Future Generation Computer Systems 22 (2006) 423–429 Numerical methods for computing SVD in the D-orthogonal group T. Politi a, ∗ , A. Pugliese b a Dipartimento di Matematica, Politecnico di Bari, Via Amendola 126/B, I-70126 Bari, Italy b School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA Available online 5 January 2005 Abstract In this paper, we consider the problem to compute a special kind of singular value decomposition of a square matrix A = UΣV , where U and V belong to the same D-orthogonal group, i.e. they are orthogonal with respect to a real diagonal orthogonal matrix D, while Σ is a real diagonal positive definite matrix. In this work, we propose an algebraic method and derive a continuous approach using the projected gradient technique. The differential systems given by the continuous approach are solved using a standard integration solver together with a projection technique obtained computing the D-orthogonal factor of the hyperbolic QR decomposition. © 2004 Elsevier B.V. All rights reserved. Keywords: Singular value decomposition; Quadratic groups; Hypernormal matrices 1. Introduction In this work, we consider the problem of the nu- merical computation of the singular values in the D- orthogonal group, where a real n × n matrix A is de- composed as A = UΣV (1) with Σ nonnegative diagonal matrix and where U and V are two n × n orthogonal matrices with respect the D- orthogonal (or pseudo-orthogonal) inner product, that is U T DU = D, V T DV = D where D denotes the fol- ∗ Corresponding author. E-mail addresses: politi@poliba.it (T. Politi), pugliese@math.gatech.edu (A. Pugliese) lowing diagonal matrix D =  I k -I n-k  , with 1 ≤ k<n. The diagonal elements of Σ are called D-singular values of A. In the previous definition, we can also assume any diagonal sign matrix D belong- ing to the set D k of the diagonal matrices with exactly k elements equal to +1 and the other n - k entries equal to -1. In all cases, D -1 = D. A detailed study on D-orthogonal (or pseudo-orthogonal, or hypernor- mal) matrices may be found in [5,7]. The decomposi- tion (1) is called D-singular value decomposition of A (or D-SVD). In particular, when k = n - 1, it is called 0167-739X/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.future.2004.11.025