Volume- 3
Issue- 3
Year- 2016
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D.A.GISMALLA , D.A.GISMALLA
In this survey , our aim is to represent to the reader a fascinating and a beautiful approach called Continued Fraction Technique (C.F.T.) to evaluate infinite series by transformation a series to its corresponding equivalent C.F. algorithm. First , We introduce the basic ideas of continued fraction to see how they arise out of high school division and also from solving equations .Second , theorems for Transformation infinite series to their equivalent C.F. are given .Some of these Transformation are to transform alternating series and series having an infinite product of terms. We observe one can use algebra to simplify the evaluations of the n-th continued fraction term to its equivalent one to generate the - transformation easily as in Theorem 2.3. Third , Matlab program software is written to compute some types of series efficiently. We , conclude with a computational remark showing the difficulties in computing some C.F. algorithms due to overflow or underflow for rounding errors , however this will not slamming C.F. for some types of certain problems it can't be avoided specially when evaluating Bessel's functions and its zeros or Hypergeometric series in [ 4] ,pp.272
[1] Josef Kallrath Basf OSEF KALLRATH On Rational Function Techniques and Pade Approximants , An Overview ,AG ZX/ZC - C13, D-67056 Ludwigshafen, GERMANY e-mail: kallrath@zx.basf-ag.de September 16, 2002
[2] P.Hanggi ,F.Rossel , and D.Trautmann Evaluations of infinite Series by Use of Continued Fraction Expansions : A numerical Study . Journal of Computational Physics , Vol. 37 , No.2 Sep.1980
[3] Leonhard Euler On the Transformation of Infinite Series to Continued Fractions Translated by Daniel W. File1 Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210 July 11, 2004
[4] Sergey Khrushchev Orthogonal Polynomials and Continued Fractions From Euler’s Point of View Atilim University, Turkey Cambridge University Press
[5] Lange ,L.J. An Elegant Contiued Fraction for ,the American Mathematical Monthly ,106(1999) ,pp.456-458.
[6] Osler ,T;J. Morphing Lord Brouncker's Contiued Fraction for pi into the Product of Wallis ,the Mathematical gazette 95(2011) ,pp.17-22
[7] Osler ,T;J , Lord Brouncker's forgotten sequence of Contiued Fraction for pi. International Journal of Mathematical Education in Science and Technology , Volume 41, Isuue 1 Januray 2020 ,pages 105-110
[8]L. N. Trefethen, "Matlab programs for CF approximation," in C. K. Chui, L.L. Schumaker, and J. D. Ward, eds., Approximation Theory V, Academic Press, 1986, pp. 599-60
[9] Barber, C. B., D.P. Dobkin, and H.T. Huhdanpaa, "The Quickhull Algorithm for Convex Hulls," ACM Transactions on Mathematical Software, Vol. 22, No. 4, Dec. 1996, p. 469-483. Available in HTML format at http://www.acm.org/pubs/citations/journals/toms/19 96-22-4/p469-barber/ and in PostScript format at ftp://geom.umn.edu/pub/software/qhull-96.ps.
[10] National Science and Technology Research Center for Computation and Visualization of Geometric Structures (The Geometry Center), University of Minnesota. 1993.
[11] D. Elliott, The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function, Mathematics of Computation, 18 (1964), 274-284.
[12] Walter Gautsch, Orthogonal Polynomials : Computation and Approximation. Oxford University , Agust 2004
[13] H.Rutishauser Description of a;gol 60 , Springer ,1967
[14] H.S. Wali , Analytic Theory of Continued Fraction , AMS Chelsea \publishing 1948 ; 433pp; reprinted 1973 , first AMS printing 2000
[15] Dafalla Awadalla Gismalla , Numerical Methods and Algorithms using Matlab Programming Language , LAMBERT Academic Publishing https://www.lappublishing.com/site/how-to-publish/14
[16] D. A. Gismalla , Chebyshev Approximation for COS (½ ×—X4) &SIN (½ , Proceedings of the International Conference on Computing, pp.37, ICC 2010, INDIA
[17]D.A.Gismalla ,et al .Acceleration of Convergence of Series for Certain Multiple Integrals. I. J.C.M, Vol. 24, pp 55-68, 1987.
[18]Abramowitz,M ,and Stegun ,I.A. 1964 , Handbook of Mathematical Functions , Applied Mathematics Series , Volum 55 (Washington : National Bureau of Standards; reprinted 1968 by Dover Publication New York )
[19] E.Hernendez ,K.Commedfordy , M. J. Perez_ quiles. MATLAB GUI for computing Bessel functions using fraction algorithm Revista Brasilerira de E nsino de Fasica , v.33,n.1,1303 (2011)
[20] Walter Gander and Pedro Connent Polynomial Recurrence ,Newton Correction and Continued Fraction , Of Computational Science, Institute ETH ZUTICH
[21] A.M. Cohen Chebyshev Approximations for cos(½ x2 pi) and sin ( ½ x2 pi) with Applications, Applied Numerical mathematics 6(1990) , 159- 167 , NORTH _HOLLAND
[22] C.W.Clenshaw , Chebyshev Series for Mathematical Functions, N.P.L. Mathematical Table 5 (HMSO, LONDON, 1962).
[23] Pi from a Continued Fraction http://www.cs.utsa.ed/~wagner/pi/ruby/pi works.html
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