A procedure for finding the quadratic factors for the complex conjugate roots of a polynomial
with real coefficients.
![[x-(a+ib)][x-(a-ib)]=x^2+2ax+(a^2+b^2)=x^2+Bx+C.](/images/equations/BairstowsMethod/NumberedEquation1.svg) |
(1)
|
Now write the original polynomial as
![P(x)=(x^2+Bx+C)Q(x)+Rx+S](/images/equations/BairstowsMethod/Inline2.svg) |
(2)
|
![R(B+deltaB,C+deltaC) approx R(B,C)+(partialR)/(partialB)dB+(partialR)/(partialC)dC](/images/equations/BairstowsMethod/Inline3.svg) |
(3)
|
![S(B+deltaB,C+deltaC) approx S(B,C)+(partialS)/(partialB)dB+(partialS)/(partialC)dC](/images/equations/BairstowsMethod/Inline4.svg) |
(4)
|
![(partialP)/(partialC)=0=(x^2+Bx+C)(partialQ)/(partialC)+Q(x)+x(partialR)/(partialC)+(partialS)/(partialC)](/images/equations/BairstowsMethod/Inline5.svg) |
(5)
|
![-Q(x)=(x^2+Bx+C)(partialQ)/(partialC)+x(partialR)/(partialC)+(partialS)/(partialC)](/images/equations/BairstowsMethod/Inline6.svg) |
(6)
|
![(partialP)/(partialB)=0=(x^2+Bx+C)(partialQ)/(partialB)+xQ(x)+x(partialR)/(partialB)+(partialS)/(partialB)](/images/equations/BairstowsMethod/Inline7.svg) |
(7)
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![-xQ(x)=(x^2+Bx+C)(partialQ)/(partialB)+x(partialR)/(partialB)+(partialS)/(partialB).](/images/equations/BairstowsMethod/Inline8.svg) |
(8)
|
Now use the two-dimensional Newton's method to
find the simultaneous solutions.
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References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 277 and 283-284, 1989.Referenced
on Wolfram|Alpha
Bairstow's Method
Cite this as:
Weisstein, Eric W. "Bairstow's Method."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BairstowsMethod.html
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