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Tytuł artykułu

Bounds for sine and cosine via eigenvalue estimation

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Define n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the first super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the first super- and subdiagonal are minus one. Then, denoting by λ(·) the largest eigenvalue, [...] Using certain lower bounds for the largest eigenvalue, we provide lower bounds for these expressions and, further, lower bounds for sin x and cos x on certain intervals. Also upper bounds can be obtained in this way.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
1
Opis fizyczny
Daty
wydano
2014-01-01
otrzymano
2013-10-04
zaakceptowano
2014-03-02
online
2014-04-09
Twórcy
autor
  • Department of Mathematics, University of Coimbra EC Santa Cruz, 3001-501 Coimbra, Portugal, kovacec@mat.uc.pt
Bibliografia
  • [1] D. Caccia, Solution of Problem E 2952, Amer. Math. Monthly 93 (1986), 568-569.
  • [2] N. D. Cahill, J. R. D’Errico and J. P. Spence, Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13-19.
  • [3] K. Fan, O. Taussky and J. Todd, Discrete analogs of inequalities of Wirtinger, Monatsh. Math. 59 (1955), 73-90.
  • [4] M. Fekete and G. Pólya, Über ein Problem von Laguerre. In G. Pólya, Collected Papers, Vol. II: Location of zeros, Ed. by R. Boas, MIT Press, 1974, p. 2.
  • [5] R. A. Horn and C. R. Johnson, Matrix Analysis, Second Edition, Cambridge Univ. Pr., 2013.
  • [6] S. Hyyrö, J. K. Merikoski and A. Virtanen, Improving certain simple eigenvalue bounds, Math. Proc. Camb. Phil. Soc. 99 (1986), 507-518.
  • [7] M.-K. Kuo, Re_nements of Jordan’s inequality, J. Ineq. Appl. 2011 (2011), Art. 130, 6 pp.
  • [8] D. London, Two inequalities in nonnegative symmetric matrices, Pacific J. Math. 16 (1966), 515-536.
  • [9] G. V. Milovanovic, I. Ž. Milovanovic, On discrete inequalities of Wirtinger’s type, J. Math. Anal. Appl. 88 (1982), 378-387.[Crossref]
  • [10] N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutsch. Verl. Wiss., 1963.
  • [11] F. Qi, D.-W. Niu and B.-N. Guo, Refinements, generalizations, and applications of Jordan’s inequality and related problems, J. Ineq. Appl. 2009 (2009), Art. ID 271923, 52 pp.[WoS]
  • [12] R. Redheffer, Problem 5642, Amer. Math. Monthly 75 (1968), 1125.
  • [13] R. Redheffer, Correction, Amer. Math. Monthly 76 (1969), 422.
  • [14] D. E. Rutherford, Some continuant determinants arising in physics and chemistry, I, Proc. Royal Soc. Edinburgh 62A (1947), 229-236.
  • [15] J. Sándor, On the concavity of sin x/x, Octogon Math. Mag. 13 (2005), 406-407.
  • [16] J. Sándor, Selected Chapters of Geometry, Analysis and Number Theory: Classical Topics in New Perspectives, Lambert Acad. Publ., 2008.
  • [17] D. Y. Savio and E. R. Suryanarayan, Chebychev polynomials and regular polygons, Amer. Math. Monthly 100 (1993), 657-661.
  • [18] J. P. Williams, A delightful inequality, Solution of Problem 5642, Amer. Math. Monthly 76 (1969), 1153-1154.
  • [19] A. Y. Özban, A new refined form of Jordan’s inequality and its applications, Appl. Math. Lett. 19 (2006), 155-160. [Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_spma-2014-0003
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