LOGO
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]
KOMPUTASI NUMERIK TERAPAN
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
KOMPUTASI NUMERIK TERAPAN
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
KOMPUTASI NUMERIK TERAPAN
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
KOMPUTASI NUMERIK TERAPAN
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
KOMPUTASI NUMERIK TERAPAN
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?
KOMPUTASI NUMERIK TERAPAN
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?
KOMPUTASI NUMERIK TERAPAN
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
KOMPUTASI NUMERIK TERAPAN
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
KOMPUTASI NUMERIK TERAPAN
+
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