Computation of the Euler Angles of a Symmetric $3 \times 3$ Matrix

Closed form formulas for computing the eigenvectors of a symmetric $3 \times 3$ matrix are presented. The matrix of the eigenvectors is computed as a product of three rotations through Euler angles. The formulas require approximately 90 arithmetic operations, six trigonometric evaluations, and two root evaluations. These formulas may be applied as a subroutine in a parallel one-sided Jacobi-type method in which three rather than two columns, as is the case in the standard Jacobi method, are operated on in each step.