A Constrained Optimization Approach to Bairstow’s Method

Bairstow’s method employs synthetic division to express a polynomial p(x) of degree n in the form p(x)=q(x)(x2+Bx+C)+R(B,C)x+S(B,C), where q(x) is the quotient polynomial of degree n−2, and R(B,C), S(B,C) are the remainder coefficients that depend nonlinearly on the quadratic parameters B and C. The original algorithm proposed by Bairstow uses Newton–Raphson method to solve R(B,C)=S(B,C)=0; it requires initial guesses within very narrow attraction basins for ill-conditioned polynomials and fails at singular Jacobian matrices. To address these issues, we reformulate Bairstow’s method as a constrained optimization problem that maximizes C2 as the objective function subject to the constraints R(B,C)=S(B,C)=0. While modern, highly optimized, non-linear solvers (available in commercial software like MATLAB) have largely superseded classical iterative polynomial rootfinding techniques, our reformulated Bairstow approach offers distinct advantages for selective root extraction and application-specific constraints. Specifically, the optimization formulation enables the extraction of specific roots of interest rather than computing all roots simultaneously, naturally accommodates additional constraints for application-specific factorization x (such as discriminant conditions for real versus complex root extraction). The C2 objective automatically selects the quadratic factor with the largest root magnitude, enhancing numerical stability during deflation. Numerical experiments validate the approach on polynomials with degree bigger than 10 including cases with simple real roots, multiple roots, mixed real and complex roots, and Chebyshev polynomials, achieving machine precision accuracy with robust handling of the discriminant constraint.