The Jacobi rotations method is not too bad. It usually converges
in between 18 *N*^{3} to 30 *N*^{3} operations. There is no way
to get out of the *N*^{3} dependence, but the coefficient
can be reduced to
(for diagonalization without eigenvectors),
and that can be
a very considerable saving, especially if *N* is very large.

The most efficient known technique for finding eigenvalues and
eigenvectors of a symmetric matrix is the combination of
the Householder reduction, which reduces a symmetric real matrix
to a tridiagonal form, followed by the so called QR or a QL
algorithm that can diagonalize a tridiagonal matrix within
about 30 *N*^{2} steps without eigenvectors. If eigenvectors
are required then the number of operations grows to 3 *N*^{3}.

The QR or a QL method is still an iterative method. But the orthogonal transformation employed preserves:

- symmetry, and
- tridiagonal form

The Householder reduction on the other hand is a finite
procedure, i.e., not an iterative one at all. A symmetric
matrix can be reduced to a tridiagonal form within a finite
well defined number of steps: *N*-2 orthogonal transformations.