Some Linear Transformations with MATLAB Applications, Especially On Infinite Integrals
D.A.GISMALLA , OHAMED H.A.ELHEBER
In this paper , We have considered , listed and described some linear transformations with their applications to sum infinite series accurately to a reasonable number of decimal places , e.g. at least to 10 decimal places of accuracy for most of the applications .We give attentions and especial consideration for evaluating an infinite integral section∮ 5 . This can be done by many linear transformations that can be found in the literature. However , here, We apply two fundamental Linear transformations called Euler's Transformations in section∮ 3. , Euler-Maclaurin Summation Formula in section∮ 4 and Levin's transform in . ∮ 5.1.The Matlab program written for Euler's Transformation uses the diagonal elements for forward difference table as can be found in the Matlab File ForwadDifrenceTable.m in Fig.(6(b) ) or obtained directly from ∆ = (−)( ) () as in the Matlab File EulerTransform.m in Fig.( 5 ) . This means the Matlab program written for Euler's Transformations uses slightly two different approaches and the reader should distinguish carefully between them. The most attractive in this paper the generalized linear transformations which are called Chebyshev Weights and Salzer's Means in section ∮ 2. For these two methods some fundamental background and the theory of generalized linear transformations are given to furnished to the reader how to follow the program to understand first the constraints on regularity or irregularity for these two method or similar ones .This means that the convergence of the method can be achieved only if these transformations are regular along any path. The path can be along zigzag ,column or a row. To be confident first that our program will be straight and accurately right ,We have used eqn.(3) and eqn.(4) to ensure regularity along any rows according to Toplitz limit theorem as in Theorem 3. Lastly in section ∮ 5 ,We consider the most robot method Levin's transform that have been pointed in many references such as [ 2] and . Levin's transform have been analyzed and modified in many different ways to suit poorly slowly convergent series as in [ 2] .
Toeplitz limit. Chebyshev, Salzer and Levin's Transformations. Gaussian Quadtrue.
 Dafalla Awadalla Gismalla , Numerical Methods and Algorithms using Matlab Programming Language , LAMBERT Academic Publishing https://www.lap-publishing.com/site/howto-publish/14
 D.A.Gismalla ,et al .Acceleration of Convergence of Series for Certain Multiple Integrals. International Journal of Computing Mathematics I. J.C.M, Vol. 24, pp 55-68, 1987.
 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 )
 Clark-Erik Frberg , Numerical Mathematics, Theory and Computer , Application. The Benjamin /Cummings Publishing Company, Inc , 1985
 Jet Wimp , Sequence Transformations and Their applications , Mathematics in Sciences and Engineering , Vol.154, 1981 ,Academic Press, London ,New York.
 D.A. Gismalla, Survey on Transformations for Infinite Series to Continued Fractions with MATLAB Program for Computing Some, International Journal of Innovative Research in Engineering & Management(IJIERM) ISSN:2350-0557,Volume 3 , Issue 3 , May 2016 http://www.ijirem.org/IJIREM/Arch.aspx?rep=12
 D.A.Gismalla On Evaluations Some Special Functions with Pade Approximant in Mupad Interface and Mathematica , International Journal Engineering Research & Management(IJERM) ISSN:2349-2058 , Volume 3 ,Issue 5 ,May 2016 https://www.ijerm.com/vol/Volume-03- Issue-05
 D.A.Gismalla & Mohamed H. A. ELHEBIR,Factorizations , Multiplications and Views on Parallel Computing. International Journal of Innovative Research in Engineering & Management (IJIREM) ISSN: 2350-0557, Volume-3, Issue-6, November-2016 http://www.ijirem.org/IJIREM/Arch.aspx?rep=15
 Peyerimhoff.A. (1969)." Lecture Notes of Summability" . Lecture Notes In Mathematics
 Hardy ,G.H.(1956). " Divergent Series." Oxford Univ. Press , London and
 Methods for the Evaluations of N-dimensional Integrals, (a Thesis for the Ph.D. , Software was written using FORTRAN Language with Full Machine Accuracy), University of Wales, U.K. 1984 http://www.philadelphia.edu.jo/newlibrary/thesisdissertation/406-mathematics/25864-4382
[D.A.GISMALLA , OHAMED H.A.ELHEBER (2017) Some Linear Transformations with MATLAB Applications, Especially On Infinite Integrals IJIREM Vol-4 Issue-1 Page No-547-557] (ISSN 2350 - 0557). www.ijirem.org