On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies

Abstract

Abstract. The paper analyzes special cyclic Jacobi methods for symmetric matrices of order 4. Only those cyclic pivot strategies that enable full parallelization of the method are considered. These strategies, unlike the serial pivot strategies, can force the method to be very slow or very fast within one cycle, depending on the underlying matrix. Hence, for the global convergence proof one has to consider two or three adjacent cycles. It is proved that for any symmetric matrix A of order 4 the inequality S(A [2] ) ≤ (1 − 10 −5 )S(A) holds, where A [2] results from A by applying two cycles of a particular parallel method. Here S(A) stands for the Frobenius norm of the strictly upper-triangular part of A. The result holds for two special parallel strategies and implies the global convergence of the method under all possible fully parallel strategies. It is also proved that for every ǫ > 0 and n ≥ 4 there exist a symmetric matrix A(ǫ) of order n and a cyclic strategy, such that upon completion of the first cycle of the appropriate Jacobi method the inequality S(A [1] ) > (1 − ǫ)S(A(ǫ)) holds.

Jacobi method for symmetric 4 × 4 matrices converges for every cyclic pivot strategy

“…Let us list the orderings from C 0 ∩ C (4) sp . First we list those from C 0 ∩ C (4) c : (1,4), (2,4), (3,4), (1,4), (2,4), (3,4), (1,4), (3,4), (2,4), (1,4), (3,4), (2,4), (2,4), (1,4), (3,4), (2,4), (1,4), (3,4), (2,4), (3,4), (1,4), (2,4), (3,4), (1,4), (3,4), (1,…”

“…The orderings that belong to C 0 ∩ ← − C r (4) are the following: (2,4), (3,4), (2,4), (3,4), (3,4).…”

“…Furthermore, the result implies that any cyclic J-Jacobi method (see [15,9]) for solving the generalized eigenvalue problem Ax = λJx is globally convergent for n = 4. The global convergence of the cyclic Jacobi methods for symmetric matrices has been considered in [4,5,10,13,14,11] and lately in [1,2,6,7]. These latest results have sufficiently enlarged the class of "convergent" strategies to make the general proof in the case n = 4 possible.…”

“…The convergence properties of the Jacobi method greatly vary for different pivot strategies [2]. Say, for some parallel strategies from Section 3.3 and some initial symmetric matrices, the quotient S(A (N ) )/S(A) can be arbitrary close to 1 [2,Proposition 5.2], and in the second cycle that quotient can be as small as 10 −26 [2, Example 5.1]. Hence, for convergence considerations one has to take into account more than one cycle.…”

“…In Section 3 different classes of cyclic pivot strategies are analyzed. Specifically, the generalized serial pivot strategies from [1,7] are studied in Subsections 3.1 and 3.2 and the parallel strategies from [1,2] in Subsection 3.3. In Section 4 the proof of the main theorem is presented.…”