TOPICS
Search

Bairstow's Method

A procedure for finding the quadratic factors for the complex conjugate roots of a polynomial P(x) with real coefficients.

[x-(a+ib)][x-(a-ib)]=x^2+2ax+(a^2+b^2)=x^2+Bx+C.
(1)

Now write the original polynomial as

P(x)=(x^2+Bx+C)Q(x)+Rx+S
(2)
R(B+deltaB,C+deltaC) approx R(B,C)+(partialR)/(partialB)dB+(partialR)/(partialC)dC
(3)
S(B+deltaB,C+deltaC) approx S(B,C)+(partialS)/(partialB)dB+(partialS)/(partialC)dC
(4)
(partialP)/(partialC)=0=(x^2+Bx+C)(partialQ)/(partialC)+Q(x)+x(partialR)/(partialC)+(partialS)/(partialC)
(5)
-Q(x)=(x^2+Bx+C)(partialQ)/(partialC)+x(partialR)/(partialC)+(partialS)/(partialC)
(6)
(partialP)/(partialB)=0=(x^2+Bx+C)(partialQ)/(partialB)+xQ(x)+x(partialR)/(partialB)+(partialS)/(partialB)
(7)
-xQ(x)=(x^2+Bx+C)(partialQ)/(partialB)+x(partialR)/(partialB)+(partialS)/(partialB).
(8)

Now use the two-dimensional Newton's method to find the simultaneous solutions.

Explore with Wolfram|Alpha

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

Subject classifications