Un algoritmo cuasi-Newton para resolver la ecuación cuadrática matricial

Mauricio Macías, Héctor J. Martínez, Rosana Pérez

Resumen


En este artículo proponemos un algoritmo cuasi-Newton para resolver una ecuación cuadráti a matricial, el cual reduce el costo computacional del método Newton-Schur, tradicionalmente usado para resolver dicha ecuación. Demostramos que el algoritmo propuesto es local y hasta cuadráticamente convergente. Presentamos pruebas numéricas que ratifican los resultados teóricos desarrollados.

Palabras clave: Función cuadrática matricial, operador de Fréchet diferenciable, método de Newton-Schur, convergencia cuadrática.

Para citar este artículo: M. Macías, H.J. Martínez, R. Pérez, Un algoritmo cuasi-Newton para resolver la  ecuación cuadrática matricial, Rev. Integr. Temas Mat. 34 (2016), No. 2, 187-206.


Texto completo:

PDF

Referencias


Bai Z-Z., Guo X-X and Yin J-F., "On two iteration methods for the quadratic matrix equations", Inter. J. Numer. Anal. Model. 2 (2005), suppl., 144-122.

Berhanu M., "The polynomial eigenvalue problem", Thesis (Ph.D.), University of Manchester, 2005, 219 p.

Bermúdez A., Durán R.G., Rodríguez R. and Solomin J., "Finite element analysis of a quadratic eingenvalue problem arising in dissipative acoustics", SIAM J. Numer. Anal. 38 (2000), No. 1, 267-291.

Butler G.J., Johnson C.R. and Wolkowicz H., "Nonnegative solutions of a quadratic matrix equation arising from comparison theorems in ordinary differential equations", SIAM J. Algebraic Discrete Methods, 6 (1985), No. 1, 47-53.

Clark J.V., Zhou N. and Pister K.S.J., "Modified nodal analysis for MEMS with multi-energy domains", International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, San Diego, CA, March, 2000.

Dávila C.E., "A subspace approach to estimation of autoregressive parameters from noisy measurements", IEEE Trans. Signal Process. 46 (1998), No. 2, 531-534.

Davis G.J., "Numerical solution of a quadratic matrix equation", SIAM J. Sci. Statist. Comput. 2 (1981), No. 2, 164-175.

Dennis J.E., Jr., and Schnabel R.B., Numerical methods for unconstrained optimization and nonlinear equations. Prentice Hall, Inc., Englewood Cliffs, NJ, 1983.

Dennis J.E., Jr., Traub J.F. and Weber R.P., "The algebraic theory of matriz polynomials", SIAM J. Numer. Anal. 13 (1976), No. 6, 831-845.

Gao Y-H., "Newton's method for the quadratic matrix equation", Appl. Math. Comput. 182 (2006), No. 2, 1772-1779.

Golub G.H. and Van Loan C.F., Matrix computations, Third ed. Jonhs Hopkins University Press, Baltimore, MD, 1996.

Golub G.H., Nash S. and Van Loan C.F., "A Hessenberg-Schur method for the problem AX+XB=C", IEEE Trans. Automat. Control. 24 (1979), No. 6, 909-913.

Guo C-H., "On a quadratic matrix equation associated whit an M-matrix", IMA J. Numer. Anal. 23 (2003), No. 1, 11-27.

Higham N.J., Functions of matrices. Theory and computations, SIAM, Philadelphia, PA, 2008.

Higham N.J., "Stable iterations for the matrix square root", Numer. Algorithms 15 (1997), No. 2, 227-242.

Higham N.J. and Kim H-M., "Numerical analysis of a quadratic matrix equation", IMA J. Numer. Anal. 20 (2000), No. 4, 499-519.

Higham N.J. and Kim H-M., "Solving a quadratic matrix equation by Newton's method with exact line searches", SIAM J. Matrix Anal. Appl. 23 (2001), No. 2, 303-316.

Horn R.A. and Johnson C.R., Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.

Kay S.M., "Noise compensation for autoregressive spectral estimates", IEEE Trans. Acoust. Speech Signal Process. ASSP-28 (1980), No. 3, 292-303.

Lancaster P., Lambda-matrices and vibrating systems, Pergamon Press, Oxford-New York-París, 1966.

Laub A.J., "Efficient multivariable frequency response computations", 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, 39-41, December, 1980.

Liu L-D., "Perturbation analysis of a quadratic matrix equation associated with an M-matrix", J. Comput. Appl. Math. 260 (2014), 410-419.

Liu L-D. and Lu X., "Two kinds of condition numbers for the quadratic matrix equation", Appl. Math. Comput. 219 (2013), No. 16, 8759-8769.

Macías E.M., "Métodos secantes de cambio mínimo para el cálculo de ceros de funciones de matrices", Tesis de maestría, Universidad del Cauca, 2013.

Macías M., Martínez H.J. and Pérez R., "Sobre la convergencia de un método secante para ecuaciones matriciales no lineales", Rev. Integr. Temas Mat. 32 (2014), No. 2, 181-197.

Monsalve M. and Raydan M., "A secant method for nonlinear matrix problems", in Numerical linear algebra in signals, systems and control, Springer (2011), 387-402.

Monsalve M. and Raydan M., "Newton's method and secant methods: a longstanding relationship from vectors to matrices", Port. Math. 68 (2011), No. 4, 431-475.

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.

Seo S-H., Seo J-H. and Kim H-M , "Newton's method for solving a quadratic matrix equation whith special coefficient matrices", Honam Math. J. 35 (2013), No. 3, 417-433.

Smith H.A., Singh R.K. and Sorensen D.C., "Formulation and solution of the non-linear, damped eigenvalue problem for skeletal systems", Internat. J. Numer. Methods Engrg. 38 (1995), No. 18, 3071-3085.

Tisseur F., "Backward error and condition of polynomial eigenvalue problems", Linear Algebra Appl. 309 (2000), No. 1-3, 339-361.

Tisseur F. and Meerbergen K., "The quadratic eigenvalue problem", SIAM Rev. 43 (2001), No. 2, 235-286.

Thomson W.T., Theory of vibration with applications, Fourth ed., CRC Press, 1996.

Watkins D.S., Fundamentals of matrix computations. Second ed, Pure and Applied Mathe-matics, Wiley-Interscience, New York, 2002.

Xu H.G. and Lu L.Z., "Properties of a quadratic matrix equation and the solution of the continuos-time algebraic Riccati equation", Linear Algebra Appl. 222 (1995), 127-145.

Zhou N., Clark J.V. and Pister K.S.J., "Nodal simulation for MEMS design using SUGAR v0.5", International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, Santa Clara, CA, 308-313, 1988.




DOI: http://dx.doi.org/10.18273/revint.v34n2-2016006

Universidad Industrial de Santander

Carrera 27 - Calle 9 Ciudad Universitaria. Teléfono: 6344000 ext: 2316 - 2595

Bucaramanga, Colombia