On Kogbetliantz's SVD algorithm in the presence of clusters (Q1092610)

In extending Jacobi's method to \textit{E. Kogbetliantz}'s singular value decomposition (KSVD) algorithm [Quart. Appl. Math. 13, 123-132 (1955; Zbl 0066.101)] for an \(m\times n\) matrix A, \(m\geq n\), more properties carry over for (square) triangular matrices than for general matrices; notably, the span of the singular values remains bounded in the iteration, a fact crucial to convergence properties of the KSVD. Let A be a (complex) triangular matrix with a cluster C of k singular values, of width \(\eta\). Let the singular values of A be at least \(2\delta\) distant from each other, except those in C. Let the diagonal entries which converge to the elements of C occupy successive positions on the main diagonal, then for A the KSVD algorithm is ultimately quadratic; precisely, if \(\| A^{(r)}_{off}\|_ F<\delta,\eta =c\| A^{(r)}_{off}\|^ 2_ F/\delta,\) then after the next \(N=n(n-1)\) rotations, \(\| A_{off}^{(r+N)}\|_ F<c'\| A^{(r)}_{off}\|^ 2_ F/\delta,\) where \(c'=[(k-1)^ 2(4+c)^ 2+8]^{1/2}.\) Some numerical results are given.