Convergence of a Block‐Oriented Quasi‐Cyclic Jacobi Method

Abstract

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<j R UBCE(̺) ij . Proposition 2.21. Let π = (n 1 , . . . , n m ) be a partition of n, O ∈ O(P m ) and suppose J 2 ≤ µ π,̺ for all J ∈ J UBCE(̺) O, where µ ̺ depends on π and ̺. ThenThe assertion holds provided that in both appearances the spectral norm is replaced by the spectral radius.for some orthogonal UBCE(̺) matrices U k , 0 ≤ k ≤ T − 1, of appropriate sizes.If we show that J T ∈ J UBCE(̺) O , first claim will follow from J 2 = J T 2 ≤ µ π,̺ , while the part about the spectral radius will be a consequence of spr( J ) = spr( J T ) ≤ µ π,̺ . Note thatBy Corollary 2.17 we know that [R ij ( U k )] T ∈ R UBCE(̺) ij , hence J T ∈ J UBCE(̺) O . 2.4.1. Permutation equivalence and the block Jacobi operators. Here we derive a similar result for the block Jacobi operators J O and J O(p) . Theorem 2.22. Let π = (n 1 , . . . , n m ) be a partition of n and take O ∈ O(P m ). Let p be a permutation of S m and set O = O(p).(i) If J 2 ≤ µ π,̺ for all J ∈ J UBCEπ(̺) O , where µ π,̺ only depends on π and ̺, then J 2 ≤ µ π,̺ for any J ∈ J UBCEπ p (̺) O .(ii) If J 1 J 2 · · · J d+1 2 ≤ µ π,̺ , for all J 1 , . . . , J d+1 ∈ J UBCEπ(̺) O , where µ π,̺ only depends on π and ̺, then J 1 J 2 · · · J d+1 2 ≤ µ π,̺ for any J 1 , . . . , J d+1 ∈ J UBCEπ p (̺) O .

Implicit standard Jacobi gives high relative accuracy

“…In future work, we will consider how to speed up the new algorithm preserving its three fundamental properties: guaranteed error bounds, preservation of the symmetry, and using only orthogonal transformations. This may require much more sophisticated ideas in the spirit of the ones presented in [20,21,26,27] for the accurate computation of the Singular Value Decomposition.…”

“…Finally, the use of block versions of the implicit Jacobi algorithm apparently would inherit the same accuracy properties and may make the algorithm much faster. See [26,27] for references on block Jacobi procedures.…”

Convergence to diagonal form of block Jacobi-type methods

“…Nowadays, one can find just a few estimates (e.g. [29,45]) and they are proved only for the case π = (1, 1, . .…”

Quadratic convergence of a special quasi-cyclic Jacobi method

“…[8]). Let A be a symmetric matrix of order n. The method performs a sequence of similarity transformations…”

“…Therefore, we call it (cf. [8]), the special quasi-cyclic block-oriented method. The quasi-sweep contains M ordinary Jacobi steps, where…”