Bairstow method (Method-2). Example-2 f(x)=x^4+x^3+2x^2+x+1 and r=0.5, s=0.5

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Home > Numerical methods calculators > Bairstow method example

Bairstow method example ( Enter your problem )

( Enter your problem )

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

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

6. (Method-2). Example-2 `f(x)=x^4+x^3+2x^2+x+1` and `r=0.5, s=0.5`

Find all roots of polynomial using Bairstow method
`f(x)=x^4+x^3+2x^2+x+1` and `r=0.5, s=0.5`

Solution:
`x^4+x^3+2x^2+x+1=0`

In this problem the coefficients are `a_0=1,a_1=1,a_2=2,a_3=1,a_4=1`

Let the initial approximation `p_0=0.5` and `q_0=0.5`

`1^(st)` iteration :

`-p_0` `-0.5`	`a_0` `1`	`a_1` `1`	`a_2` `2`	`a_3` `1`	`a_4` `1`
`-q_0` `-0.5`		`-p_0*b_0` `-0.5`	`-p_0*b_1` `-0.25`	`-p_0*b_2` `-0.625`	`-p_0*b_3` `-0.0625`
			`-q_0*b_0` `-0.5`	`-q_0*b_1` `-0.25`	`-q_0*b_2` `-0.625`
	`b_0` `1`	`b_1=a_1-p_0*b_0` `0.5`	`b_2=a_2-p_0b_1-q_0b_0` `1.25`	`b_3=a_3-p_0b_2-q_0b_1` `0.125`	`b_4=a_4-p_0b_3-q_0b_2` `0.3125`
		`-p_0*c_0` `-0.5`	`-p_0*c_1` `0`	`-p_0*c_2` `-0.375`
			`-q_0*c_0` `-0.5`	`-q_0*c_1` `0`
	`c_0` `1`	`c_1=b_1-p_0*c_0` `0`	`c_2=b_2-p_0c_1-q_0c_0` `0.75`	`c_3=b_3-p_0c_2-q_0c_1` `-0.25`

`Delta p=-(b_4*c_1-b_3*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.0938)/(0.5625)=0.1667`

`Delta q=-(b_3*(c_3-b_3)-b_4*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.2812)/(0.5625)=0.5`

`p_1=p_0+Delta p=0.5+0.1667=0.6667`

`q_1=q_0+Delta q=0.5+0.5=1`

`2^(nd)` iteration :

`-p_1` `-0.6667`	`a_0` `1`	`a_1` `1`	`a_2` `2`	`a_3` `1`	`a_4` `1`
`-q_1` `-1`		`-p_1*b_0` `-0.6667`	`-p_1*b_1` `-0.2222`	`-p_1*b_2` `-0.5185`	`-p_1*b_3` `-0.0988`
			`-q_1*b_0` `-1`	`-q_1*b_1` `-0.3333`	`-q_1*b_2` `-0.7778`
	`b_0` `1`	`b_1=a_1-p_1*b_0` `0.3333`	`b_2=a_2-p_1b_1-q_1b_0` `0.7778`	`b_3=a_3-p_1b_2-q_1b_1` `0.1481`	`b_4=a_4-p_1b_3-q_1b_2` `0.1235`
		`-p_1*c_0` `-0.6667`	`-p_1*c_1` `0.2222`	`-p_1*c_2` `0`
			`-q_1*c_0` `-1`	`-q_1*c_1` `0.3333`
	`c_0` `1`	`c_1=b_1-p_1*c_0` `-0.3333`	`c_2=b_2-p_1c_1-q_1c_0` `0`	`c_3=b_3-p_1c_2-q_1c_1` `0.4815`

`Delta p=-(b_4*c_1-b_3*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.0412)/(0.1111)=0.3704`

`Delta q=-(b_3*(c_3-b_3)-b_4*c_2)/(c_2^2-c_1*(c_3-b_3))=-(0.0494)/(0.1111)=-0.4444`

`p_2=p_1+Delta p=0.6667+0.3704=1.037`

`q_2=q_1+Delta q=1-0.4444=0.5556`

`3^(rd)` iteration :

