Arnoldi methods for numerical continuation of stationary points

Article Sidebar

Publicación 23. Vol 4. No 3. (Español (España)) HTML (Español (España)) EPUB (Español (España)) XML (Español (España))
Published: Jul 3, 2020
DOI: https://doi.org/10.33262/cienciadigital.v4i3.1385

Main Article Content

Zenaida Natividad Castillo Marrero
Chimborazo Higher Polytechnic School
Gustavo Adolfo Colmenares Pacheco
Yachay Experimental Research University
Víctor Oswaldo Cevallos Vique
Chimborazo Higher Polytechnic School

Abstract

This work presents an analysis of a continuation algorithm with an embedded Arnoldi eigen solver for the numerical computation of solutions of nonlinear systems G (x, α) = 0, where x is a vector in Rn and α is a parameter which takes values in a given interval. This technique allows us to detect and predict particular solutions when computing the eigenvalues of the associate Jacobian matrix, and simultaneously to get the solution of linear systems in each iteration. This method could be applied to problems of electrical engineering, chemical reactions or coating process. The main idea is to embed an Arnoldi eigenvalue solver in a continuation algorithm to compute solutions of a nonlinear system in order to get additional information of these solutions, such as the stability or the periodicity. An example of the usability of this technique is presented, with preliminary results on models used in the bibliography of this topic. The results are encouraged, and show the reliability of this approach in the accurate detection of critical points.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Article Details

How to Cite
Castillo Marrero, Z. N., Colmenares Pacheco, G. A., & Cevallos Vique, V. O. (2020). Arnoldi methods for numerical continuation of stationary points. Ciencia Digital, 4(3), 378-390. https://doi.org/10.33262/cienciadigital.v4i3.1385
Issue
Vol. 4 No. 3 (2020): Design & Evaluation
Section
Artículos
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Crossref
Scopus
Google Scholar
Europe PMC

References

Baglama, J., Calvetti, D., Reichel, L., & Ruttan, A. (1998). Computation of a few small eigenvalues of a large matrix with application to liquid crystal modeling. Journal of Computational Physics, 146(1), 203-226.
Bindel, D., Friedman, M., Govaerts, W., Hughes, J., & Kuznetsov, Y. A. (2008). CL MATCONTL: a continuation toolbox for large equilibrium problems in MATLAB.
Calvetti, D., & Reichel, L. (1999). A block-Lanczos method for large continuation problems. Numerical Algorithms, 21(1-4), 109-118.
Calvetti, D., & Reichel, L. (2000). Iterative methods for large continuation problems. Journal of computational and applied mathematics, 123(1-2), 217-240.
Castillo, Z. (2004). A new algorithm for continuation and bifurcation analysis of large scale free surface flows. PhD thesis, Rice University, Houston, Texas.
Demmel, J. W. (1997). Applied numerical linear algebra. Society for Industrial and Applied Mathematics. Berkeley, California, first edition.
Golub, G. H., & Van Loan, C. F. (1996). Matrix Computations. 3rd. edn. ed. The Johns Hopkins University.
Govaerts, W. J. (2000). Numerical methods for bifurcations of dynamical equilibria. Society for Industrial and Applied Mathematics.
Prigogine, I., & Lefever, R. (1974). Stability and self-organization in open systems (pp. 215-240). Wiley, New York.
Saad, Y. (2011). Numerical methods for large eigenvalue problems: revised edition. Society for Industrial and Applied Mathematics.
Seydel, R. (2009). Practical bifurcation and stability analysis (Vol. 5). Springer Science & Business Media.
Sorensen, D. C. (1997). Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations. In Parallel Numerical Algorithms (pp. 119-165). Springer, Dordrecht.
Trefethen, L. N., & Bau III, D. (1997). Numerical linear algebra (Vol. 50). Siam.