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6.2. THE JACOBI–DAVIDSON METHOD 83
which remain in the inner part of the eigenvalue spectrum. A solution for this is
to use the factorization of a shifted version of the matrix to solve linear systems of
equations. However, such a factorization can be too expensive or even unfeasible,
which make these methods unattractive. The Jacobi–Davidson method introduced
and implemented by Sleijpen and van der Vorst [39],[94], called JDQR or JDQZ, avoids
this problem by using an iterative approach to solve this system of equations. In
this setting an approximate and therefore ‘cheap’ solution is accurate enough to have
a fast convergence. Nowadays, the Jacobi–Davidson method has a well-understood
convergence behavior and is considered one of the best eigenvalue solvers, especially
for interior eigenvalues.
6.2 The Jacobi–Davidson method
Suppose a large and sparse N × N real or complex matrix A is given. The Jacobi–
Davidson method computes one or more relevant eigenvalues / eigenvector pairs (λ, x)
of the matrix A which satisfy:
Ax = λx. (6.1)
Here ‘relevant’ refers to the location of an eigenvalue in a specific part of the eigenvalue
spectrum: for example the eigenvalue with the smallest magnitude, the largest real
part or the eigenvalue close to some target value τ. In the case of global polynomial
optimization the most relevant eigenvalue would be the smallest real eigenvalue of
the matrix A pλ . The deviation of an approximate eigenpair (θ, u) to a true eigenpair
(λ, x) is measured by the norm of the residual r, defined as:
r = Au − θu. (6.2)
The Jacobi–Davidson method is a subspace method which means that approximate
eigenvectors are sought in a subspace U [58]. Each iteration of such a method
consist of two main phases: the subspace expansion phase and the subspace extraction
phase. Both phases cooperate to obtain a fast and accurate convergence towards the
relevant eigenvalues.
6.2.1 Subspace expansion
In the subspace expansion phase, we start from an approximate eigenpair (θ, u) of
(λ, x), where u is an element of a low-dimensional subspace U and where θ is the
Rayleigh quotient of u:
θ = u∗ Au
u ∗ (6.3)
u
Note that this definition of θ implies that r ⊥ u:
u ∗ r = u ∗ (Au − θu) =u ∗ Au − θu ∗ u =0. (6.4)
Obviously, one may require u to be normalized to have norm 1 without loss of generality.