Research and Innovation Articles
On the convergence of a secant method for nonlinear matrix equations
Published 2014-10-31
Keywords
- Matrix function,
- Fréchet operator,
- Fréchet differentiable,
- secant method,
- nonlinear matrix equation
- superlinear convergence ...More
How to Cite
Macías C., M., Martínez, H. J., & Pérez, R. (2014). On the convergence of a secant method for nonlinear matrix equations. Revista Integración, Temas De matemáticas, 32(2), 181–197. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/4382
Abstract
In this paper we develop a general theory of convergence of a secant method to solve nonlinear matrix equations. In addition, we give sufficient conditions in order to this method provide a local and superlinearly convergent algorithm.
To cite this article: E.M. Macías, H.J. Martínez, R. Pérez, Sobre la convergencia de un método secante para ecuaciones matriciales no lineales, Rev. Integr. Temas Mat. 32 (2014), no. 2, 181-197.
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References
- Acevedo R., Pérez R. y Arenas F., “El método DL para resolver sistemas de ecuaciones no lineales”, Matemáticas: Enseñanza Universitaria 16 (2008), 23-36.
- Bittani S., Laub A.J. and Willems C., The Riccati equation, Springer-Verlag, Berlin, 1991.
- Davis J.G., “Numerical solution of a quadratic matrix equation”, SIAM J Sci and Stat. Comput. 2 (1981), no. 2, 164-175.
- Dennis J.E. and Schnabel R.B., Numerical methods for unconstrained optimization and nonlinear equations, Prentice-Hall, New Jersey, 1983.
- Dennis J.E. and Walker H.F., “Convergence theorems for least-change secant update methods”, SIAM J. Numer. Anal. 18 (1981), no. 6, 949-987.
- Golub H.G., Nash S. and Van Loan C., “A Hessenberg-Schur method for the problem AX + XB = C”, IEEE Trans. Automat. Control 24 (1979), no. 6, 909-913.
- Hashemi B. and Dehghan M., “Efficient computation of enclosures for the exact solvents of a quadratic matrix equation”, Electron. J. Linear Algebra 20 (2010), 519-536.
- Higham N.J., Functions of matrices theory and computations, SIAM, 2008.
- Higham N.J. and Kim H., “Numerical analysis of a quadratic matrix equation”, IMA J. Numer. Anal. 20 (2000), no. 4, 499-519.
- Higham N.J. and Kim H., “Solving a quadratic matrix equation by Newton’s methods with exact line searches”, SIAM J. Matrix Anal. Appl. 23 (2001), no. 2, 303-316.
- Kreyszig E., Introductory functional analysis with applications, Wiley & Sons, Canada, 1978.
- Lancaster P. and Rodman L., Algebraic Riccati equations, The Clarendon Press, Oxford University Press, New York, 1995.
- Martínez J.M., “On the relation between two local convergence theories of least-change secant updates
- methods”, Math. Comp. 59 (1992), no. 200, 457-481.
- Martínez J.M. and Santos S.A., “Métodos computacionais de otimização”, 20 Colóquio Brasileiro de Matemática, IMPA 1995, p. 87.
- Monsalve M. and Raydan M., “Newton’s method and secant methods: A long-standing relationship from vectors to matrices”, Port. Math. 68 (2011), no. 4, 431-475.
- Monsalve M. and Raydan M., “A secant method for nonlinear matrix problems”, Chapter
- of Numerical Linear Algebra in Signals, Systems and Control, P. Van Dooren et al. (eds.), Springer Verlag. 80, 2011, 387-412.
- Parks P.C., “AM Lyapunov’s stability theory–100 years on”, IMA J. Math. Control Inform. 9 (1992), no. 4, 275-303.
- Pérez R. y Díaz T., Minimización sin restricciones, Editorial Universidad del Cauca, 2010.
- Tisseur F. and Meerbergen K., “The quadratic eigenvalue problem”, SIAM Rev. 43 (2001), no. 2, 235-286.