About the Rational Krylov Toolbox

Overview and Download
Version History
Planned Features
Acknowledgments
References

Overview and Download

In its latest version the Rational Krylov Toolbox [2] contains

an implementation of Ruhe's rational Krylov sequence method [6], allowing for various orthogonalization options, including user-defined inner products, exploitation of complex-conjugate shifts, and rerunning [4],
algorithms for the implicit and explicit relocation of the poles of a rational Krylov space [2],
an implementation of RKFIT [3,4], a robust algorithm for rational L2 approximation, including automated degree reduction, and
the RKFUN class [4] allowing for numerical computations with rational functions, including support for MATLAB Variable Precision Arithmetic and the Advanpix Multiple Precision toolbox [1].

To automatically download and install the Rational Krylov Toolbox, simply copy and paste the following three lines to your MATLAB command window:

 unzip('http://guettel.com/rktoolbox/rktoolbox.zip'); 
 cd('rktoolbox'); addpath(fullfile(cd)); 
 addpath(fullfile(cd,'utils')); savepath

Alternatively, the toolbox can be downloaded manually from http://guettel.com/rktoolbox/rktoolbox.zip. To install it, unpack the zip file and add the main folder 'rktoolbox' and the subfolder 'utils' to your MATLAB path.

Versions

Version	Release	Changes
2.0 (latest)	2015-06-13	RAT_KRYLOV: new pencil structure for (A,B) supported, allowing the use of function handles; allows for rerunning an existing rational Arnoldi decomposition; allows to expand an existing decomposition; parameter structure; supports classical and modified GS with or without reorthogonalisation; can do one iteration of iterative refinement for the linear system solves; fixed bug with near 0 poles; HH2TH: can now deal with real-valued quasi-pencils; RKFIT: generalized to nondiagonal approximants; automated degree reduction; allows for multiple "functions" F and a block of vectors in B; allows for weighting matrices D; changed to relative misfit; added output structure for more insight; rational function now returned as a RKFUN object; added utility function to convert data to real form if possible; RKFUN: more efficient evaluation via rerunning; implemented as a class with 24 new methods; including root-finding, conversion to partial fraction form, and differentiation; OTHER: new utils subfolder; added several examples and tests
1.0	2015-02-04	First version of the toolbox

Planned features

The Rational Krylov Toolbox is under continuous development and new features will be added over time. Here is our current todo list:

Make move_poles_expl preserve real pencils with complex conjugate eigenpairs.
Add matrix function codes [5] to the toolbox, like invsqrtmv and logmv currently available at http://guettel.com/markovfunmv/.
Add more unit tests for all functionalities.
Add an optional waitbar to rat_krylov.
Add utility file for solving nonlinear eigenvalue problems.
Allow the first input argument of rkfit to be a RKFUN object.

Acknowledgments

This web site was generated using MATLAB's publish command. The convenient 2-line Matlab code for automated download and installation of this toolbox was adapted from a similar code on the Chebfun website.

References

[1] Advanpix LLC., Multiprecision Computing Toolbox for MATLAB, ver 3.8.3.8882, Tokyo, Japan, 2015. http://www.advanpix.com.

[2] M. Berljafa and S. Güttel, A Rational Krylov Toolbox for MATLAB, MIMS EPrint 2014.56, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, 2014. Available at http://eprints.ma.man.ac.uk/2209/.

[3] M. Berljafa and S. Güttel, Generalized rational Krylov decompositions with an application to rational approximation, accepted for publication in SIAM J. Matrix Anal. Appl., 2015. MIMS EPrint 2014.59, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, 2014. Available at http://eprints.ma.man.ac.uk/2278/.

[4] M. Berljafa and S. Güttel, The RKFIT algorithm for nonlinear rational approximation, MIMS EPrint 2015.38, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, 2014. Available at http://eprints.ma.man.ac.uk/2309/.

[5] S. Güttel and L. Knizhnerman, A black-box rational Arnoldi variant for Cauchy--Stieltjes matrix functions, BIT Numer. Math., 53(3):595--616, 2013.

[6] A. Ruhe, Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils, SIAM J. Sci. Comput., 19(5):1535--1551, 1998.