On Kogbetliantz's SVD Algorithm in the Presence of Clusters

We consider matrices with off-diagonal blocks of small norm and derive tight bounds for the approximation of their singular values by those of their diagonal blocks. These results are used to show that triangular matrices with clusters of singular values must possess a principal submatrix of “nearly” diagonal form. From the latter we then derive results pertaining to the quadratic convergence of Kogbetliantz’s algorithm for computing the SVD, in the presence of clusters.

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