A Numerically Stable Framework for Nonic Equations via Cubic Tschirnhaus Transformation, Newton Refinement and Reduction to a Cubic

This paper presents a numerically stable framework for solving real-coefficient nonic equations (degree 9). The method is based on a cubic Tschirnhaus transformation y = x^3 + \alpha x^2 + \beta x + \gamma that approximately eliminates the y^8, y^7, and y^6 terms. The parameters (\alpha, \beta, \gamma) are determined by solving a 3\times3 nonlinear system using Newton's method with an LU/SVD fallback to handle ill-conditioned Jacobians. After refinement, the transformed nonic reduces to the near-simplified form y^9 + A y^6 + B y^3 + C \approx 0 with residuals below 10^{-14}. The substitution w = y^3 then yields a cubic w^3 + A w^2 + B w + C = 0, which is solved using a backward-stable cubic solver. Back-substitution via stable cubic solvers produces all nine roots. The paper provides a full theoretical justification, including: · Local existence of the transformation parameters (via the inverse function theorem),· Quadratic convergence of Newton’s method,· Root perturbation bounds justifying the neglect of the residual \epsilon(y),· The multiplicity structure of the 27 back-substitution candidates,· Backward stability analysis under IEEE 754 double-precision arithmetic. Numerical experiments on 10 test cases — including factored polynomials, perfect powers, roots of unity, random real roots, near-degenerate polynomials, extreme scaling (roots from 10^{-4} to 9000), Wilkinson-type perturbations, and Chebyshev polynomials — demonstrate relative errors below 10^{-14} in all well-conditioned cases. Comparisons with Jenkins–Traub and companion matrix methods show competitive or superior performance, especially in extreme scaling regimes where classical methods overflow or underflow. This framework extends the author’s previous unified stable solvers for degrees 3–8 and 10, and it is a step toward a complete numerically stable treatment of polynomials up to degree 10. Keywords: Nonic equations; cubic Tschirnhaus transformation; Newton refinement; cubic solver; numerical stability; catastrophic cancellation; polynomial root-finding; backward stability; Wilkinson polynomial; Chebyshev polynomial.