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

TL;DR

This paper introduces a numerically stable, closed-form method for computing eigenvalues of real 3x3 matrices, improving accuracy and speed over traditional formulas and existing libraries.

Contribution

It develops a stable evaluation approach based on matrix invariants, with proven error bounds and a fast algorithm outperforming LAPACK in speed for challenging cases.

Findings

The new method maintains errors within stability bounds for well-conditioned matrices.

The proposed algorithm is approximately ten times faster than LAPACK.

It achieves comparable accuracy while significantly improving computational efficiency.

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.