The Jacobi-Davidson algorithm is given in Algorithm 8.1. This algorithm attempts to compute the generalized Schur pairs , for which the ratio is closest to a specified target value in the complex plane. The algorithm includes restart in order to limit the dimension of the search space, and deflation with already converged left and right Schur vectors.
To apply this algorithm we need to specify a starting vector , a tolerance , a target value , and a number that specifies how many eigenpairs near should be computed. The value of specifies the maximum dimension of the search subspace. If it is exceeded then a restart takes place with a subspace of dimension .
On completion the generalized eigenvalues close to are delivered, and the corresponding reduced Schur form , , where and are by orthogonal and , are by upper triangular. The generalized eigenvalues are the on-diagonals of and . The computed form satisfies , , where is the th column of .
The accuracy of the computed reduced Schur form depends on the
distance between the target value and the eigenvalue
If we neglect terms of
order machine precision and of order , then we have
where the constants
and are given by
We will now explain the successive main phases of the algorithm.
We expand the subspaces , , , and . denotes the matrix with the current basis vectors for the search subspace as its columns. The other matrices are defined in a similar obvious way.
Note that the scalars can also be computed from the scalars , and the orthogonalization constants of in step (10).
The QZ decomposition for the pair of by matrices can be computed by a suitable routine for dense matrix pencils from LAPACK.
We have chosen to compute the generalized Petrov pairs, which makes the algorithm suitable for computing interior generalized eigenvalues of , for which is close to a specified .
For algorithms for reordering the generalized Schur form, see [448,449,171].
Detection of all wanted eigenvalues cannot be guaranteed; see note (13) for Algorithm 4.17 (p. ).
Of course, the correction equation can be solved by any suitable process, for instance, a preconditioned Krylov subspace method that is designed to solve unsymmetric systems. However, because of the different projections, we always need a preconditioner (which may be the identity operator if nothing else is available) that is deflated by the same skew projections so that we obtain a mapping between and itself. Because of the occurrence of and , one has to be careful with the usage of preconditioners for the matrix . The inclusion of preconditioners can be done as in Algorithm 8.2. Make sure that the starting vector for an iterative solver satisfies the orthogonality constraints . Note that significant savings per step can be made in Algorithm 8.2 if is kept the same for a (few) Jacobi-Davidson iterations. In that case columns of can be saved from previous steps. Also the matrix can be updated from previous steps, as well as its decomposition.
It is not necessary to solve the correction equation very accurately. A strategy, often used for inexact Newton methods , also works well here: increase the accuracy with the Jacobi-Davidson iteration step, for instance, solve the correction equation with a residual reduction of in the th Jacobi-Davidson iteration ( is reset to 0 when a Schur vector is detected).
In particular, in the first few initial steps, the approximate eigenvalue may be very inaccurate, and then it does not make sense to solve the correction equation accurately. In this stage it can be more effective to temporarily replace by or to take for the expansion of the search subspace [335,172].
For the full theoretical background of this method, as well as details on the deflation technique with Schur vectors, see .