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 AM  Vol.5 No.16 , September 2014
Compound Means and Fast Computation of Radicals
Abstract: In last decades, several algorithms were developed for fast evaluation of some elementary functions with very large arguments, for example for multiplication of million-digit integers. The present paper introduces a new fast iterative method for computing values  with high accuracy, for fixed  and . The method is based on compound means and Padé approximations.
Cite this paper: Šustek, J. (2014) Compound Means and Fast Computation of Radicals. Applied Mathematics, 5, 2493-2517. doi: 10.4236/am.2014.516241.
References

[1]   Householder, A.S. (1970) The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill, New York.

[2]   Hardy, G.H., Littlewood, J.E. and Pólya, G. (1952) Inequalities. Cambridge University Press, Cambridge.

[3]   Borwein, J.M. and Borwein, P.B. (1987) Pi and the AGM. John Wiley & Sons, Hoboken.

[4]   Gauss, C.F. (1866) Werke. Göttingen.

[5]   Matkowski, J. (1999) Iterations of Mean-Type Mappings and Invariant Means. Annales Mathematicae Silesianae, 12, 211-226.

[6]   Karatsuba, A. and Ofman, Yu. (1962) Umnozhenie mnogoznachnykh chisel na avtomatakh. Doklady Akademii nauk SSSR, 145, 293-294.

[7]   Schönhage, A. and Strassen, V. (1971) Schnelle Multiplikation Großer Zahlen. Computing, 7, 281-292.
http://dx.doi.org/10.1007/BF02242355

[8]   Fürer, M. (2007) Faster Integer Multiplication. Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, 11-13 June 2007, 55-67.

[9]   Brent, R.P. (1975) Multiple-Precision Zero-Finding Methods and the Complexity of Elementary Function Evaluation. In: Traub, J.F., Ed., Analytic Computational Complexity, Academic Press, New York, 151-176.

[10]   Knuth, D.E. (1998) The Art of Computer Programming. Volume 2: Seminumerical Algorithms. Addison-Wesley, Boston.

[11]   Brezina, K. (2012) Smísené Pruměry. Master Thesis, University of Ostrava, Ostrava.

[12]   Karatsuba, A. (1995) The Complexity of Computations. Proceedings of the Steklov Institute of Mathematics, 211, 169-183.

[13]   Pan, V.Ya. (1961) Nekotorye skhemy dlya vychisleniya znacheni polinomov s veshchestvennymi koeffitsientami. Problemy Kibernetiki, 5, 17-29.

[14]   Wilf, H. and Zeilberger, D. (1990) Rational Functions Certify Combinatorial Identities. Journal of the American Mathematical Society, 3, 147-158.
http://dx.doi.org/10.1090/S0894-0347-1990-1007910-7

[15]   Jarník, V. (1984) Diferenciální pocet 1. Academia, Praha.

 
 
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