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

2025-10-31

AI Summary

Conventional trigonometric closed-form methods for computing eigenvalues of 3×3 real symmetric matrices suffer from numerical instability in the presence of multiple eigenvalues. Method: This paper proposes a robust closed-form solution leveraging the trace, deviatoric invariants, and discriminant—integrating Cardano–Viète algebraic principles, invariant analysis, and error propagation theory. We derive, for the first time, a tight forward error bound for closed-form eigenvalue computation of 3×3 matrices and design a high-precision J₂ algorithm that ensures controllable error under well-conditioned eigenvector bases. Contribution/Results: Experimental evaluation demonstrates accuracy comparable to LAPACK, with approximately 10× speedup; observed errors strictly satisfy the theoretical forward error bound. The method significantly enhances computational efficiency and robustness in high-reliability applications requiring guaranteed numerical stability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: 3D Computer Vision
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Trigonometric formulas for eigenvalues of $3 imes 3$ 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 $3 imes 3$ matrices via four invariants: the trace $I_1$, the deviatoric invariants $J_2$ and $J_3$, and the discriminant $Δ$. 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.

Problem

• Numerically stable eigenvalue evaluation for 3×3 matrices
• Overcoming instability in trigonometric formulas for repeated eigenvalues
• Developing fast closed-form solutions with tight error bounds

Innovation

• Numerically stable closed-form evaluation of eigenvalues
• Uses four invariants: trace, deviatoric invariants, discriminant
• Ten times faster than LAPACK with comparable accuracy