10 research outputs found
Bounds for D-finite closure properties
We provide bounds on the size of operators obtained by algorithms for
executing D-finite closure properties. For operators of small order, we give
bounds on the degree and on the height (bit-size). For higher order operators,
we give degree bounds that are parameterized with respect to the order and
reflect the phenomenon that higher order operators may have lower degrees
(order-degree curves)
Efficient Algorithms for Mixed Creative Telescoping
Creative telescoping is a powerful computer algebra paradigm -initiated by
Doron Zeilberger in the 90's- for dealing with definite integrals and sums with
parameters. We address the mixed continuous-discrete case, and focus on the
integration of bivariate hypergeometric-hyperexponential terms. We design a new
creative telescoping algorithm operating on this class of inputs, based on a
Hermite-like reduction procedure. The new algorithm has two nice features: it
is efficient and it delivers, for a suitable representation of the input, a
minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the
telescoper it produces.Comment: To be published in the proceedings of ISSAC'1
Simplifying Multiple Sums in Difference Fields
In this survey article we present difference field algorithms for symbolic
summation. Special emphasize is put on new aspects in how the summation
problems are rephrased in terms of difference fields, how the problems are
solved there, and how the derived results in the given difference field can be
reinterpreted as solutions of the input problem. The algorithms are illustrated
with the Mathematica package \SigmaP\ by discovering and proving new harmonic
number identities extending those from (Paule and Schneider, 2003). In
addition, the newly developed package \texttt{EvaluateMultiSums} is introduced
that combines the presented tools. In this way, large scale summation problems
for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics) can be
solved completely automatically.Comment: Uses svmult.cls, to appear as contribution in the book "Computer
Algebra in Quantum Field Theory: Integration, Summation and Special
Functions" (www.Springer.com