# (approximately) compute absolute largest eigenvalue of symmetrix 3x3 matrix

I need to compute (an approximation may be good enough) the largest (by absolute value) eigenvalue of a real symmetric 3x3 matrix many ($10^{6-12}$) times. Is there anything better than just computing the eigenvalues (say as described here) and then finding the absolute largest?

• @PVAL defined "many". Definitely many more than 20. – Walter Dec 2 '15 at 22:25

For a $3\times3$ matrix, there is nothing wrong with computing all three eigenvalues since the resulting characteristic polynomial is of low degree, and most computer algebra systems are happy to find the roots of low degree polynomials for you.
• I'm only interested in $3\times3$ matrices (as stated). Solving for all 3 eigenvalues can be done using the exact formula for the roots of a 3rd order polynomial, but does call $\arccos$ once and $\cos$ twice, which are computationally expensive. – Walter Dec 2 '15 at 22:22