\(qd\) block algorithm (Q608503)

The authors consider the block \(qd\) algorithm for block tridiagonal positive definite symmetric matrices, where the blocks are \(\ell \times \ell\) matrices. It is shown that the eigenvalues \(\lambda_i^{(k)}\), \(k \in N\), of the first block on the block diagonal of the decomposition obtained in the \(k\)th step of the \(qd\) algorithm constitute for all \(i=1,2, \dots, \ell\) a strictly increasing sequence, i.e.~\(\lambda_i^{(k)} < \lambda_i^{(k+1)}\), \(k \in N\). The eigenvalues of the last block constitute a strictly decreasing sequence. Furthermore, the convergence of the block \(qd\) algorithm is proved.