On Jacobi Methods for Singular Value Decompositions

Abstract

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.

On sharp quadratic convergence bounds for the serial Jacobi methods

“…[18,8]). However, if A is triangular and the pivot strategy is row-or column-cyclic, then the SVD Jacobi methods are as efficient as the corresponding symmetric Jacobi methods (see [9,3,10]). Namely, if A is triangular, then each A (k) is block-triangular whose diagonal blocks are again block-triangular with triangular diagonal blocks.…”

“…[10]). Their global convergence has been proved in [9] and I-5] whereas their quadratic convergence has been proved in [3] and [10]. Here we use the technique from Sect.…”

Numerical Linear Algebra

“…The final form of the QR algorithm for computing SVD, which has been the preferred SVD method for dense matrices up to now, is due to Golub and Reinsch (1970); see Anderson et al (1999), Björck (1996) or Gentle (1998) for the description of the algorithm and some modifications. An alternative approach based on Jacobi algorithm was given by Hari and Veselić (1987). Latest contributions to the pool of computational methods for SVD, including von Matt (1995), Demmel et al (1999) and Higham (2000), aim to improve the accuracy of singular values and computational speed using recent advances in the QR decomposition.…”

Linear convergence of the row cyclic Jacobi and Kogbetliantz methods

“…Heath et al [12] and Hari and Veseli6 [10] have shown that after a sweep of the row cyclic algorithm an upper (lower) triangular matrix becomes a lower (upper) triangular matrix. …”

“…It is given by ffZew(0)=z2(10 , 1)+z2(10, 2)+z2(10, 3)+z2 (10,4) =z2(4, 1)+z2 (7,2) Thus, it is always possible to satisfy the condition (3.4). We note that the angles ~, l= 1, 5, 8, t0 do not have to be constrained as they do not occur in our analysis.…”