The rational Krylov subspace method computes eigenvalues and eigenvectors of
a regular pencil,
Rational Krylov starts as a shifted and inverted Arnoldi iteration
with shift and starting vector . It computes an
orthonormal basis using the basic recursion,
Now the idea behind the rational Krylov algorithm is to continue with a new shift , when we want to get approximations to the eigenvalues close to that new shift. The tricky thing is to avoid throwing away all the information gathered in the basis , computed using the old shift . This is indeed possible if we replace the basis with a new basis , which spans the same subspace as but can be interpreted as the orthogonal basis of an Arnoldi recursion (8.18) with the new shift . At the same time the trapezoidal (Hessenberg) matrix is replaced by a new matrix of the same form, .
Rewrite the recursion (8.18) as
It remains to get rid of the factor on the left-hand side,
and this is done in the following way. Make a QR decomposition
of and get
We once again stress that all these transformations are effected without actually performing any operations on the large matrices and . We may even accumulate all the and factors and avoid forming explicitly, thus avoiding all extra work on long vectors. Also note that even if we get an Arnoldi recursion, it does not start at the original starting vector but at , which is a linear combination of all the vectors computed during the whole rational Krylov algorithm.
In a practical implementation, this is combined with locking,
purging, and implicit restart. First run shifted and inverted Arnoldi
with the first shift . When an appropriate number
of eigenvalues around have converged, say after steps,
lock these converged eigenvalues
and purge those that are altogether outside the interesting
region, leaving a -step Arnoldi recursion (8.18).
Then introduce the new shift and follow
the description in this section to get a new basis which
Start at the new shift by operating on the last vector of this new basis
Notice that in step (3) we make just operations with the shifted and inverted matrix, the first basis vectors are those saved from the previous shift . The lock, purge, and deflation operations in step (4) are very similar to those for implicitly restarted Arnoldi as described in §7.6; we have actually used a nonhermitian version of thick restart . In step (5) we choose the new shift as a value in the complex plane where we are interested in studying the behavior of the matrix pencil (8.17). However, a choice too close to an eigenvalue will result in a nearly singular matrix . Moreover, a new shift at a nearly converged Ritz value will make the upper triangular factor in step (6) close to singular.
When the second matrix in the original pencil (8.17) is positive definite, we may choose to make the basis orthogonal. If then is also Hermitian, so is , and we may work with tridiagonal matrices; see .
This formulation of rational Krylov is different from the one used earlier , where the same set of basis vectors is used throughout and the pencil (8.17) is transformed into two Hessenberg matrices. We have found that this new formulation gives a more natural way to continue with a new shift and signal when we risk losing accuracy due to linear dependence.
See the report  for a numerical example from model reduction.