A new algorithm for computing eigenpairs of matrices

In this paper, we have proposed a hybrid of restarted Arnoldi algorithm and particle swarm optimization (PSO) method for calculating eigenvalues and eigenvectors of a nonsymmetric matrix which is called the PSO-RA algorithm. Numerical examples are used to show the good numerical properties.

[1]  Ahmet Yildirim,et al.  Effects of partial slip on the peristaltic flow of a MHD Newtonian fluid in an asymmetric channel , 2010, Math. Comput. Model..

[2]  Ge Xiurun,et al.  An improved PSO-based ANN with simulated annealing technique , 2005, Neurocomputing.

[3]  H. Saberi Najafi,et al.  Weighted restarting method in the weighted Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix , 2006, Appl. Math. Comput..

[4]  Russell C. Eberhart,et al.  A new optimizer using particle swarm theory , 1995, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science.

[5]  R. Lehoucq.  Analysis and implementation of an implicitly restarted Arnoldi iteration , 1996 .

[6]  W. Arnoldi.  The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[7]  Ronald B. Morgan,et al.  On restarting the Arnoldi method for large nonsymmetric eigenvalue problems , 1996, Math. Comput..

[8]  Maurice Clerc,et al.  The particle swarm - explosion, stability, and convergence in a multidimensional complex space , 2002, IEEE Trans. Evol. Comput..

[9]  Jafar Biazar,et al.  A hybrid of the restarted Arnoldi and electromagnetism meta-heuristic methods for calculating eigenvalues and eigenvectors of a non-symmetric matrix , 2007, Appl. Math. Comput..

[10]  H. Saberi Najafi,et al.  A modification on minimum restarting method in the Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix , 2006, Appl. Math. Comput..

[11]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[12]  Y. Saad,et al.  Numerical Methods for Large Eigenvalue Problems , 2011 .

[13]  Syed Tauseef Mohyud-Din,et al.  Traveling wave solutions of Whitham–Broer–Kaup equations by homotopy perturbation method , 2010 .

[14]  Y. Saad,et al.  Numerical solution of large nonsymmetric eigenvalue problems , 1989 .

[15]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[16]  Russell C. Eberhart,et al.  Parameter Selection in Particle Swarm Optimization , 1998, Evolutionary Programming.

[18]  Ahmet Yildirim,et al.  The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets , 2010, Comput. Math. Appl..

[19]  Syed Tauseef Mohyud-Din,et al.  Analytical solutions to the pulsed Klein-Gordon equation using Modified Variational Iteration Method (MVIM) and Boubaker Polynomials Expansion Scheme (BPES) , 2010, Comput. Math. Appl..

[20]  Yue Shi,et al.  A modified particle swarm optimizer , 1998, 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360).

[21]  James Kennedy,et al.  Small worlds and mega-minds: effects of neighborhood topology on particle swarm performance , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[22]  H. Saberi Najafi,et al.  A new restarting method in the Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix , 2004, Appl. Math. Comput..

[23]  Avram Sidi,et al.  DGMRES: A GMRES-type algorithm for Drazin-inverse solution of singular non-symmetric linear systems , 2001 .

[24]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[25]  Kesheng Wu,et al.  Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods , 1998, SIAM J. Sci. Comput..