Some Linear Transformations with MATLAB Applications, Especially On Infinite Integrals

D.A.GISMALLA; OHAMED H.A.ELHEBER

Abstract

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 sectionâˆ® 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 sectionâˆ® 3. , Euler-Maclaurin Summation Formula in sectionâˆ® 4 and Levin's transform in . âˆ® 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 âˆ† = (−)( ) () 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 âˆ® 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 âˆ® 5 ,We consider the most robot method Levin's transform that have been pointed in many references such as [ 2] and [12]. Levin's transform have been analyzed and modified in many different ways to suit poorly slowly convergent series as in [ 2] .

Keywords

Toeplitz limit. Chebyshev, Salzer and Levin's Transformations. Gaussian Quadtrue.

References

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Cites this article as

D.A.GISMALLA, O. H. , "Some Linear Transformations with MATLAB Applications, Especially On Infinite Integrals", International Journal of Innovative Research in Engineering and Management (IJIREM), Vol-4, Issue-1, Page No-547-557, 2017. Available from: