Search

We analyze two algorithms for computing the symplectic factorization A = LLT of a given symmetric positive definite symplectic matrix A. The first algorithm W1 is an implementation of the HHT factorization from Dopico and Johnson (SIAM J. Matrix Anal. Appl. 31(2):650–673, 2009), see Theorem 5.2. The second one is a new algorithm W2 that uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We present a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB. A particular emphasis is given on symplecticity properties of the computed matrices in floating-point arithmetic. [ABSTRACT FROM AUTHOR]

In this paper, we consider the backward error and condition number of the indefinite linear least squares (ILS) problem. For the normwise backward error of ILS, we adopt the linearization method to derive the tight estimations for the exact normwise backward errors. We derive the explicit expressions of the normwise, mixed and componentwise condition numbers for the linear function of the solution for ILS. The tight upper bounds for the derived mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical power method for estimating matrix 1-norm [N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, PA, 2002, Chapter 15] during using the QR-Cholesky method [S. Chandrasekaran, M. Gu, and A.H. Sayed, A stable and efficient algorithm for the indefinite linear least-squares problem, SIAM J. Matrix Anal. Appl. 20(2) (1999), pp. 354–362] for solving ILS. Moreover, we revisit the previous results on condition numbers for ILS and linear least squares problem. The numerical examples show that the derived condition numbers can give sharp perturbation bound with respect to the interested component of the solution. And the linearization estimations are effective for the normwise backward errors. [ABSTRACT FROM AUTHOR]

The spectral norm of an even-order tensor is defined and investigated. An equivalence between the spectral norm of tensors and matrices is given. Using derived representations of some tensor expressions involving the Moore–Penrose inverse, we investigate the perturbation theory for the Moore–Penrose inverse of tensor via Einstein product. The classical results derived by Stewart (SIAM Rev 19:634–662, 1977) and Wedin (BIT 13:217–232, 1973) for the matrix case are extended to even-order tensors. An implementation in the Matlab programming language is developed and used in deriving appropriate numerical examples. [ABSTRACT FROM AUTHOR]