Using the discriminant in a numerically stable symmetric 3 × 3 direct eigenvalue solver

Analytical methods for solving eigenvalue problems involving real symmetric (Figure presented.) matrices are computationally efficient compared to iterative approaches, but not numerically robust when two of the eigenvalues coalesce. Analysis of the associated characteristic polynomial reveals an alternative form for the definition of the discriminant in terms of a sum of squares which is not susceptible to the numerical difficulties of conventional methods. Expressions for the angle used in the definition of the eigenvalues are derived in terms of simple combinations of the matrix invariants and the discriminant. The proposed method improves over previously proposed robust analytical schemes and is comparable in speed with the standard analytical method, making it a viable alternative.