On rational functions related to algorithms for a computation of roots. I

• Autorzy: Baran Mirosław

Identyfikatory

DOI

10.5604/01.3001.0013.7274

• Języki publikacji: EN

Abstrakty

EN

We discuss a less known but surprising fact: a very old algorithm for computing square root known as the Bhaskara-Brouncker algorithm contains another and faster algorithms. A similar approach was obtained earlier by A.K. Yeyios [8] in 1992. By the way, we shall present a few useful facts as an essential completion of [8]. In particular, we present a direct proof that k – th Yeyios iterative algorithm is of order k. We also observe that Chebyshev polynomials Tn and Un are a special case of a more general construction. The most valuable idea followed this paper is contained in applications of a simple rational function Φ(w; z) = z-w/z+w.

Słowa kluczowe

EN

algorithm; iterative method; polynomial; recurrence relation

PL

algorytm; metoda iteracyjna; wielomian; relacja powtarzalności

• Wydawca: Państwowa Wyższa Szkoła Zawodowa w Tarnowie
• Czasopismo: Science, Technology and Innovation
• Rocznik: 2019
• Tom: Vol. 7, no. 4
• Strony: 17--25
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Baran Mirosław

• Faculty of Mathematics, Physics and Technical Science, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland

Bibliografia

• [1] D. Braess, Nonlinear approximation theory, Springer Ser. Comput. Math. Springer, New York (1986).
• [2] L. Fox, I.B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, New York, Toronto (1968).0.
• [3] E.S. Gawlik, Zolotariev iterations for the matrix square root, SIAM J. Matrix Anal. Appl. 40 (2) (2019), 696–719.
• [4] J.C. Mason, D.C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC (2003).
• [5] T.J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, JohnWiley, New York. (2nd ed. of Rivlin) (1990).
• [6] H. Rutishauser, Betrachtungen zur Quadratwurzeliteration, Monath. f. Math. 67 (1963) 452–464.
• [7] J.F. Traub, Iterative Methods for the Solution of Equations , Prentice- Hall, Englewood Cliffs, NY (1083).
• [8] A.K. Yeyios, On two sequences of algorithms for approximating square roots, J. of Comp. Appl. Math. 40 (1992), 63–72.