Previous |  Up |  Next


Title: A fixed point method to compute solvents of matrix polynomials (English)
Author: Marcos, Fernando
Author: Pereira, Edgar
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 135
Issue: 4
Year: 2010
Pages: 355-362
Summary lang: English
Category: math
Summary: Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented. (English)
Keyword: fixed point method
Keyword: matrix polynomial
Keyword: matrix differential equation
MSC: 34M99
MSC: 65H10
idZBL: Zbl 1224.34010
idMR: MR2681009
DOI: 10.21136/MB.2010.140826
Date available: 2010-11-24T08:23:22Z
Last updated: 2020-07-29
Stable URL:
Reference: [1] Davis, G. J.: Numerical solution of a quadratic matrix equation.SIAM J. Scient. Computing 2 (1981), 164-175. Zbl 0467.65021, MR 0622713, 10.1137/0902014
Reference: [2] Dennis, E., Traub, J. F., Weber, R. P.: On the Matrix Polynomial, Lambda-Matrix and Block Eigenvalue Problems.Computer Science Department, Technical Report, Cornell University, Ithaca, New York and Carnegie-Mellon University, Pittsburgh, Pennsylvania (1971).
Reference: [3] Dennis, J. E., Traub, J. F., Weber, R. P.: The algebraic theory of matrix polynomials.SIAM J. Numer. Anal. 13 (1976), 831-845. Zbl 0361.15013, MR 0432675, 10.1137/0713065
Reference: [4] Dennis, J. E., Traub, J. F., Weber, R. P.: Algorithms for solvents of matrix polynomials.SIAM J. Numer. Anal. 15 (1978), 523-533. Zbl 0386.65012, MR 0471278, 10.1137/0715034
Reference: [5] Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials.Academic Press, New York (1982). Zbl 0486.15008, MR 0662418
Reference: [6] Higham, N. J., Kim, H. M.: Solving a quadratic matrix equation by Newton's method with exact line searchers.SIAM J. Matrix Anal. Appl. 23 (2001), 303-316. MR 1871314, 10.1137/S0895479899350976
Reference: [7] Higham, N. J., Kim, H. M.: Numerical analysis of a quadratic matrix equation.IMA J. Numer. Anal. 20 (2000), 499-519. Zbl 0966.65040, MR 1795295, 10.1093/imanum/20.4.499
Reference: [8] Holmes, R. B.: A formula for the spectral radius of an operator.Am. Math. Mon. 75 (1968), 163-166. Zbl 0156.38202, MR 0227783, 10.2307/2315890
Reference: [9] Kratz, W., Stickel, E.: Numerical solution of matrix polynomial equations by Newton's method.IMA J. Numer. Anal. 7 (1987), 355-369. Zbl 0631.65040, MR 0968530, 10.1093/imanum/7.3.355
Reference: [10] Lancaster, P.: Lambda-Matrices and Vibrating Systems.Pergamon Press, New York (1966). Zbl 0146.32003, MR 0210345
Reference: [11] Lancaster, P.: A fundamental theorem on lambda matrices with applications II. Difference equations with constant coefficients.Linear Algebra Appl. 18 (1977), 213-222. Zbl 0388.15004, MR 0485917, 10.1016/0024-3795(77)90052-0
Reference: [12] Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edition.Academic Press, New York (1985). MR 0792300
Reference: [13] Pereira, E., Vitória, J.: Deflation of block eigenvalues of block partitioned matrices with an application to matrix polynomials of commuting matrices.Comput. Math. Appl. 42 (2001), 1177-1188. MR 1851235, 10.1016/S0898-1221(01)00231-0
Reference: [14] Pereira, E., Serodio, R., Vitória, J.: Newton's method for matrix polynomials.Int. J. Math. Game Theory Algebra 17 (2008), 183-188. Zbl 1177.65065, MR 2353584
Reference: [15] Shih, M., Wu, J.: Asymptotic stability in the Schauder fixed point theorem.Stud. Math. 2 (1998), 143-148. Zbl 0924.47044, MR 1636415, 10.4064/sm-131-2-143-148
Reference: [16] Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem.SIAM Rev. 43 (2001), 235-286. Zbl 0985.65028, MR 1861082, 10.1137/S0036144500381988
Reference: [17] Tsai, J. S. H., Shieh, L. S., Shen, T. T. C.: Block power method for computing solvents and spectral factors of matrix polynomials.Comput. Math. Appl. 16 (1988), 683-699. MR 0973957, 10.1016/0898-1221(88)90004-1


Files Size Format View
MathBohem_135-2010-4_2.pdf 229.7Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo