Here we give a "one-page" overview of the Jacobi-Davidson method and literature.
The Topics page contains the same references
and scheme, but less explaining text.
See also the Bibliography page for more detailed information.
Origin
The Jacobi-Davidson method was proposed in 1996 by
Gerard Sleijpen and
Henk van der Vorst;
see [SVo96].
A reprint of this SIMAX paper appeared in 2000 in SIAM Review [SVo00].
The method combines ideas of the Davidson method [Dav75]
and Jacobi's method [Jac45], [Jac46],
stemming from 1975 and the 1840s (!), respectively.
Jacobi-Davidson as subspace method
Jacobi-Davidson is a subspace method for the eigenvalue problem, which
means that approximations to eigenpairs are sought from a subspace.
The main two stages in a subspace method are the
subspace extraction (how to get useful approximations from the subspace)
and the subspace expansion (how to expand the subspace).
Subspace extraction
The original paper [SVo96]
discusses the standard Rayleigh-Ritz and the harmonic Rayleigh-Ritz
[Mor91],
[PPV95],
[SVo96],
[Bea98],
[Ste01][SEs03]
as subspace extractions.
But the method can also be combined with refined Rayleigh-Ritz
[Jia97],
[FJi05],
or variations on the harmonic and refined Rayleigh-Ritz method
[Hoc05],
[Hocpp].
Subspace expansion
The subspace expansion of Jacobi-Davidson is done by approximately
solving a correction equation (cf. the "logo" at the top of this page).
See [Not02], [SEl02],
[Sta07a],
and [HNo09] for studies when to stop the inner iteration.
Instead of the usual vector projections, subspace projections have also been tried
for the correction equation [GSl99];
see also [SVM98].
Convergence properties have been studied in for instance
[Esh02],
[HSl03],
[Ovt03a],
[Ovt03b],
[FSp07].
Other important issues
Apart from the subspace extraction and subspace expansion,
in practice the Jacobi-Davidson may often need restarts
[SSa98],
and deflation
[FSV98b],
[Sta07b].
The possibility of preconditioning of the correction equation
is one of the key aspects and main advantages of Jacobi-Davidson: see
[Bas99],
[GSV00],
[GSV03],
[BPS03],
[SWu04],
[AGe05].
Parallelization is discussed in
[NPl99],
[NPl00],
[ABG06].
However the relatively common view that Jacobi-Davidson type methods can only
be competitive if a preconditioner is available may not be true in general---in
particular for generalized eigenvalue problems.
The parameters and options in Jacobi-Davidson (such as minimal and maximal dimension of the
search spaces, tolerance and/or number of iterations for the inner iteration,
the use of an initial Krylov space, thick restarts, etc)
may often have a reasonably large influence on its performance.
Relations and comparisons with other methods
Apart from the connection with Jacobi's method and Davidson's method
(see also [Not05]),
the Jacobi-Davidson method also has relations with
the iterative improvement of approximate eigenvalues
[DMW83],
[Don83];
trace minimization
[SWi82],
[STo00];
differential-geometric approaches
[EAS98],
[LEl02],
[ASV04],
[BAG06],
[ABG06b],
[AMS08];
(inexact) Rayleigh quotient iteration
[SEl02],
[Not03];
(inexact) rational Krylov
[LMe99];
and the Riccati equation
[Bra03a],
[Bra03b].
Jacobi-Davidson can also be seen as an inexact Newton method
[SVo96b],
[FSV98a],
[HNo07].
Some more practically oriented comparisons with other methods are provided in
[AGe99],
[Mor00],
[AGe05],
[AHL05].
In conclusion
In Jacobi-Davidson, the matrix is accessed only by
a matrix-vector product, which makes the method "matrix-free".
No exact (shift-and-)inverses are necessary; the correction equation needs
not be solved very accurately and may be preconditioned.
This makes the method attractive for interior eigenvalues of large sparse matrices,
in particular when a preconditioner is available.
