Remember that the equation:

defines up to a constant. That is, you can add the same constant to all and still have the same

The rate with which off-diagonal elements converge to zero in the
**QL** sequence
is given by

which explains why it is so hard to kill off-diagonal terms that correspond to degenerate eigenvalues. If two eigenvalues and are very close then convergence can be slow. This convergence can be accelerated by decomposing:

instead of

So, we can always reconstruct

But the convergence in this procedure will be given by

and by choosing we can, in principle, speed it up enormously.

But the difficulty is that we don't know what is going to be!

A common strategy is to compute the eigenvalues of the leadingdiagonal submatrix ofAand then setk_{s}to thethat is closer toa_{11}.

One can show that the convergence here is cubic or at worst quadratic if eigenvalues are degenerate.

Although in general the
**QL** decomposition
is obtained by a sequence of Householder transformations, for a
tridiagonal matrix Jacobi rotations
**P**_{pq} can be
used and are cheaper. A sequence of

will eliminate

and

The resulting