A Revised Quadratic Tschirnhaus Transformation for Solving Quartic Equations with Backward Stability

This paper presents a unified numerically stable framework for solving quartic equations with real coefficients. The quadratic Tschirnhaus transformation y = z² + mz + n is applied to the depressed quartic z⁴ + pz² + qz + r = 0 to eliminate both the z³ and z terms, yielding a biquadratic form y⁴ + Py² + Q = 0. Unlike classical presentations that require the user to re-derive the transformation, this paper provides direct closed-form formulas for all parameters in terms of the original coefficients a, b, c, d, e. A user only needs to substitute the coefficients into the given formulas and follow a simple algorithmic procedure. Key contributions:1. Direct formulas for p, q, r, m, n, P, Q in terms of a, b, c, d, e2. An improved biquadratic formula that avoids catastrophic cancellation when P² ≈ 4Q3. Integration with our previously developed backward-stable cubic solver 1,2,34. Complete algorithmic procedure with decision tree5. Ten numerical examples (5 detailed + 5 summary)6. Statistical benchmarking on 1000 randomly generated quartics7. Comparison with Jenkins-Traub, Durand-Kerner, and companion matrix methods8. Five detailed practical applications in the appendix (optics, structural mechanics, thermodynamics, control theory, chemical equilibrium) The proposed method is scale-invariant, backward stable under IEEE 754 double-precision arithmetic, and achieves relative errors below 10⁻¹⁴ in all test cases, including near-degenerate configurations where classical methods lose up to 10⁻⁴ accuracy. Keywords: Quartic equations, Tschirnhaus transformation, Numerical stability, Direct formulas, Catastrophic cancellation, Ferrari method, Descartes-Euler method. Related works in this series:1 Numerically stable solutions for cubic equations: Special cases (b² = 3ac and b² 3ac3 A unified numerically stable framework for cubic equations