## The importance of using the “omega calculus” in computer algebra

#### Citation

Snopce, H., Spahiu, I. ve Aliu, A. (2009). The importance of using the “omega calculus” in computer algebra. Maltepe Üniversitesi. s. 187.#### Abstract

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.

#### Source

International Conference of Mathematical Sciences#### Collections

- Makale Koleksiyonu [586]

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