A robust algorithm for finding the eigenvalues and eigenvectors of 3 × 3 symmetric matrices

Many concepts in continuum mechanics are most easily understood in principal coordinates; using these concepts in a numerical analysis requires a robust algorithm for finding the eigenvalues and eigenvectors of 3 × 3 symmetric matrices. A robust algorithm for solving this eigenvalue problem is presented along with an analysis of the algorithm. The special case of two or three nearly identical eigenvalues is examined in detail using an asymptotic analysis. Numerical results are shown that compare this algorithm with existing methods found in the literature. The behavior of this algorithm is shown to be more reliable than the other methods with a minimal computational cost.