zbMATH — the first resource for mathematics

A robust algorithm for finding the eigenvalues and eigenvectors of \(3 \times 3\) symmetric matrices. (English) Zbl 1197.65037
Summary: 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 \(3 \times 3\) 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.

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Full Text: DOI
[1] Franca, L.P., An algorithm to compute the square root of a 3×3 positive definite matrix, Comput. math. appl., 18, 5, 459-466, (1989) · Zbl 0686.65019
[2] Hartmann, S., Computational aspects of the symmetric eigenvalue problem of second order tensors, Tech. mech., 23, 2-4, 283-294, (2003)
[3] Higham, N.J., Stable iterations for the matrix square root, Numer. algor., 15, 227-242, (1997) · Zbl 0884.65035
[4] Hill, R., Aspects of invariance in solid mechanics, Adv. appl. mech., 18, 1-75, (1978) · Zbl 0475.73026
[5] Golub, G.H.; van Loan, C.F., Matrix computations, (1983), The Johns Hopkins University Press Baltimore, MD · Zbl 0559.65011
[6] Malvern, L.E., Introduction to the mechanics of a continuous medium, (1969), Prentice-Hall Englewood Cliffs, NJ · Zbl 0181.53303
[7] Rashid, M.M., Incremental kinematics for finite element applications, Int. J. numer. methods engrg., 36, 3937-3956, (1993) · Zbl 0788.73073
[8] Simo, J.C.; Hughes, T.J.R., Computational inelasticity, (1998), Springer-Verlag New York, NY · Zbl 0934.74003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.