`-p_2` `-1.037`	`a_0` `1`	`a_1` `1`	`a_2` `2`	`a_3` `1`	`a_4` `1`
`-q_2` `-0.5556`		`-p_2*b_0` `-1.037`	`-p_2*b_1` `0.0384`	`-p_2*b_2` `-1.5378`	`-p_2*b_3` `0.5364`
			`-q_2*b_0` `-0.5556`	`-q_2*b_1` `0.0206`	`-q_2*b_2` `-0.8238`
	`b_0` `1`	`b_1=a_1-p_2*b_0` `-0.037`	`b_2=a_2-p_2b_1-q_2b_0` `1.4829`	`b_3=a_3-p_2b_2-q_2b_1` `-0.5172`	`b_4=a_4-p_2b_3-q_2b_2` `0.7125`
		`-p_2*c_0` `-1.037`	`-p_2*c_1` `1.1139`	`-p_2*c_2` `-2.1168`
			`-q_2*c_0` `-0.5556`	`-q_2*c_1` `0.5967`
	`c_0` `1`	`c_1=b_1-p_2*c_0` `-1.0741`	`c_2=b_2-p_2c_1-q_2c_0` `2.0412`	`c_3=b_3-p_2c_2-q_2c_1` `-2.0372`

`Delta p=-(b_4*c_1-b_3*c_2)/(c_2^2-c_1*(c_3-b_3))=-(0.2904)/(2.5337)=-0.1146`

`Delta q=-(b_3*(c_3-b_3)-b_4*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.6683)/(2.5337)=0.2637`

`p_3=p_2+Delta p=1.037-0.1146=0.9224`

`q_3=q_2+Delta q=0.5556+0.2637=0.8193`

`4^(th)` iteration :

`-p_3` `-0.9224`	`a_0` `1`	`a_1` `1`	`a_2` `2`	`a_3` `1`	`a_4` `1`
`-q_3` `-0.8193`		`-p_3*b_0` `-0.9224`	`-p_3*b_1` `-0.0715`	`-p_3*b_2` `-1.0231`	`-p_3*b_3` `0.0799`
			`-q_3*b_0` `-0.8193`	`-q_3*b_1` `-0.0635`	`-q_3*b_2` `-0.9087`
	`b_0` `1`	`b_1=a_1-p_3*b_0` `0.0776`	`b_2=a_2-p_3b_1-q_3b_0` `1.1092`	`b_3=a_3-p_3b_2-q_3b_1` `-0.0867`	`b_4=a_4-p_3b_3-q_3b_2` `0.1712`
		`-p_3*c_0` `-0.9224`	`-p_3*c_1` `0.7793`	`-p_3*c_2` `-0.9863`
			`-q_3*c_0` `-0.8193`	`-q_3*c_1` `0.6922`
	`c_0` `1`	`c_1=b_1-p_3*c_0` `-0.8449`	`c_2=b_2-p_3c_1-q_3c_0` `1.0692`	`c_3=b_3-p_3c_2-q_3c_1` `-0.3807`

`Delta p=-(b_4*c_1-b_3*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.052)/(0.8947)=0.0581`

`Delta q=-(b_3*(c_3-b_3)-b_4*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.1576)/(0.8947)=0.1761`

`p_4=p_3+Delta p=0.9224+0.0581=0.9805`

`q_4=q_3+Delta q=0.8193+0.1761=0.9954`

`5^(th)` iteration :

`-p_4` `-0.9805`	`a_0` `1`	`a_1` `1`	`a_2` `2`	`a_3` `1`	`a_4` `1`
`-q_4` `-0.9954`		`-p_4*b_0` `-0.9805`	`-p_4*b_1` `-0.0191`	`-p_4*b_2` `-0.9663`	`-p_4*b_3` `-0.014`
			`-q_4*b_0` `-0.9954`	`-q_4*b_1` `-0.0194`	`-q_4*b_2` `-0.981`
	`b_0` `1`	`b_1=a_1-p_4*b_0` `0.0195`	`b_2=a_2-p_4b_1-q_4b_0` `0.9855`	`b_3=a_3-p_4b_2-q_4b_1` `0.0143`	`b_4=a_4-p_4b_3-q_4b_2` `0.005`
		`-p_4*c_0` `-0.9805`	`-p_4*c_1` `0.9424`	`-p_4*c_2` `-0.9143`
			`-q_4*c_0` `-0.9954`	`-q_4*c_1` `0.9567`
	`c_0` `1`	`c_1=b_1-p_4*c_0` `-0.9611`	`c_2=b_2-p_4c_1-q_4c_0` `0.9325`	`c_3=b_3-p_4c_2-q_4c_1` `0.0567`

