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
denote the accepted eigenvector
approximations and let us assume that these vectors are orthonormal. The
matrix
has the vectors
as its columns. In order to
find the next eigenvector
, we apply the Jacobi-Davidson
algorithm to the deflated matrix^{}