2,217 research outputs found
A Fast Algorithm for MacMahon's Partition Analysis
This paper deals with evaluating constant terms of a special class of
rational functions, the Elliott-rational functions. The constant term of such a
function can be read off immediately from its partial fraction decomposition.
We combine the theory of iterated Laurent series and a new algorithm for
partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega
calculus, which (partially) avoids the "run-time explosion" problem when
eliminating several variables. We discuss the efficiency of our algorithm by
investigating problems studied by Andrews and his coauthors; our running time
is much less than that of their Omega package.Comment: 22 page
Smaller SDP for SOS Decomposition
A popular numerical method to compute SOS (sum of squares of polynomials)
decompositions for polynomials is to transform the problem into semi-definite
programming (SDP) problems and then solve them by SDP solvers. In this paper,
we focus on reducing the sizes of inputs to SDP solvers to improve the
efficiency and reliability of those SDP based methods. Two types of
polynomials, convex cover polynomials and split polynomials, are defined. A
convex cover polynomial or a split polynomial can be decomposed into several
smaller sub-polynomials such that the original polynomial is SOS if and only if
the sub-polynomials are all SOS. Thus the original SOS problem can be
decomposed equivalently into smaller sub-problems. It is proved that convex
cover polynomials are split polynomials and it is quite possible that sparse
polynomials with many variables are split polynomials, which can be efficiently
detected in practice. Some necessary conditions for polynomials to be SOS are
also given, which can help refute quickly those polynomials which have no SOS
representations so that SDP solvers are not called in this case. All the new
results lead to a new SDP based method to compute SOS decompositions, which
improves this kind of methods by passing smaller inputs to SDP solvers in some
cases. Experiments show that the number of monomials obtained by our program is
often smaller than that by other SDP based software, especially for polynomials
with many variables and high degrees. Numerical results on various tests are
reported to show the performance of our program.Comment: 18 page
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