Iteration schemes for the divide-and-conquer eigenvalue solver (Q1330380)

The heart of the recent divide-and-conquer approaches is a method to update the eigenvalues of a matrix after a rank-1 update, which would in this case connect again parts of the matrix which had been deliberately broken in the divide stage. Here a globally convergent iteration scheme is proposed. The updating problem is split into the stages separation, isolation and approximation. The first stage separates the eigenvalues from the rest of the spectrum, the second stage isolates single eigenvalues, and the last stage approximates it up to the desired precision. The proposed method has the further advantage of having a lower sequential algorithmic complexity than other known methods. The results are demonstrated on some numerical examples.