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6.5. A JACOBI–DAVIDSON METHOD FOR COMMUTING MATRICES 105
input : A device to compute A pλ x and A xi x for arbitrary vectors x, where
the action with A xi is (much) cheaper than the action with A pλ ;
The starting vector v 1 ;
The tolerance ε;
The maximum number of iterations maxIter
output: An approximate eigenpair (θ, u) ofA pλ
1 t = v ;
2 V 0 =[];
3 for k ← 1 to maxIter do
4 RGS(V k−1 ,t) → V k ;
5 Compute kth columns of W k = A pλ V k ;
6 Compute kth rows and columns of H k = Vk ∗A
p λ
V k = Vk ∗W
k;
7 Extract a Ritz pair (θ, c);
8 v = V k c;
9 r = W k c − θv;
10 Stop if ||r|| ≤ ε;
11 Compute η = v ∗ A xi v;
12 Solve (approximately) t ⊥ v from:
13 (I − vv ∗ )(A xi − ηI)(I − vv ∗ )t = −r
14 end
Algorithm 3: JDCOMM: A Jacobi–Davidson type method for commuting matrices
Remark 6.3. The JDCOMM method requires ‘a device to compute A pλ x and A xi x for
arbitrary vectors x’ as some of the input variables (see Algorithm 3). Both devices
can be implemented with an nD-system as presented in Chapter 5. This makes it
possible to let the JDCOMM method operate in a matrix-free fashion too.
Section 7.5 of the next chapter describes four experiments in which the JDCOMM
method is used to compute the global minima of some multivariate polynomials.