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For the polynomial $$p(z)=3z^5-7z^4-5z^3+z^2-8z+2$$ apply Bairstow's method with the initial point $(u,v)=(3,1)$. Compute the corrections $\delta u$ and $\delta v$.

Since $u=3, v=1$, then we have $z^2-3z-1$. If we divide $p(z)$ by $z^2-3z-1$, then we have $$p(z)=(z^2-3z-1)(3z^3+2z^2+4z+15)+41(z-3)+170.$$ By the Bairstow's method, we let $b_2=15$, $b_3=4$, $b_4=2$, $b_5=3$, $b_0=170$, and $b_1=41$.

However once I get to this point I am unsure of compute the Jacobian.

Username Unknown
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1 Answers1

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So you found that the second approximate factor to $a(z)=z^2−3z−1$ is $$ b(z)=3z^3+2z^2+4z+15. $$ You need to solve $$ \Delta a(z)·b(z)=r(z) \pmod{ a(z)} $$ where $r(z)=p(z)-a(z)b(z)=41(z−3)+140$. This gets easier by first computing the remainder of $b(z)$ modulo $a(z)$, $$ b(z)=(z^2−3z−1)(3z+11)+40z+26. $$ So it remains to solve $$ (-Δu·z-Δv)(40z+26)=41z+17\pmod{z^2−3z−1}\\ -Δu·(40(3z+1)+26z)-Δv·(40z+26)=41z+17 $$ which results in the linear system $$ \begin{bmatrix} 146&40\\ 40&26 \end{bmatrix}· \begin{bmatrix} -Δu\\-Δv \end{bmatrix} = \begin{bmatrix} 41\\17 \end{bmatrix} $$ Solving this results in $Δu=-193/1098$ and $Δv=-421/1098$ and the remainder of the updated factor $a(z)=z^2-(u+Δu)z-(v+Δv)$ is around $r(z)\approx 6.05798·z+4.93328$, so smaller than the original.

If you do this computation repeatedly you should get the iteration steps

                 u                                   v                                 Δu                               Δv 
[  3.0000000000000000000000000000 ,  1.0000000000000000000000000000 ] [ -0.17577413479052823315118397086, -0.38342440801457194899817850637 ]
[  2.82422586520947176684881602912,  0.6165755919854280510018214936 ] [  0.18068915178461465967821961276, -0.76964397103124847113723999388 ]
[  3.00491501699408642652703564188, -0.1530683790458204201354185003 ] [  0.21839555807980775993869699470, -0.64219297328632886129259028856 ]
[  3.22331057507389418646573263658, -0.7952613523321492814280087888 ] [ -0.02328190181886663525821053714,  0.06857388080402612581518772222 ]
[  3.20002867325502755120752209949, -0.7266874715281231556128210666 ] [ -0.00055188208069591099502978794,  0.00162856128477162344612529787 ]
[  3.19947679117433164021249231148, -0.7250589102433515321666957687 ] [ -0.00000028208555627734095531324,  0.00000083294961402081994655938 ]
[  3.19947650908877536287153699821, -0.7250580772937375113467492094 ] [ -7.29932855082287833370E-14     ,  2.15590017434042941586E-13      ]
[  3.19947650908870236958602876940, -0.7250580772935219213293151664 ] [ -4.8720049504E-27               ,  1.4388822908E-26                ]
Lutz Lehmann
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