link to my thesis

6.5. A JACOBI–DAVIDSON METHOD FOR COMMUTING MATRICES 103
the N eigenvalues. Therefore it is a plausible choice to compute eigenvalues of the
matrix A pλ by means of a Jacobi–Davidson eigenvalue solver.
This section describes a Jacobi–Davidson eigenvalue solver called the JDCOMM
method. The first important feature of this solver is that it targets on just one
eigenvalue of the matrix A pλ : it tries to compute the smallest real eigenvalue of the
matrix A pλ as it corresponds to the value of the global minimum of a polynomial
p λ . Available options of iterative eigenvalue solvers are to target on eigenvalues with
the smallest or largest magnitudes or smallest or largest real parts. Computing the
smallest real eigenvalue is our own developed option. The second important feature
of the JDCOMM eigenvalue solver is that it takes advantage of the fact that the matrix
A pλ commutes with the other matrices A xi ,...,A xn and the fact that the matrices
A xi ,...,A xn are much sparser than the matrix A pλ . This Jacobi–Davidson type
method for commuting matrices, JDCOMM, is able to outperform a normal Jacobi–
Davidson method by iterating with one of the very sparse matrices A xi in the inner
loop and only iterate with the less sparse matrix A pλ in the outer loop. In general,
in Jacobi–Davidson type methods, most of the work is done in the inner loops and
therefore a speed up is expected.
6.5.2 The JDCOMM expansion phase
As a basis for the developed JDCOMM method we use the algorithm of the JD method
described in [58]. The pseudocode of this JD method is given in Algorithm 2 of this
thesis. To make it possible for the JDCOMM method to iterate with the sparse matrices
A xi in the inner loop and iterate with the less sparse matrix A pλ in the outer loop,
the expansion phase of the JD method, as described in Section 6.2.1, is extended and
modified.
The Jacobi–Davidson expansion phase of the JDCOMM method works as follows.
Suppose we have an approximate eigenpair (θ, v) for the matrix A pλ , where v has
unit norm and where θ is the Rayleigh quotient of A pλ and v defined as θ = v ∗ A pλ v.
The deviation of an approximate eigenpair (θ, v) to a true eigenpair of the matrix
A pλ is similarly as in Equation (6.2) measured by the norm of the residual r, defined
here as:
r = A pλ v − θv. (6.60)
We now look for an update t ⊥ v such that the updated vector v + t is a true
eigenvector of the matrix A pλ :
Rearranging the terms gives:
A pλ (v + t) =λ(v + t). (6.61)
(A pλ − θI)t = −r +(λ − θ)v +(λ − θ)t. (6.62)
Analogously as in Section 6.2.1, we derive the Jacobi–Davidson correction equation
for the JDCOMM method: discarding the (λ − θ)t term, which is asymptotically secondorder
(or third-order if A pλ is symmetric), and projecting out the unknown quantity