Abstract
In this paper, we prove a more general result concerning the location of the eigenvalues of a matrix polynomial in an annulus from which we deduce an interesting result due to Higham and Tisseur [11]. Several other known results have been extended to matrix polynomials, which in particular include extension and generalization of a classical result of Cauchy [4]. We also present two examples of matrix polynomials to show that the bounds obtained are close to the actual bounds.
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C. Affane-Aji, S. Biaz and N. K. Govil, On annuli containing all the zeros of a polynomial, Math. Comp. Modelling 52 (2010), 1532–1537.
A. Aziz and A. Qayoom, Estimates for the modulii of the zeros of a polynomial, Math. Inequ. Appl. 9(1) (2006), 107–116.
M. Bidkham and E. Shashahani, An annulus for the zeros of polynomials, Appl. Math. Lett. 24 (2011), 122–125.
A. L. Cauchy, Exercices de Mathématiques, IV, Année de Bure Fréres, Paris, 1829.
A. Dalal and N. K. Govil, On region containing all the zeros of a polynomial, Appl. Math. Comp. 219 (2013), 9609–9614.
A. Dalal and N. K. Govil, Annulus containing all the zeros of a polynomial, Appl. Math. Comp. 249 (2014), 429–435.
B. Datt and N. K. Govil, On the location of zeros of a polynomial, J. Approx. Theory 24 (1978), 78–82.
J. L. Diaz-Barrero, An annulus for the zeros of polynomials, J. Math. Anal. Appl. 273 (2002), 349–352.
J. L. Diaz-Barrero and J. J. Egozcue, Bounds for the moduli of zeros, Appl. Math. Lett. 17 (2004), 993–996.
I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.
N. J. Higham and F. Tisseur, Bounds for the eigenvalues of matrix polynomials, Linear Algebra Appl. 358 (2003), 5–22.
A. Joyal, G. Labelle and Q. I. Rahman, On the location of zeros of polynomials, Canad. Math. Bull. 10 (1967), 53–63.
S. H. Kim, On the moduli of the zeros of a polynomial, Amer. Math. Month. 112 (2005), 924–925.
P. Lancaster, Lambda-matrices and vibrating system, Pergamon, Oxford, 1966.
M. Marden, Geometry of Polynomials, Mathematical Surveys, vol. 3, IInd ed., Amer. Math. Soc. Providence, RI, USA, 1966.
A. Melmam, Bounds for eigenvalues of matrix polynomials with applications to scalar polynomials, Linear Algebra Appl., 504 (2016), 190–203.
A. Melmam, Cauchy-like and Pellet-like results for polynomials, Linear Algebra Appl., 505 (2016), 174–193.
G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities and Zeros, World Scientific, Singapore, 1994.
V. Noferini, M. Sharify and F. Tisseur, Tropical roots as approximations to eigenvalues of matrix polynomials, SIAM J. Matrix Anal. Appl., 36 (1)(2015) 138-157.
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press Inc., New York, 2002.
Y. J. Sun and J. G. Hsieh, A note on circular bound of polynomial zeros, IEEE Trans. Circuits Syst. I 43 (1996), 476-478.
D. T-H-Bình, L. C-Trình and N. T-Dúc, On the location of eigenvalues of matrix polynomials, (2017) arXiv:1703.00747v1.
F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Review, 43 (2)(2001) 235-286.
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The authors are extremely grateful to the referee for his valuable suggestions.
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Communicated by Kaushal Verma.
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Hans, S., Raouafi, S. Annulus containing all the eigenvalues of a matrix polynomial. Indian J Pure Appl Math 53, 405–412 (2022). https://doi.org/10.1007/s13226-021-00211-8
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DOI: https://doi.org/10.1007/s13226-021-00211-8