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Bairstow method example ( Enter your problem )
  1. (Method-1). Algorithm Formula : `b_0=a_0+rb_1+sb_2`
  2. (Method-1). Example-1 `f(x)=x^4-3x^3+3x^2-3x+2` and `r=0.1,s=0.1`
  3. (Method-1). Example-2 `f(x)=x^4-2x^3+6x^2-2x+5` and `r=-1,s=-1`
  4. (Method-2). Algorithm Formula : `b_2=a_2-pb_1-qb_0`
  5. (Method-2). Example-1 `f(x)=x^3+x^2-x+2` and `r=-0.9, s=0.9`
  6. (Method-2). Example-2 `f(x)=x^4+x^3+2x^2+x+1` and `r=0.5, s=0.5`

4. (Method-2). Algorithm Formula : `b_2=a_2-pb_1-qb_0`
(Previous example)
6. (Method-2). Example-2 `f(x)=x^4+x^3+2x^2+x+1` and `r=0.5, s=0.5`
(Next example)

5. (Method-2). Example-1 `f(x)=x^3+x^2-x+2` and `r=-0.9, s=0.9`





Find all roots of polynomial using Bairstow method
`f(x)=x^3+x^2-x+2` and `r=-0.9, s=0.9`


Solution:
`x^3+x^2-x+2=0`

In this problem the coefficients are `a_0=1,a_1=1,a_2=-1,a_3=2`

Let the initial approximation `p_0=-0.9` and `q_0=0.9`


`1^(st)` iteration :

`-p_0`
`0.9`
`a_0`
`1`
`a_1`
`1`
`a_2`
`-1`
`a_3`
`2`
`-q_0`
`-0.9`
`-p_0*b_0`
`0.9`
`-p_0*b_1`
`1.71`
`-p_0*b_2`
`-0.171`
`-q_0*b_0`
`-0.9`
`-q_0*b_1`
`-1.71`
`b_0`
`1`
`b_1=a_1-p_0*b_0`
`1.9`
`b_2=a_2-p_0*b_1-q_0*b_0`
`-0.19`
`b_3=a_3-p_0*b_2-q_0*b_1`
`0.119`
`-p_0*c_0`
`0.9`
`-p_0*c_1`
`2.52`
`-q_0*c_0`
`-0.9`
`c_0`
`1`
`c_1=b_1-p_0*c_0`
`2.8`
`c_2=b_2-p_0*c_1-q_0*c_0`
`1.43`


`Delta p=-(b_3*c_0-b_2*c_1)/(c_1^2-c_0*(c_2-b_2))=-(0.651)/(6.22)=-0.1047`

`Delta q=-(b_2*(c_2-b_2)-b_3*c_1)/(c_1^2-c_0*(c_2-b_2))=-(-0.641)/(6.22)=0.1031`

`p_1=p_0+Delta p=-0.9-0.1047=-1.0047`

`q_1=q_0+Delta q=0.9+0.1031=1.0031`


`2^(nd)` iteration :

`-p_1`
`1.0047`
`a_0`
`1`
`a_1`
`1`
`a_2`
`-1`
`a_3`
`2`
`-q_1`
`-1.0031`
`-p_1*b_0`
`1.0047`
`-p_1*b_1`
`2.014`
`-p_1*b_2`
`0.011`
`-q_1*b_0`
`-1.0031`
`-q_1*b_1`
`-2.0108`
`b_0`
`1`
`b_1=a_1-p_1*b_0`
`2.0047`
`b_2=a_2-p_1*b_1-q_1*b_0`
`0.011`
`b_3=a_3-p_1*b_2-q_1*b_1`
`0.0002`
`-p_1*c_0`
`1.0047`
`-p_1*c_1`
`3.0234`
`-q_1*c_0`
`-1.0031`
`c_0`
`1`
`c_1=b_1-p_1*c_0`
`3.0093`
`c_2=b_2-p_1*c_1-q_1*c_0`
`2.0313`


`Delta p=-(b_3*c_0-b_2*c_1)/(c_1^2-c_0*(c_2-b_2))=-(-0.0327)/(7.0357)=0.0047`

`Delta q=-(b_2*(c_2-b_2)-b_3*c_1)/(c_1^2-c_0*(c_2-b_2))=-(0.0215)/(7.0357)=-0.0031`

`p_2=p_1+Delta p=-1.0047+0.0047=-1`

`q_2=q_1+Delta q=1.0031-0.0031=1`


`3^(rd)` iteration :

`-p_2`
`1`
`a_0`
`1`
`a_1`
`1`
`a_2`
`-1`
`a_3`
`2`
`-q_2`
`-1`
`-p_2*b_0`
`1`
`-p_2*b_1`
`2`
`-p_2*b_2`
`0`
`-q_2*b_0`
`-1`
`-q_2*b_1`
`-2`
`b_0`
`1`
`b_1=a_1-p_2*b_0`
`2`
`b_2=a_2-p_2*b_1-q_2*b_0`
`0`
`b_3=a_3-p_2*b_2-q_2*b_1`
`0`
`-p_2*c_0`
`1`
`-p_2*c_1`
`3`
`-q_2*c_0`
`-1`
`c_0`
`1`
`c_1=b_1-p_2*c_0`
`3`
`c_2=b_2-p_2*c_1-q_2*c_0`
`2.0001`


`Delta p=-(b_3*c_0-b_2*c_1)/(c_1^2-c_0*(c_2-b_2))=-(0)/(7.0001)=0`

`Delta q=-(b_2*(c_2-b_2)-b_3*c_1)/(c_1^2-c_0*(c_2-b_2))=-(0)/(7.0001)=0`

`p_3=p_2+Delta p=-1=-1`

`q_3=q_2+Delta q=1=1`

Approximate root `p=-1` and `q=1`

Hence extracted quadratic factor `=x^2+px+q=x^2-x+1`


This material is intended as a summary. Use your textbook for detail explanation.
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4. (Method-2). Algorithm Formula : `b_2=a_2-pb_1-qb_0`
(Previous example)
6. (Method-2). Example-2 `f(x)=x^4+x^3+2x^2+x+1` and `r=0.5, s=0.5`
(Next example)





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