Computation of eigenvalues of a real, symmetric 3 × 3 matrix with particular reference to the pernicious case of two nearly equal eigenvalues

Standard, closed-form solutions for eigenvalues of symmetric, real-valued (Formula presented.) matrices are susceptible to loss of significance in floating point computation in the case of two nearly equal eigenvalues. The non-negative discriminant of the cubic characteristic polynomial vanishes as these two eigenvalues merge. Turning points of the cubic curve reveal two non-negative invariant factors of the discriminant. These factors are defined in the favorable form of sums of squares by the Cayley–Hamilton theorem, to be used as numerically stable invariants instead of the determinant. Full numerical stability of the procedure is attained by expressing the angular coordinate in terms of the tangent function rather than the cosine, enhanced by range reduction obtained by considering the half angle. This approach combines robust performance in floating point computation with run times that are on par with conventional schemes.