Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Numerical Algorithms
Published in: Numerical Algorithms 4/2020

04-03-2020 | Original Paper

Extended nonsymmetric global Lanczos method for matrix function approximation

Authors: A. H. Bentbib, M. El Ghomari, K. Jbilou

Published in: Numerical Algorithms | Issue 4/2020

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Extended Krylov subspace methods are attractive methods for computing approximations of matrix functions and other problems producing large-scale matrices. In this work, we propose the extended nonsymmetric global Lanczos method for solving some matrix approximation problems. The derived algorithm uses short recursive relations to generate bi-orthonormal bases, with respect to the Frobenius inner product, of the corresponding extended Krylov subspaces \({K^{e}_{m}}(A,V)\) and \({K^{e}_{m}}(A^{T},W)\). Here, A is a large nonsymmetric matrix; V and \(W\in \mathbb {R}^{n\times s}\) are two blocks. New algebraic properties of the proposed method are developed and applications to approximation of both WTf(A)V and trace(WTf(A)V ) are given. Numerical examples are presented to show the performance of the extended nonsymmetric global Lanczos for these problems.
previous article On the calculation of the poles of multivariate meromorphic functions using the symbolic-numeric two-point qd-algorithm
next article Leja, Fejér-Leja and ℜ-Leja sequences for Richardson iteration

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Abidi, O., Heyouni, M., Jbilou, K.: On some properties of the extended block and global Arnoldi methods with applications to model reduction. Numer. Algorithms 75, 285–304 (2017)MathSciNetCrossRef
2.
Barkouki, H., Bentbib, A.H., Heyouni, M., Jbilou, K.: An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systems. Calcolo 55, 13–36 (2018)MathSciNetCrossRef
3.
Baroni, S., Gebauer, R., Malcioglu, O.B., Saad, Y., Umari, P., Xian, J.: Harnessing Molecular Excited States with Lanczos Chains, vol. 22. Art. Id. 074204 (2010)
4.
Bellalij, M., Reichel, L., Rodriguez, G., Sadok, H.: Bounding matrix functionals via partial global block Lanczos decomposition. Appl. Numer. Math. 94, 127–139 (2015)MathSciNetCrossRef
5.
Bellalij, M., Jbilou, K., Sadok, H.: New convergence results on the global GMRES method for diagonalizable matrices. J. Comput. Appl. Math. 219, 350–358 (2008)MathSciNetCrossRef
6.
Bouyouli, R., Jbilou, K., Sadaka, R., Sadok, H.: Convergence properties of some block Krylov subspace methods for multiple linear systems. J. Comput. Appl. Math. 196, 498–511 (2006)MathSciNetCrossRef
7.
Bowler, D.R., Miyazaki, T.: O(n) methods in electronic structure calculations. Rep. Prog. Phys. 75, 43 (2012). Art. Id. 036503CrossRef
8.
9.
Druskin, V., Knizhnerman, L.: Extended Krylov subspaces: approximation of the matrix square root and related functions. SIAM J. Matrix Anal. Appl. 19, 755–771 (1998)MathSciNetCrossRef
10.
Estrada, E.: The Structure of Complex Networks: Theory and Applications. Oxford University Press, Oxford (2011)CrossRef
11.
Fenu, C., Reichel, L., Rodriguez, G., Sadok, H.: GCV for Tikhonov regularization by partial SVD. BIT 57, 1019–1039 (2017)MathSciNetCrossRef
12.
Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton University Press, Princeton (2010)CrossRef
13.
Han, I., Malioutov, D., Shin, J.: Large-scale log-determinant computation through stochastic chebyshev expansions. Proceedings of The 32nd International Conference on Machine Learning 37, 908–917 (2015)
14.
Hansen, P.C.: Rank-deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)CrossRef
15.
Heyouni, M.: Extended Arnoldi methods for large low-rank Sylvester matrix equations. Appl. Numer. Math. 60, 1171–1182 (2010)MathSciNetCrossRef
16.
HIGHAM, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)
17.
Jbilou, K., Sadok, A., Messaoudi, H.: Global FOM and GMRES algorithms for matrix equations. Appl. Numer Math. 31, 49–63 (1999)MathSciNetCrossRef
18.
Jbilou, K., Sadok, H., Tinzeft, A.: Oblique projection methods for multiple linear systems. Elect. Trans. Num. Anal. 20, 119–138 (2005)MATH
19.
Knizhnerman, VL., Simoncini, A: New investigation of the extended Krylov subspace method for matrix function evaluations. Numer. Linear Algebra Appl. 17, 615–638 (2010)MathSciNetMATH
20.
21.
Reichel, L., Rodriguez, G., Tang, T.: New block quadrature rules for the approximation of matrix functions. Linear Algebra Appl. 502, 299–326 (2016)MathSciNetCrossRef
22.
Saad, Y., Chelikowsky, J., Shontz, S.: Numerical methods for electronic structure calculations of materials. SIAM Rev. 52, 3–54 (2010)MathSciNetCrossRef
23.
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)CrossRef
24.
Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29, 1268–1288 (2007)MathSciNetCrossRef
25.
Schweitzer, M.: A two-sided short-recurrence extended Krylov subspace method for nonsymmetric matrices and its relation to rational moment matching. Numer. Algorithms 76, 1–31 (2016)MathSciNetCrossRef
Metadata
Title
Extended nonsymmetric global Lanczos method for matrix function approximation
Authors
A. H. Bentbib
M. El Ghomari
K. Jbilou
Publication date
04-03-2020
Publisher
Springer US
Published in
Numerical Algorithms / Issue 4/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-020-00896-8

Other articles of this Issue 4/2020

Numerical Algorithms 4/2020 Go to the issue

Original Paper

On the calculation of the poles of multivariate meromorphic functions using the symbolic-numeric two-point qd-algorithm

Preface

Special volume on mathematical modeling with applications

Original Paper

Leja, Fejér-Leja and ℜ-Leja sequences for Richardson iteration

Original Paper

Identification of the source function for a seawater intrusion problem in unconfined aquifer

Original Paper

Some unusual results on extrapolation methods

Original Paper

Mixed finite element methods for the Oseen problem

Premium Partner

    Image Credits