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6.2. THE JACOBI–DAVIDSON METHOD 85
for the original problem is computed as u = Uc. The projected low-dimensional
eigenproblem is usually solved by a direct method. The vector u is called a Ritz
vector. Moreover, because of (6.8) r ⊥ u and θ is the Rayleigh quotient of u. Thus,
(θ, Uc) =(θ, u) is the back-transformed eigenpair which acts as an approximation to
(λ, x) of the original problem (6.1).
An important property of the Jacobi–Davidson method is that it allows the user
to focus on a subset of the eigenvalues via the specification of a target, as further
discussed in Section 6.3 of this chapter.
In some cases the Rayleigh-Ritz extraction approach can give poor approximations.
This issue is addressed by the Refined Rayleigh-Ritz extraction method. Given
an approximate eigenpair (θ, c) of the small projected eigenproblem (6.9), this refined
approach solves:
ĉ =
argmin
c∈C k ,||c||=1
|| (AU − θU) c || 2 2. (6.10)
The back-transformed eigenvector û = Uĉ, called the refined Ritz vector, usually
gives a much better approximation than the ordinary Ritz vector u (see [94]). A new
approximate eigenvalue ˆθ can be computed using the Rayleigh quotient (6.3) of û.
Note that both the subspace expansion and the subspace extraction phase of
the Jacobi–Davidson method can be performed in a matrix-free fashion as all the
operations in which the matrix A is involved are matrix-vector products, which can
be computed without having the matrix A explicitly at hand. This enables the use
of the matrix-free nD-systems approach discussed in Chapter 5.
6.2.3 Remaining important issues
In the two previous subsections the main phases of the Jacobi–Davidson method, the
subspace extension phase and the subspace extraction phase, are presented. In this
subsection we discuss the important remaining issues of the Jacobi–Davidson method.
First, during the iterations the search space U grows. When this space reaches the
maximum dimension j max the method is restarted with a search space of dimension
j min containing the j min most promising vectors. A ‘thick’ restart indicates that
the method is restarted with the approximate eigenvectors from the previous j min
iterations. The quantities j min and j max are user specified parameters. Second, before
the first iteration the search space U is empty. One can decide to start with a Krylov
search space of the matrix A of dimension j min to reach the wanted eigenvalue in less
iteration steps. Finally, when one wants to compute more than one eigenvalue the
so-called ‘deflation step’ becomes important. In this step the matrix A is deflated
to a matrix  such that it no longer contains the already computed eigenpair (λ, x):
 =(I − xx ∗ ) A (I − xx ∗ ).
All the parameter options mentioned in this paragraph, and also the assumed tolerance
and the maximum allowed number of iterations, have a considerable influence
on the performance of the method.