2025/10/31 by Habera, Michal, Zilian, Andreas
#65 #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Software (cs.MS) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2511.00292
Trigonometric formulas for eigenvalues of 3 × 3 matrices that build on Cardano's and Viète's work on algebraic solutions of the cubic are numerically unstable for matrices with repeated eigenvalues. This work presents numerically stable, closed-form evaluation of eigenvalues of real, diagonalizable 3 × 3 matrices via four invariants: the trace I1, the deviatoric invariants J2 and J3, and the discriminant Δ. We analyze the conditioning of these invariants and derive tight forward error bounds. For J2 we propose an algorithm and prove its accuracy. We benchmark all invariants and the resulting eigenvalue formulas, relating observed forward errors to the derived bounds. In particular, we show that, for the special case of matrices with a well-conditioned eigenbasis, the newly proposed algorithms have errors within the forward stability bounds. Performance benchmarks show that the proposed algorithm is approximately ten times faster than the highly optimized LAPACK library for a challenging test case, while maintaining comparable accuracy.