An iterative method for computing a symplectic SVD-like decomposition

Abstract

In this paper, we present an effective iterative method for computing symplectic SVD-like decomposition for a 2n-by-m rectangular real matrix A. The main purpose here is a block-power iterative method with the ortho-symplectic SR decomposition in the normalization step. We compute an k-block symplectic SVD-like decomposition, namely \(A_k=S_{k} \Sigma _{k} V_{k}^{T}\) where \(S_{k}\in \mathbb {R}^{2n\times 2k}\) is symplectic and \(V_{k}\in \mathbb {R}^{m\times 2k}\) is orthogonal. For large matrices, we usually have to compute only part of eigenvalues. The main interest of the proposed method is to efficiently compute the desired number of ordered in magnitude eigenvalues of structured matrices.