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84 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS
The main issue of this phase is how to expand the current search space U with a
new direction to get a better approximation in the next iteration. The key idea here is
to look for an orthogonal direction s ⊥ u such that (u+s) satisfies A(u+s) =λ (u+s),
or in other words such that (u + s) is proportional to the normalized eigenvector x.
This implies that:
which can be rewritten as:
As − θs =(−Au + θu)+(λu − θu)+(λs − θs) (6.5)
(A − θI)s = −r +(λ − θ)u +(λ − θ)s. (6.6)
Because ||s|| and (λ − θ) both are expected to be small and much smaller than the
other terms in this equation, the last term on the right-hand side of (6.6) is neglected.
Furthermore, the term (λ−θ)u is projected out with the orthogonal projection matrix
(I − uu ∗ ). Note that (I − uu ∗ )u = 0 (for ||u|| = 1), (I − uu ∗ )r = r (since r ⊥ u)
and (I − uu ∗ )s = s (upon requiring s ⊥ u). This gives rise to the Jacobi–Davidson
correction equation:
(I − uu ∗ )(A − θI)(I − uu ∗ )s = −r where s ⊥ u. (6.7)
In a Jacobi–Davidson method the vector s is solved approximately (with modest
accuracy) from (6.7) in a matrix-free fashion using, for example, methods as GMRES
of BiCGStab. A preconditioner can be used in this case to prevent for (possible) illconditioning
and to (possibly) speed up the convergence. The vector s is then used
to expand the search space U: the vector s is orthogonalized against all the vectors
in U and added to it.
6.2.2 Subspace extraction
In the subspace extraction phase, the main goal is to compute an approximate eigenpair
(θ, u) for the matrix A with u ∈U. The extraction approach described here is
the Rayleigh-Ritz approach (see [58]).
Let U be an N × k matrix whose columns constitute an orthonormal basis of U
and where k ≪ N. Any vector u ∈U can be written as u = Uc, a linear combination
of the columns of U with c a small k-dimensional vector. To compute an approximate
eigenvector u, the Rayleigh-Ritz approach proceeds by computing c by imposing the
Ritz-Galerkin condition:
r = Au − θu = AUc − θUc ⊥ U (6.8)
which holds if and only if (θ, c) is an eigenpair of the k-dimensional projected eigenproblem:
U ∗ AU c = θc (6.9)
(the orthonormality of U is used here: U ∗ U = I k ). Once this low-dimensional eigenproblem
is solved and an eigenpair (θ, c) is available, the approximate eigenvector