A Kogbetliantz-type algorithm for the hyperbolic SVD

Abstract

In this paper a two-sided, parallel Kogbetliantz-type algorithm for the hyperbolic singular value decomposition (HSVD) of real and complex square matrices is developed, with a single assumption that the input matrix, of order n, admits such a decomposition into the product of a unitary, a non-negative diagonal, and a J-unitary matrix, where J is a given diagonal matrix of positive and negative signs. When J = ±I, the proposed algorithm computes the ordinary SVD.The paper's most important contribution-a derivation of formulas for the HSVD of 2 × 2 matrices-is presented first, followed by the details of their implementation in floating-point arithmetic. Next, the effects of the hyperbolic transformations on the columns of the iteration matrix are discussed. These effects then guide a redesign of the dynamic pivot ordering, being already a well-established pivot strategy for the ordinary Kogbetliantz algorithm, for the general, n × n HSVD. A heuristic but sound convergence criterion is then proposed, which contributes to high accuracy demonstrated in the numerical testing results. Such a J-Kogbetliantz algorithm as presented here is intrinsically slow, but is nevertheless usable for matrices of small orders. Keywords hyperbolic singular value decomposition • Kogbetliantz algorithm • Hermitian eigenproblem • OpenMP multicore parallelization Mathematics Subject Classification (2010) 65F15 • 65Y05 • 15A18 This work has been supported in part by Croatian Science Foundation under the project IP-2014-09-3670 "Matrix Factorizations and Block Diagonalization Algorithms" (MFBDA).

Batched computation of the singular value decompositions of order two by the AVX-512 vectorization

“…Should INVSQRT not be available, there are two remedies, both starting from sec α = √ 1 + tan 2 α. The first, faster one computes cos α = 1/ sec α, while the second, possibly more accurate one due to requiring one rounding less than the first [4], does not require the cosine at all, and instead replaces all multiplications by it with divisions by the secant. Now r ′ 12 has to be made real and non-negative by multiplying R ′′ by D from the right, obtaining R ′ as in (8), and then r ′ 22 to undergo a similar procedure Batched computation of the SVDs of order two by the AVX-512 vectorization 15…”

“…Let a finite sequence A = (A [k] ) k , where 1 ≤ k ≤ n, of complex 2 × 2 matrices be given, and let the corresponding sequences U = (U [k] ) k , V = (V [k] ) k of 2×2 unitary matrices be sought for, as well as a sequence Σ = (Σ [k] ) k of 2 × 2 diagonal matrices with the real and non-negative diagonal elements, such that A [k] = U [k] Σ [k] (V [k] ) * , i.e., for each k, the right hand side of the equation is the singular value decomposition (SVD) of the left hand side. This batch of 2 × 2 SVD computational tasks arises naturally in, e.g., parallelization of the Kogbetliantz algorithm [1] for the 2n × 2n SVD [2][3][4]. A parallel step of the algorithm, repeated until convergence, amounts to forming and processing such a batch, with each A [k] assembled column by column from the elements of the iteration matrix at the suitably chosen pivot positions (p k , p k ), (q k , p k ), (p k , q k ), and (q k , q k ).…”