Approximation of Matrices and a Family of Gander Methods for Polar Decomposition

Abstract

Two matrix approximation problems are considered: approximation of a rectangular complex matrix by subunitary matrices with respect to unitarily invariant norms and a minimal rank approximation with respect to the spectral norm. A characterization of a subunitary approximant of a square matrix with respect to the Schatten norms, given by Maher, is extended to the case of rectangular matrices and arbitrary unitarily invariant norms. Iterative methods, based on the family of Gander methods and on Higham's scaled method for polar decomposition of a matrix, are proposed for computing subunitary and minimal rank approximants. Properties of Gander methods are investigated in details. (2000): 65F30, 15A18. AMS subject classification

On a sub‐Stiefel Procrustes problem arising in computer vision

“…j .A//. The orthogonal matrices U A and V A from the SVD are not unique in the general case (for a relation between orthogonal matrices from two singular value decompositions of A, see Lemma in [19]; compare Problem B.11 in [20, p. 334] and [21]). Let A D P † A Q T be another SVD of A, let † A D diag.…”

Asynchronous Task-Based Polar Decomposition on Single Node Manycore Architectures

“…The polar decomposition algorithm has been well studied in the last three decades in terms of complexity and numerical robustness/accuracy [14], [15], [16], [17], [18], [19], [20]. Initially designed with Newton's method based on an explicit matrix inversion calculation, numerical instability was reported, especially in presence of ill-conditioned matrices.…”

Perturbation analysis for best approximation and the polar factor by subunitary matrices

“…Moreover, the perturbation bounds of Q k and H k can be obtained by using the results for the generalized polar decomposition (see Ref. [15]). …”

“…In Ref. [15], Laszkkiewicz and Zietak proved that for all subunitary matrices Q ∈ U m×n , the following inequality holds:…”

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