Matrix pencil method for rectangular polynomial two-parameter eigenvalue problem

Niranjan Bora; Bharati Borgohain

doi:10.5269/bspm.78848

Niranjan Bora Dibrugarh University, Assam, India
Bharati Borgohain Research Scholar

Abstract

Rectangular multiparameter eigenvalue problems (RMEP) consisting of a single multivariate polynomial have received interest among the researchers due to their applications in diverse scientific domain, particularly in optimal least square model problems. A common method for determining the optimal least squares of linear time-invariant dynamical systems (LTI) and autoregressive moving average (ARMA) models are obtained from the solution of the rectangular polynomial two-parameter eigenvalue problems (RPTEP). This makes it necessary to find effective solution methods for this particular kind of eigenvalue problem. Linearizing the matrix polynomial associated with RMEP followed by the conversion to a known form of linear two-parameter eigenvalue problem, and then using the Vandermonde compression is the currently available method in the literature. In this paper, we present a two-parameter matrix pencil method to obtain the solution of the RPTEP, that can be used as a ready reference to compute the solution of LTI and ARMA. Numerical works are performed to verify the computational efficiency of the method.

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Author Biography

Bharati Borgohain, Research Scholar

Department of Mathematics, Dibrugarh University, Assam, India.

References

Adhikari, B., Vector space of linearizations for the quadratic two-parameter matrix polynomial, Linear Multilinear Algebra. 61(5), 603-616, (2013).

Alsubaie, F. F., H2 Optimal Model Reduction for Linear Dynamic Systems and the Solution of Multiparameter Matrix Pencil Problems, PhD thesis, Department of Electrical and Electronic Engineering, Imperical College, London, (2019).

Bora, N., Buragohain, B. and Sharma, B. Linearizations of cubic two-parameter eigenvalue problem and its vector space, Contemp. Math. 5(1), 930-948, (2024).

Das, B. and Bora, S., Vector spaces of generalized linearizations for rectangular matrix polynomials, Electron. J. Linear Algebra. 35, 116-155, (2019).

Dong, B., The Homotopy Method for the Complete Solution of Quadratic Two-parameter Eigenvalue Problems, J. Sci. Comput. 90, 18, (2022).

Dopico, F. M., Lawrence, P. W., Perez, J. and Dooren, P. V., Block Kronecker linearizations of matrix polynomials and their backward errors, Numer. Math. 140, 373–426, (2018).

Faßbender, H., and Saltenberger, P., Block Kronecker ansatz spaces for matrix polynomials, Linear Algebra Appl. 542, 118-148, (2018).

Ghosh, A., Linearization, sensitivity and backward perturbation analysis of multiparameter eigenvalue problems, Ph.D. Thesis, Department of Mathematics, Indian Institute of Technology, India (2017).

Gungah, S. K., Alsubaie, F. F. and Jaimoukha, I. M., On the Two-Parameter Matrix Pencil Problem, SIAM J. Matrix Anal. Appl. 45(3), 1287-1309, (2024).

Hochstenbach, M. E., Muhic, A., and Plestenjak, B., Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems, J. Comput. Applied Math. 288, 251-263, (2015).

Hochstenbach, M. E., Muhic, A., and Plestenjak, B., On linearizations of the quadratic two-parameter eigenvalue problems, Linear Algebra Appl. 436, 2725-2743, (2012).

Hochstenbach, M. E., Kosir, T. and Plestenjak, B., Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models, Numer. Linear Algebra Appl. 31(2), Article No. e2540, (2024).

Jarlebring, E., Critical delays and polynomial eigenvalue problems, J. Comput. Applied Math. 224(1), 296-306, (2009).

Jarlebring, E., and Hochstenbach, M. E., Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations, Linear Algebra Appl. 431(3-4), 369–380, (2009).

Kosir, T. and Plestenjak, B., On the singular two-parameter eigenvalue problem II, Linear Algebra Appl. 649, 433-451, (2022).

Khazanov, V. B., On spectral properties of multiparameter polynomial matrices, J. Math. Sci. 89, 1775–1800, (1998).

Kublanovskaya, V. N., Solving eigenvalue problems for two-parameter polynomial matrices. Methods and algorithms. Part I., Russian J. Numer. Anal. Math. Model. 9(2), 111-120, (1994).

Kublanovskaya, V. N., On some factorizations of two-parameter polynomial matrices. J. Math. Sci. 86, 2866–2879, (1997).

Kublanovskaya, V. N., Methods and algorithms of solving spectral problems for polynomial and rational matrices, J. Math. Sci. 96, 3085-3287, (1999).

Kublanovskaya, V. N., Solution of spectral problems for polynomial matrices, J. Math. Sci. 127(3), 2024–2032, (2005).

Meerbergen, K., Schr¨oder, C., and Voss, H., A Jacobi–Davidson method for two-real-parameter nonlinear eigenvalue problems arising from delay-differential equations, Linear Algebra Appl. 20(5), 852-868, (2013).

Moor, B. D., Least squares realization of LTI models is an eigenvalue problem, Paper presented at: 18th European control conference IEEE. 2270–2275, (2019).

Moor, B. D., Least squares optimal realisation of autonomous LTI systems is an eigenvalue problem, Communication Info. Syst. 20(2), 163-207, (2020).

Muhic, A. and Plestenjak, B., On the quadratic two-parameter eigenvalue problem and its linearization, Linear Algebra Appl. 432, 2529-2542, (2010).

Plestenjak, B., Numerical methods for nonlinear two-parameter eigenvalue problems, BIT Numer. Math. 56, 241-262, (2016).

Plestenjak, B., MultiParEig https://www.mathworks.com/matlabcentral/fileexchange/47844-multipareig, MATLAB Central File Exchange. Retrieved March 30, 2024.

Rodriguez, J. I., Du, J. H., You, Y. and Lim, L. H., Fiber product homotopy method for multiparameter eigenvalue problems, Numer. Math. 148, 853-888, (2021).

Shapiro, B. and Shapiro, M., On eigenvalues of rectangular matrices, Proc. Steklov Inst. Math. 267(1), 248-255, (2009).

Teran, F. D., Dopico, F. M. and Mackey, D. S., Fiedler companion linearizations for rectangular matrix polynomials, Linear Algebra Appl. 437(3), 957-991, (2012).

Teran, F. D., Dopico, F. M. and Mackey, D. S., Spectral equivalence of matrix polynomials and the Index Sum Theorem, Linear Algebra . Appl. 459, 264-333, (2014).

Vermeersch, C. and Moor, B. D., Globally optimal least-squares ARMA model identification is an eigenvalue problem, IEEE Control Syst. Lett. 3(4), 1062-1067, (2019).

Vermeersch, C. and Moor, B. D., Two complementary block Macaulay matrix algorithms to solve multiparameter eigenvalue problems, Linear Algebra Appl. 654(1), 177-209, (2022).