On real logarithms of nearby matrices and structured matrix interpolation
type: Item
instance of: scholarly article
title: On real logarithms of nearby matrices and structured matrix interpolation
zbMATH Open document ID: 0930.65045
DOI: 10.1016/S0168-9274(98)00027-0
author: Luca Dieci
author: Benedetta Morini
author: Alessandra Papini
author: Aldo Pasquali
published in: Applied Numerical Mathematics
publication date: 2000-02-01T00:00:00Z
review text: Two different, but related problems are considered: The computation of the real logarithm of a matrix, given the logarithm of a nearby matrix; the second problem is to interpolate given structured matrix data by a smooth matrix function, within a class of matrices having the same structure (structured here means symplectic, positive definite, or orthogonal). The logarithm of a nearby matrix is obtained by updating the known logarithm with a Taylor series expansion or a Newton iteration method. The Fréchet derivative can be given an integral representation which could be evaluated with a quadrature formula. However, it might be advantageous to not compute the principal logarithm because this could give a discontinuity if we want to keep the imaginary parts of the eigenvalues of the logarithm in \((-\pi,\pi)\). For the interpolation problem, the structure of the interpolant can be guaranteed by transforming the data, interpolating the transformed data by a structured polynomial matrix, and then transforming back. This method guarantees the smoothness of the interpolant. The transform could be a Cayley transform, but also a log transform with inverse transform exp. This relates the two problems. Theoretical analysis and error estimates are given, but also implementation details and several numerical examples are included.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65D05
zbMATH DE Number: 1311283
zbMATH Keywords: matrix logarithm
zbMATH Keywords: structured matrices
zbMATH Keywords: Taylor series expansion
zbMATH Keywords: Newton iteration method
zbMATH Keywords: error estimates
zbMATH Keywords: numerical examples
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
cites work: Polynomial and rational matrix interpolation: theory and control applications
cites work: Consideration on computing real logarithms of matrices, Hamiltonian logarithms, and skew-symmetric logarithms
cites work: Real Hamiltonian logarithm of a symplectic matrix
cites work: Computational Techniques for Real Logarithms of Matrices
cites work: Q4040931
cites work: Adaptive Linear Equation Solvers in Codes for Large Stiff Systems of ODEs
cites work: Padé error estimates for the logarithm of a matrix
cites work: Condition Estimates for Matrix Functions
cites work: Note on real matrices and linear dynamical systems with periodic coefficients
cites work: The Sensitivity of the Matrix Exponential
description: scientific article; zbMATH DE number 1311283