next up previous contents index Next: Computing Interior
Eigenvalues Up: Jacobi-Davidson Methods   G. Sleijpen Previous:
Generalization of Hermitian Case   Contents   Index  Schur Form
and Restart  If we want to include restarts and deflation then
matters become more complicated since non-Hermitian matrices do
not have orthonormal eigensystems in general. Since we prefer to
work with an orthonormal basis for at least the test subspace,
we compute Schur forms of the reduced matrices. Instead of
eigenvectors of the matrix $A$, we compute a partial Schur form
$A{Q}_{k}={Q}_{k}{R}_{k}$, where ${Q}_{k}$ is an $n$ by ${k}$
orthonormal matrix and ${R}_{k}$ is ${k}$ by ${k}$ upper
tridiagonal. A scheme that accommodates for Schur forms is given
in Algorithm 7.18. This scheme includes the possibility for
restart when the dimension of the current subspace exceeds
${m}_{\max}$.  The scheme computes ${k}_{\max}$ eigenvalues
close to a target $\tau $ in the complex plane. Here we have to
be necessarily inexact, since the