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    The importance of using the “omega calculus” in computer algebra
    (Maltepe Üniversitesi, 2009) Snopce, Halil; Spahiu, Ilir; Aliu, Azir
    In his book ”Combinatory Analysis”, Percy A. MacMahon developed the so called ”Omega calculus”. In this contribution we emphasize the importance of the ”Omega Calculus”. Using the properties of this tool, we investigate the possible aplication in computer algebra.We investigate how the methods presented by Macmahon’s can be applied to the problem of enumerating lattice points in convex polyhedron. A lot of Scientific and Engineering problems require the solution of large systems of linear equations of the form Ax=b in an effective manner. LU-Decomposition offers good choices for solving this problem. QR Factorization has implementation in various problems of linear algebra. Discrete Fourier transformation can be implemented in different problems regarding the signal and image processing, pattern recognition etc. We investigate a possible optimization of these problems finding the lower bound of processing elements (PEs) required by a schedule as a function of n. From a given algorithm, defining a corresponding index space, we consider that the elements of that index space are lattice points inside 3-dimensional convex polyhedron. The faces of the polyhedron are defined by the inequalities which are the consequence of the given algorithm. From these inequalities augmenting by the condition of linear schedule for the corresponding dag, we convert the geometrical interpretation of the problem, into a combinatorial interpretation, exactly into finding of solutions to the system of Diophantine equations. Then we run the Mathematica program DiophantineGF.m. This program calculates the generating function from which is possible to find the number of solutions to the system of Diophantine equalities, which in fact gives the lower bound for the number of processors needed for achieving a given schedule. We give a mathematical explanation and then we confirm the conclusion taking a random example.
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    Some characteristics of systolic arrays
    (Maltepe Üniversitesi, 2009) Spahiu, Ilir; Snopce, Halil; Aliu, Azir
    We investigate a possible optimization of some linear algebra problems which can be solved by parallel processing using the special arrays called systolic arrays. In this paper are used some special types of transformations for the designing of this arrays. We show the characteristics of each one giving the examples of their implementation as well. The main focus is on discussing the advantages of these arrays in parallel computation of matrix product, with special approach to the designing of systolic array for matrix multiplication and discrete Fourier transformation. Multiplication of large matrices requires a lot of computational time and its complexity is O(n 3 ). There are developed many algorithms (both sequential and parallel) with the purpose of minimizing the time of calculations. Systolic arrays are good suited for these purpose. In this paper we show that using a appropriate composite function, the given index space can be mapped in another index space suitable for systolic array. This mapping implicates in finding more optimal arrays for doing the calculations of this type. We show that this can be implemented on the designing of optimal systolic array for Discrete Fourier transformation.