the roots of polynomial equations by determine quadratic factors. Polynomial equation is written as follows:

11

21 ...)( nnn axaxaxPn

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If x2-rx-s is quadratic factor which we want to find,, so:

11

21 ...)( nnn axaxaxPn

113

22

12 )()...)((

nnnnn brxbbxbxbsrxx

LOGO Bairstow Method

If x2 - rx - s quadratic factor that is correct , then bn = 0 and bn+1= 0 . Relationship between ai and bi can be done with multiplication and identity :

bn and bn +1 is a function of r and s where bn = 0 dan bn+1 = 0

b1 = a1a1 = b1

a2 = b2 – rb1 b2 = a2 + rb1

a3 = b3 – rb2 – sb1 b3 = a3 + rb2 + sb1

:: :

:

an = bn – rbn-1 – sb1bn = an + rbn-1 + sbn-2

an+1 = bn+1 – rbn – sbn-1bn+1 = an+1 + rbn + sbn-1

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LOGO How to use Bairstow Method?

22

1

21

.)(

..

nnn

nnnn

ccc

cbcbr Step

1

Choose a value of initial approach R and S, then select the value of tolerance

Determine b(i) and c(i) as below:b1 = a1 c1 = b1

b2 = a2 – rb1 c2 = b2 + rc1

b3 = a3 – rb2 – sb1 c3 = b3 + rc2 + sc1

: :: :bn = an – rbn-1 – sbn-2 cn = bn + rcn-1 + scn-2

bn+1 = an+1 – rbn – sbn-1 cn+1 = bn+1 + rcn + scn-1

Determine b(i) and c(i) as below:b1 = a1 c1 = b1

b2 = a2 – rb1 c2 = b2 + rc1

b3 = a3 – rb2 – sb1 c3 = b3 + rc2 + sc1

: :: :bn = an – rbn-1 – sbn-2 cn = bn + rcn-1 + scn-2

bn+1 = an+1 – rbn – sbn-1 cn+1 = bn+1 + rcn + scn-1

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Step2

LOGO How to use Bairstow Method?

22

1

21

.)(

..

nnn

nnnn

ccc

cbcbr Step

3

Determine:DENOM=(cn-1)2-cn*cn-2

If DENOM = 0, we can set R = R+1;S=S+1, then back to the step 2.But, if DENOM #0,so we can continue to the step 5

If DENOM = 0, we can set R = R+1;S=S+1, then back to the step 2.But, if DENOM #0,so we can continue to the step 5

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Step4

LOGO

Step5

Step5

22

1

11

.)(

..

nnn

nnnn

ccc

cbcbs

22

1

21

.)(

..

nnn

nnnn

ccc

cbcbr

Found new

value

We can found the new value of Δr and Δs

this equation:

How to use Bairstow Method?

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LOGO

22

1

21

.)(

..

nnn

nnnn

ccc

cbcbr Step

6After Δr and Δs obtain, we can found value of r* and s*

r* = r + Δrs* = s + Δs

How to use Bairstow Method?

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If {|Δr|} + {|Δs|} <= toleration, calculation must be stopped and last value of R and S is the final value of R and S. But, if {|Δr|} + {|Δs|} >= toleration, so we must determine b(i) and c(i)

Step7

LOGO EXAMPLEX3 -6X2 +11X-6=0 ; Toleration= 0,05 solving method:1. chosen the value of r & s r=0 & s=02. InterationInteration 1 1 -6 11 -6r=0 0 0 0s=0 0 0 1 -6 11 -6 bn bn-1

r=0 0 0 0s=0 0 0 1 -6 11 -6 cn-2 cn-1 cn

+

+

ai

bi

ci

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LOGO EXAMPLE

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LOGO

Interation 2Interation 2 11 -6-6 1111 -6-6 aaii

r=2.4 r=2.4 2.4 -8.64 13.8242.4 -8.64 13.824s=3.4 s=3.4 3.4 -12.243.4 -12.24 11 -3.6 5.76 -3.6 5.76 -4.416-4.416 b bii

bbn n b bn-1n-1 r=2.4 r=2.4 2.4 -2.88 15.0722.4 -2.88 15.072s=3.4 s=3.4 3.4 -4.083.4 -4.08 11 -1.2-1.2 6.286.28 6.576 c6.576 cii

ccn-n-11 c cn n ccnn+1+1

EXAMPLE

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+

LOGO EXAMPLE

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LOGO EXAMPLE

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LOGO EXAMPLE

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