Un algoritmo tipo Newton globalizado para resolver la ecuación cuadrática matricial

• Autores: Mauricio Macías, Héctor J. Martínez, Rosana Pérez
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 36, Nº. 2, 2018, págs. 117-132
• Idioma: español
• DOI: 10.18273/revint.v36n2-2018004
• Títulos paralelos:
  • A globalized Newton type algorithm to solve the matrix quadratic equation
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En este artículo se presenta una globalización del algoritmo cuasi-Newton local propuesto en [16] para resolver la ecuación cuadrática matricial. Se demuestra que la estrategia de globalización usada no interfiere en la tasade convergencia del algoritmo cuasi-Newton. Pruebas numéricas muestran un buen desempeño del algoritmo global propuesto.

  • English

    In this paper we propose a globalized quasi-Newton algorithm to solve the quadratic matrix equation. It is shown that the globalization strategy used does not interfere in the convergence of the local quasi-Newton algorithm. We present some numerical tests that show the good performance of the global quasi-Newton algorithm.

• Referencias bibliográficas
  • Citas [1] Bermúdez A., Durán R.G., Rodríguez R. and Salomin J., “Finite element analysis of a quadratic eingenvalue problem arising in dissipative...
  • [2] Clark J.V., Zhou N. and Pister K.S.J., “Nodal simulation for MEMS design using SUGAR v0.5”, International Conference on Modeling and Simulation...
  • [3] Clark J.V., Zhou N. and Pister K.S.J., “Modified nodal analysis for MEMS with multienergydomains”, International Conference on Modeling...
  • [4] Davila C.E., “A subspace approach to estimation of autoregressive parameters from noisy measurements”, IEEE Trans. Signal Process. 46...
  • [5] Davis J.G., “Numerical solution of a quadratic matrix equation”, SIAM J. Sci. Statict. Comput. 2 (1981), No. 2, 164–175.
  • [6] Dennis J.E. and Schnabel R.B., Numerical methods for unconstrained optimization and nonlinear equations, SIAM, Philadephia, 1996.
  • [7] Golub H.G. and Van Loan C.F., Matrix Computations, The Jonhs Hopkins University Press, Jonhs Hopskins University Press, Baltimore, MD,...
  • [8] Higham N.J., Functions of matrices theory and computations, SIAM, Philadephia, PA, 2008.
  • [9] Higham N.J. and Kim H., “Numerical analysis of a quadratic matrix equation”, IMA J. Numer. Anal. 20 (2000), No. 4, 499–519.
  • [10] Higham N.J. and Kim H., “Solving a quadratic matrix equation by Newton’s methods with exact line searches”, SIAM J. Anal. Appl. 23 (2001),...
  • [11] Kay S.M., “Noise compensation for autoregressive spectral estimates”, IEEE Trans. Acoust. Speech Signal Process. 28 (1980), No. 3, 292–303.
  • [12] Kim H.M., “Numerical methods for solving a quadratic matix equation,” Thesis (Ph.D.), Manchester University, 2000, 59–60.
  • [13] Lancaster P., Lambda-Matrices and Vibrating Systems, Pergamon Press, Oxford-New York- Paris, 1966.
  • [14] Laub A.J., “Efficient multivariable frequency response computations”, 19th IEEE Conference on Decision and Control including the Symposium...
  • [15] Long J.H., Xi-Yan and Zhang L., “Improved Newton’s method with exact line searches to solve quadratic matrix equation”, J. Comput. Appl....
  • [16] Macías M., Martínez H.J. y Pérez R., “Un algoritmo cuasi-Newton para resolver la ecuación cuadrática matricial“, Rev. Integr. Temas Mat....
  • [17] Macías M., Martínez H.J. y Pérez R., “Sobre la convergencia de un método secante para ecuaciones matriciales no lineales”, Rev. Integr....
  • [18] Monsalve M. and Raydan M., “Newton’s method and secant methods: A long-standing relationship from vectors to matrices”, Port. Math. 68...
  • [19] Monsalve M. and Raydan M., “A secant method for nonlinear matrix problems”, Numerical linear algebra in signals, systems and control,...
  • [20] Nocedal J. and Wright J.G., Numerical Optimizations, Springer-Verlag, New York, 1999.
  • [21] Reddy S.C., Schmid P.J. and Henningson D.S., “Pseudospectra of the Orr-Sommerfeld operator”, SIAM J. App. Math. 53 (1993), No. 1, 15–47.
  • [22] Smith H.A., Singh R.K. and Sorensen D.C., “Formulation and solution of the nonlinear, damped eigenvalue problem for skeletal systems”,...
  • [23] Tisseur F. and Meerbergen K., “The quadratic eigenvalue problem”, SIAM Rev. 43 (2001), No. 2, 235–286.
  • [24] Tisseur F., “Backward error and condition of polynomial eigenvalue problems”, Linear Algebra Appl. 309 (2000), No. 1-3, 339–361.
  • [25] Thomson W.T., Teoría de vibraciones con aplicaciones, CRC Press, 1996.