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).
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].
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.
Special attention has been devoted to the situation where the eigenvalue problem is Hermitian or real symmetric [SSa98], [Bas99], [Esh02], [Geu02], [Not02], [BPS03]; real [NRo07]; nonnormal [Sta02], [HSl03]; or complex symmetric [AHo04], [CAS05], [Chi05].
Block variants of Jacobi-Davidson are discussed in [Geu02], [Bra03a]. The Jacobi-Davidson method has been generalized for various other eigenvalue problems, including the generalized eigenvalue problem [SBF96], [FSV98b], [HSl03], [HNo07], [Rom08]; the polynomial eigenvalue problem [SVG96], [SBF96], [Mee00], [TMe01], [HSl03], [HNo07], [HSl08]; the multiparameter eigenvalue problem [HPl02], [HKP05]; the nonlinear eigenvalue problem [BVo04], [Vos04a], [Vos04b], [MVo04], [Vos06], [Vos07b]; the (generalized) singular value problem [Hoc01], [Hoc04], [Hoc09]; and the product eigenvalue problem [Hoc08].
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].
Use in applications
The Jacobi-Davidson has been used in various applications, including pole-zero analysis [Rom02], [RBV04], [RVM04], [Rom07]; chemistry [YPN01]; magneto-hydrodynamics (MHD) [BFV96], [NPl00]; electromagnetism, opto-electronics [AGe99], [NPl00], [AGe01], [Geu02], [SWF02], [AGe05]; oceanography [DOW01], [WDO03], [DGh05], [Dij05], [Dij06]; and continuation [Dor97].
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.
For more information on Jacobi-Davidson and other eigenvalue methods, see the Bibliography and Topics pages and in particular the books [BDD00], [Ste01], [Vor02] and the review papers [VGo97], [Vor99], [Sor02], [Vor04], [HNo06].
See the Other links page for links to some competitive methods such as ARPACK and the LOBPCG.