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104 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS
(λ − θ)v by the orthogonal projector (I − vv ∗ ) which also fixes r, gives rise to the
Jacobi–Davidson correction equation:
(I − vv ∗ )(A pλ − θI)(I − vv ∗ )t = −r where t ⊥ v. (6.63)
In the Jacobi–Davidson method, the vast majority of the computational work is spent
in solving the correction equation. Therefore, the simple but crucial idea is to make
use of the sparser matrices A xi , i =1,...,n, that have the same eigenvectors as A pλ .
Since v is an approximate eigenvector for A pλ , it is likewise an approximate eigenvector
for the matrix A xi . Let η be the Rayleigh quotient of A xi and v: η = v ∗ A xi v.
Instead of Equation (6.61) we now want to update the vector v such that we get an
eigenvector for A xi :
A xi (v + t) =µ(v + t) (6.64)
for a certain eigenvalue µ of A xi . Using the approximate value η for A xi , this leads,
similarly to Equation (6.62), to:
where the residual r is defined here as:
(A xi − ηI)t = −r +(µ − η)v +(µ − η)t, (6.65)
r = A xi v − ηv. (6.66)
By again neglecting the higher-order term (µ−η)t and projecting out the unknown
term (µ − η)v by the orthogonal projector (I − vv ∗ ), we get the Jacobi–Davidson
correction equation involving the matrix A xi :
(I − vv ∗ )(A xi − ηI)(I − vv ∗ )t = −r where t ⊥ v. (6.67)
The advantage of this correction equation compared to the correction equation in
(6.63) is that the matrix-vector products with A xi spent in approximately solving
this equation are much cheaper than matrix-vector multiplication with A pλ . Typical
practical examples indicate that the number of non-zeros of A xi is often 10% or less
than the number of non-zeros of A pλ .
6.5.3 Pseudocode of the JDCOMM method
In this subsection we give in Algorithm 3 the pseudocode of the JDCOMM method, the
Jacobi–Davidson type method for commuting matrices. Note that in the subspace
extraction phase of every (outer) iteration (Step 5 of the algorithm) we need to work
with A pλ , since we should head for the leftmost real eigenvalue of A pλ .
We note that in Step 11, we may choose between two procedures. First, we may
just compute η by computing A xi v and subsequently left-multiplying by v ∗ . This
implies an extra cost of one matrix-vector product per iteration. Alternatively, we
can store A xi V .