Rational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs, which gives rise to a near-optimal parallelization strategy that allows us to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas.


  1. rational Krylov
  2. orthogonalization
  3. parallelization

MSC codes

  1. 68W10
  2. 65F25
  3. 65G50
  4. 65F15

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Benchmark Collection, Oberwolfach Model Reduction Benchmark Collection, data collection, 2003, https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark.
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Software Environ. Tools 11, SIAM, Philadelphia, 2000.
P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: A state of the art survey, GAMM-Mitt. Ges. Angew. Math. Mech., 36 (2013), pp. 32--52.
M. Berljafa and S. Güttel, A Rational Krylov Toolbox for MATLAB, MIMS EPrint 2014.56, Manchester Institute for Mathematical Sciences, The University of Manchester, Manchester, UK, 2014, software available for download at http://guettel.com/rktoolbox/.
M. Berljafa and S. Güttel, Generalized rational Krylov decompositions with an application to rational approximation, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 894--916, https://doi.org/10.1137/140998081.
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, Manchester, UK, 2015.
R.-U. Börner, O. G. Ernst, and S. Güttel, Three-dimensional transient electromagnetic modelling using rational Krylov methods, Geophys. J. Int., 202 (2015), pp. 2025--2043.
T. A. Davis and Y. Hu, The University of Florida Sparse Matrix Collection, ACM Trans. Math. Software, 38 (2011), pp. 1--25.
V. Druskin, L. Knizhnerman, and V. Simoncini, Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation, SIAM J. Numer. Anal., 49 (2011), pp. 1875--1898, https://doi.org/10.1137/100813257.
V. Druskin, L. Knizhnerman, and M. Zaslavsky, Solution of large scale evolutionary problems using rational Krylov subspaces with optimized shifts, SIAM J. Sci. Comput., 31 (2009), pp. 3760--3780, https://doi.org/10.1137/080742403.
K. Gallivan, E. Grimme, and P. Van Dooren, A rational Lanczos algorithm for model reduction, Numer. Algorithms, 12 (1996), pp. 33--63.
L. Giraud and J. Langou, When modified Gram--Schmidt generates a well-conditioned set of vectors, IMA J. Numer. Anal., 22 (2002), pp. 521--528.
T. Göckler and V. Grimm, Uniform approximation of $\varphi$-functions in exponential integrators by a rational Krylov subspace method with simple poles, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1467--1489, https://doi.org/10.1137/140964655.
G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013.
L. Grasedyck, Existence of a low rank or ${\mathcal H}$-matrix approximant to the solution of a Sylvester equation, Numer. Linear Algebra Appl., 11 (2004), pp. 371--389.
E. Grimme, Krylov Projection Methods for Model Reduction, Ph.D. thesis, University of Illinois at Urbana-Champaign, Champaign, IL, 1997.
S. Gugercin, A. C. Antoulas, and C. Beattie, ${\mathcal H}_2$ model reduction for large-scale linear dynamical systems, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 609--638, https://doi.org/10.1137/060666123.
S. Güttel, Rational Krylov Methods for Operator Functions, Ph.D. thesis, Institut für Numerische Mathematik und Optimierung der Technischen Universität Bergakademie Freiberg, Freiberg, Germany, 2010.
S. Güttel, Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection, GAMM-Mitt. Ges. Angew. Math. Mech., 36 (2013), pp. 8--31.
S. Güttel, R. V. Beeumen, K. Meerbergen, and W. Michiels, NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems, SIAM J. Sci. Comput., 36 (2014), pp. A2842--A2864, https://doi.org/10.1137/130935045.
E. Jarlebring and H. Voss, Rational Krylov for nonlinear eigenproblems, An iterative projection method, Appl. Math., 50 (2005), pp. 543--554.
G. Lassaux and K. Willcox, Model reduction for active control design using multiple-point Arnoldi methods, AIAA Paper, 616 (2003), pp. 1--11.
R. B. Lehoucq and K. Meerbergen, Using generalized Cayley transformations within an inexact rational Krylov sequence method, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 131--148, https://doi.org/10.1137/S0895479896311220.
J.-R. Li and J. White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 260--280, https://doi.org/10.1137/S0895479801384937.
C. C. Paige, B. N. Parlett, and H. A. van der Vorst, Approximate solutions and eigenvalue bounds from Krylov subspaces, Numer. Linear Algebra Appl., 2 (1995), pp. 115--133.
A. Ruhe, Rational Krylov sequence methods for eigenvalue computation, Linear Algebra Appl., 58 (1984), pp. 391--405.
A. Ruhe, Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils, SIAM J. Sci. Comput., 19 (1998), pp. 1535--1551, https://doi.org/10.1137/S1064827595285597.
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003.
D. Skoogh, A parallel rational Krylov algorithm for eigenvalue computations, in Applied Parallel Computing Large Scale Scientific and Industrial Problems, B. K\ragström, J. Dongarra, E. Elmroth, and J. Waśniewski, eds., Lecture Notes in Comput. Sci. 1541, Springer, Berlin, 1998, pp. 521--526.
G. W. Stewart, Backward error bounds for approximate Krylov subspaces, Linear Algebra Appl., 340 (2002), pp. 81--86.
R. Van Beeumen, K. Meerbergen, and W. Michiels, Compact rational Krylov methods for nonlinear eigenvalue problems, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 820--838, https://doi.org/10.1137/140976698.
H. A. Van der Vorst, Computational methods for large eigenvalue problems, in Handb. Numer. Anal., VIII, North--Holland, Amsterdam, 2002, pp. 3--179.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: S197 - S221
ISSN (online): 1095-7197


Submitted: 9 June 2016
Accepted: 10 January 2017
Published online: 26 October 2017


  1. rational Krylov
  2. orthogonalization
  3. parallelization

MSC codes

  1. 68W10
  2. 65F25
  3. 65G50
  4. 65F15



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