Summary. Using a new technique we derive sharp quadratic convergence bounds for the serial symmetric and SVD Jacobi methods. For the symmetric Jacobi method we consider the cases of well and poorely separated eigenvalues. Our result implies the result proposed, but not correctly proved, by Van Kempen. It also extends the well-known result of Wilkinson to the case of multiple eigenvalues.

The paper studies the global convergence of the block Jacobi method for symmetric matrices. Given a symmetric matrix A of order n, the method generates a sequence of matrices by the rule A (k+1) = U T k A (k) U k , k ≥ 0, where U k are orthogonal elementary block matrices. A class of generalized serial pivot strategies is introduced, significantly enlarging the known class of weak wavefront strategies, and appropriate global convergence proofs are obtained. The results are phrased in the stronger form: S(A ′ ) ≤ cS(A), where A ′ is the matrix obtained from A after one full cycle, c < 1 is a constant and S(A) is the off-norm of A. Hence, using the theory of block Jacobi operators, one can apply the obtained results to prove convergence of block Jacobi methods for other eigenvalue problems, such as the generalized eigenvalue problem. As an example, the results are applied to the block J-Jacobi method. Finally, all results are extended to the corresponding quasi-cyclic strategies.Proof. The proof is the same as the proof of [18, Lemma 4.4].If the spectral norm is used instead of the spectral radius, then we have the following result. (2.7). Suppose that in the chain there are exactly d pairs of neighboring terms that are shift equivalent. IfThe constant µ π,̺ may depend only on π and ̺.Proof. The proof is the same as the proof of [18, Lemma 4.8(ii)]. The role of the set Ψ π from [18, Lemma 4.8(ii)] is played by the set i

Summary. This paper deals with quadratic convergence estimates for the serial J-symmetric Jacobi method recently proposed by Veseli6. The method is characterized by the use of orthogonal and hyperbolic plane rotations. Using a new technique recently introduced by Hari we prove sharp quadratic convergence bounds in the general case of multiple eigenvalues.

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An improvement of the Jacobi singular value decomposition algorithm is proposed. The matrix is first reduced to a triangular form. It is shown that the row-cyclic strategy preserves the triangularity. Further improvements lie in the convergence properties. It is shown that the method converges globally and a proof of the quadratic convergence is indicated as well. The numerical experiments confirm these theoretical predictions. Our method is about 2-3 times slower than the standard QR method but it almost reaches the latter if the matrix is diagonally dominant or of low rank.