Numerically stable evaluation of closed-form expressions for eigenvalues of $3 \times 3$ matrices

Abstract

Trigonometric formulas for eigenvalues of $3 \times 3$ matrices that build on Cardano's and Vi\`ete'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 $3 \times 3$ matrices via four invariants: the trace $I_1$, the deviatoric invariants $J_2$ and $J_3$, and the discriminant $\Delta$. We analyze the conditioning of these invariants and derive tight forward error bounds. For $J_2$ 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.