Research and Innovation Articles
Published 2018-12-12
Keywords
- Matrix function,
- secant method,
- nonlinear matrix equations,
- superlinear convergence
How to Cite
Macías, M., Martínez, H. J., & Pérez, R. (2018). A globalized Newton type algorithm to solve the matrix quadratic equation. Revista Integración, Temas De matemáticas, 36(2), 117–132. https://doi.org/10.18273/revint.v36n2-2018004
Abstract
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.
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References
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[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), No. 2, 303–316.
[11] Kay S.M., “Noise compensation for autoregressive spectral estimates”, IEEE Trans. Acoust. Speech Signal Process. 28 (1980), No. 3, 292–303.
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[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. Math. 222 (2008), No. 2, 645–654.
[16] Macías M., Martínez H.J. y Pérez R., “Un algoritmo cuasi-Newton para resolver la ecuación cuadrática matricial“, Rev. Integr. Temas Mat. 34 (2016), No. 2, 187–206.
[17] Macías M., Martínez H.J. y Pérez R., “Sobre la convergencia de un método secante para ecuaciones matriciales no lineales”, Rev. Integr. Temas Mat. 32 (2014), No. 2, 181–197.
[18] 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.
[19] Monsalve M. and Raydan M., “A secant method for nonlinear matrix problems”, Numerical linear algebra in signals, systems and control, Springer 80 (2011), 387–402.
[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”, Internat. J. Numer. Methods Engrg. 38
(1995), No. 18, 3071–3085.
[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.
[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 of Microsystems, Semiconductors, Sensors and Actuators, Santa Clara, CA. 1988, 308–313.
[3] Clark J.V., Zhou N. and Pister K.S.J., “Modified nodal analysis for MEMS with multienergydomains”, International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, San Diego, CA. March, 2000.
[4] Davila C.E., “A subspace approach to estimation of autoregressive parameters from noisy measurements”, IEEE Trans. Signal Process. 46 (1988), No. 2, 531–534.
[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, 1996.
[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), No. 2, 303–316.
[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 on Adaptive Processes, Albuquerque, USA, 39–41, December, 1980.
[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. Math. 222 (2008), No. 2, 645–654.
[16] Macías M., Martínez H.J. y Pérez R., “Un algoritmo cuasi-Newton para resolver la ecuación cuadrática matricial“, Rev. Integr. Temas Mat. 34 (2016), No. 2, 187–206.
[17] Macías M., Martínez H.J. y Pérez R., “Sobre la convergencia de un método secante para ecuaciones matriciales no lineales”, Rev. Integr. Temas Mat. 32 (2014), No. 2, 181–197.
[18] 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.
[19] Monsalve M. and Raydan M., “A secant method for nonlinear matrix problems”, Numerical linear algebra in signals, systems and control, Springer 80 (2011), 387–402.
[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”, Internat. J. Numer. Methods Engrg. 38
(1995), No. 18, 3071–3085.
[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.