`Delta p=-(b_4*c_1-b_3*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.0181)/(0.9102)=0.0199`

`Delta q=-(b_3*(c_3-b_3)-b_4*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.004)/(0.9102)=0.0044`

`p_5=p_4+Delta p=0.9805+0.0199=1.0005`

`q_5=q_4+Delta q=0.9954+0.0044=0.9999`

`6^(th)` iteration :

`-p_5` `-1.0005`	`a_0` `1`	`a_1` `1`	`a_2` `2`	`a_3` `1`	`a_4` `1`
`-q_5` `-0.9999`		`-p_5*b_0` `-1.0005`	`-p_5*b_1` `0.0005`	`-p_5*b_2` `-1.0011`	`-p_5*b_3` `0.0006`
			`-q_5*b_0` `-0.9999`	`-q_5*b_1` `0.0005`	`-q_5*b_2` `-1.0005`
	`b_0` `1`	`b_1=a_1-p_5*b_0` `-0.0005`	`b_2=a_2-p_5b_1-q_5b_0` `1.0006`	`b_3=a_3-p_5b_2-q_5b_1` `-0.0006`	`b_4=a_4-p_5b_3-q_5b_2` `0.0001`
		`-p_5*c_0` `-1.0005`	`-p_5*c_1` `1.0014`	`-p_5*c_2` `-1.0026`
			`-q_5*c_0` `-0.9999`	`-q_5*c_1` `1.0008`
	`c_0` `1`	`c_1=b_1-p_5*c_0` `-1.0009`	`c_2=b_2-p_5c_1-q_5c_0` `1.0021`	`c_3=b_3-p_5c_2-q_5c_1` `-0.0024`

`Delta p=-(b_4*c_1-b_3*c_2)/(c_2^2-c_1*(c_3-b_3))=-(0.0005)/(1.0025)=-0.0005`

`Delta q=-(b_3*(c_3-b_3)-b_4*c_2)/(c_2^2-c_1*(c_3-b_3))=-(-0.0001)/(1.0025)=0.0001`

`p_6=p_5+Delta p=1.0005=1`

`q_6=q_5+Delta q=0.9999=1`

`7^(th)` iteration :

`-p_6` `-1`	`a_0` `1`	`a_1` `1`	`a_2` `2`	`a_3` `1`	`a_4` `1`
`-q_6` `-1`		`-p_6*b_0` `-1`	`-p_6*b_1` `0`	`-p_6*b_2` `-1`	`-p_6*b_3` `0`
			`-q_6*b_0` `-1`	`-q_6*b_1` `0`	`-q_6*b_2` `-1`
	`b_0` `1`	`b_1=a_1-p_6*b_0` `0`	`b_2=a_2-p_6b_1-q_6b_0` `1`	`b_3=a_3-p_6b_2-q_6b_1` `0`	`b_4=a_4-p_6b_3-q_6b_2` `0`
		`-p_6*c_0` `-1`	`-p_6*c_1` `1`	`-p_6*c_2` `-1`
			`-q_6*c_0` `-1`	`-q_6*c_1` `1`
	`c_0` `1`	`c_1=b_1-p_6*c_0` `-1`	`c_2=b_2-p_6c_1-q_6c_0` `1`	`c_3=b_3-p_6c_2-q_6c_1` `0`

`Delta p=-(b_4*c_1-b_3*c_2)/(c_2^2-c_1*(c_3-b_3))=-(0)/(1)=0`

`Delta q=-(b_3*(c_3-b_3)-b_4*c_2)/(c_2^2-c_1*(c_3-b_3))=-(0)/(1)=0`

`p_7=p_6+Delta p=1=1`

`q_7=q_6+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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