10 research outputs found
Trading Order for Degree in Creative Telescoping
We analyze the differential equations produced by the method of creative
telescoping applied to a hyperexponential term in two variables. We show that
equations of low order have high degree, and that higher order equations have
lower degree. More precisely, we derive degree bounding formulas which allow to
estimate the degree of the output equations from creative telescoping as a
function of the order. As an application, we show how the knowledge of these
formulas can be used to improve, at least in principle, the performance of
creative telescoping implementations, and we deduce bounds on the asymptotic
complexity of creative telescoping for hyperexponential terms
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
Convolution surfaces with varying radius: Formulae for skeletons made of arcs of circles and line segments
International audienceWe develop closed form formulae for the computation of the defining fields of convolutions surfaces. The formulae are obtained for power inverse kernels with skeletons made of line segments or arcs of circle. To obtain the formulae we use Creative Telescoping and describe how this technique can be used for other families of kernels and skeleton primitives. We apply the new formulae to obtain convolution surfaces around skeletons, some of them closed curves. We showcase how the use of arcs of circles greatly improves the visualization of the surface around a general curve compared with a segment based approach