This is the companion repository to the paper
- L. Aceto, F. Durastante. Theoretical error estimates for computing the matrix logarithm by Pade-type approximants.
Abstract In this article, we focus on the error that is committed when computing the matrix logarithm using the Gauss–Legendre quadrature rules. These formulas can be interpreted as Pade approximants of a suitable Gauss hypergeometric function. Empirical observation tells us that the convergence of these quadratures becomes slow when the matrix is not close to the matrix identity, thus suggesting the usage of an inverse scaling and squaring approach for obtaining a matrix with this property. The novelty of this work is the introduction of error estimates that can be used to select a priori both the number of Legendre points needed to obtain a given accuracy and the number of inverse scaling and squaring to be performed. We include some numerical experiments to show the reliability of the estimates introduced.
The code can be obtained by doing
git clone https://github.com/Cirdans-Home/padelogarithm.git
It use code from the Chebfun project, that is included here as a Git submodule. You can install Chebfun on your machine by doing
unzip('https://github.com/chebfun/chebfun/archive/master.zip')
movefile('chebfun-master', 'chebfun'), addpath(fullfile(cd,'chebfun')), savepath
The examples in the paper were made with the 5.7.0 version, to be sure of using it you can check out the appropriate commit by doing
git submodule init
and then add to your Matlab path the version in the cloned repository folder.
The folder scalingandsquaring contains a copy as is of MATLAB's logm
function (Higham & Relton) with an added print statement that tells the
user what choices have been made for the number of scaling and squaring
step and the number of terms in the diagonal Padé expansion. This has been
used to make the comparison reported in the paper.
For information about that function see
- A. H. Al-Mohy and Nicholas J. Higham, Improved inverse scaling and squaring algorithms for the matrix logarithm, SIAM J. Sci. Comput., 34(4), (2012), pp. C153-C169.
- A. H. Al-Mohy, Nicholas J. Higham and Samuel D. Relton, Computing the Frechet derivative of the matrix logarithm and estimating the condition number, SIAM J. Sci. Comput., 35(4), (2013), C394-C410.
To focus on the error of the rational approximation more than to the numerical conditioning of the computation of the inverses, one of the example using the bound based on pseudospectra uses computation in quadruple precision. These are made available by the advanpix library.