Deflation.

When a Ritz value is close enough to an eigenvalue, the remaining part of the current subspace will already have rich components in nearby eigenpairs, since we have selected in all steps the Ritz vectors for Ritz values close to the desired eigenvalue. We can use this information as the basis for a subspace for the computation of a next eigenvector. In order to avoid the old eigenvector reentering the computational process, we make the new search vectors in the Jacobi-Davidson algorithm explicitly orthogonal to the computed eigenvectors. This technique is called explicit deflation. We will discuss this in slightly more detail.

Let $\widetilde{x}_1,\ldots,\widetilde{x}_{k-1}$ denote the accepted eigenvector approximations and let us assume that these vectors are orthonormal. The matrix $\widetilde{X}_{k-1}$ has the vectors $\widetilde{x}_j$ as its columns. In order to find the next eigenvector $\widetilde{x}_k$, we apply the Jacobi-Davidson algorithm to the deflated matrix

$$(I-\widetilde{X}_{k-1}\widetilde{X}_{k-1}^\ast)\,A\, (I-\widetilde{X}_{k-1}\widetilde{X}_{k-1}^\ast) ,$$

and this leads to a correction equation of the form

$${P}_{m}(I-\widetilde{X}_{k-1}\widetilde{X}_{k-1}^\ast)\, (... ...}\widetilde{X}_{k-1}^\ast) {P}_{m} t_j^{(m)} = -r_j^{({m})} ,$$ (59)

with ${P}_{m}\equiv (I-{u}_j^{(m)}{{u}_j^{(m)}}^{\ast})$, that has to be solved for the correction $t_j^{(m)}$ to each new eigenvector approximation ${u}_j^{(m)}$, with corresponding Ritz value $\theta_j^{(m)}$. In [172] it is shown, by numerical evidence, that the explicit deflation against the vectors represented by $\widetilde{X}_{k-1}$ is highly recommended for the correction equation, but it is not necessary to include this deflation in the computation of the projected matrix (the projection of $A$ onto the subspace spanned by the successive approximations ${v}_j$ in the search for the $k$th eigenvector). The projected matrix can be computed as ${V}_{m}^\ast A {V}_{m}$, without significant loss of accuracy.