Extending Numerically Stable Cubic Solvers to Solvable Quintics: A Unified Framework

This paper presents a numerically stable analytical framework for solving solvable quintic equations of the form x^5 + 5\alpha x^3 + 5\beta x^2 + \gamma x + \delta = 0 under the solvability condition \beta = \alpha^2/2. The method introduces scale-invariant parameters (R, K, L) that prevent overflow/underflow and reduce the problem to a cubic resolvent. The cubic resolvent is solved using backward-stable formulas developed in previous works on cubic equations. The five roots are recovered using fifth roots of unity. Extensive numerical examples, including the King quintic and near-degenerate cases, demonstrate relative errors below 10^{-13}. Practical applications in quantum mechanics, robotics, control theory, and computer graphics are discussed. The algorithm is deterministic, requires no initial guess, and avoids catastrophic cancellation.