74 research outputs found
Effective Scalar Products for D-finite Symmetric Functions
Many combinatorial generating functions can be expressed as combinations of
symmetric functions, or extracted as sub-series and specializations from such
combinations. Gessel has outlined a large class of symmetric functions for
which the resulting generating functions are D-finite. We extend Gessel's work
by providing algorithms that compute differential equations these generating
functions satisfy in the case they are given as a scalar product of symmetric
functions in Gessel's class. Examples of applications to k-regular graphs and
Young tableaux with repeated entries are given. Asymptotic estimates are a
natural application of our method, which we illustrate on the same model of
Young tableaux. We also derive a seemingly new formula for the Kronecker
product of the sum of Schur functions with itself.Comment: 51 pages, full paper version of FPSAC 02 extended abstract; v2:
corrections from original submission, improved clarity; now formatted for
journal + bibliograph
Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators
We study tight bounds and fast algorithms for LCLMs of several linear
differential operators with polynomial coefficients. We analyze the arithmetic
complexity of existing algorithms for LCLMs, as well as the size of their
outputs. We propose a new algorithm that recasts the LCLM computation in a
linear algebra problem on a polynomial matrix. This algorithm yields sharp
bounds on the coefficient degrees of the LCLM, improving by one order of
magnitude the best bounds obtained using previous algorithms. The complexity of
the new algorithm is almost optimal, in the sense that it nearly matches the
arithmetic size of the output.Comment: The final version will appear in Proceedings of ISSAC 201
The Distribution of Patterns in Random Trees
Let denote the set of unrooted labeled trees of size and let
be a particular (finite, unlabeled) tree. Assuming that every tree of
is equally likely, it is shown that the limiting distribution as
goes to infinity of the number of occurrences of as an induced subtree is
asymptotically normal with mean value and variance asymptotically equivalent to
and , respectively, where the constants and
are computable
Low Complexity Algorithms for Linear Recurrences
We consider two kinds of problems: the computation of polynomial and rational
solutions of linear recurrences with coefficients that are polynomials with
integer coefficients; indefinite and definite summation of sequences that are
hypergeometric over the rational numbers. The algorithms for these tasks all
involve as an intermediate quantity an integer (dispersion or root of an
indicial polynomial) that is potentially exponential in the bit size of their
input. Previous algorithms have a bit complexity that is at least quadratic in
. We revisit them and propose variants that exploit the structure of
solutions and avoid expanding polynomials of degree . We give two
algorithms: a probabilistic one that detects the existence or absence of
nonzero polynomial and rational solutions in bit
operations; a deterministic one that computes a compact representation of the
solution in bit operations. Similar speed-ups are obtained in
indefinite and definite hypergeometric summation. We describe the results of an
implementation.Comment: This is the author's version of the work. It is posted here by
permission of ACM for your personal use. Not for redistributio
An Extension of Zeilberger's Fast Algorithm to General Holonomic Functions
We extend Zeilberger's fast algorithm for definite hypergeometric summation to non-hypergeometric holonomic sequences. The algorithm generalizes to differential and~-cases as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy
Differential Equations for Algebraic Functions
It is classical that univariate algebraic functions satisfy linear
differential equations with polynomial coefficients. Linear recurrences follow
for the coefficients of their power series expansions. We show that the linear
differential equation of minimal order has coefficients whose degree is cubic
in the degree of the function. We also show that there exists a linear
differential equation of order linear in the degree whose coefficients are only
of quadratic degree. Furthermore, we prove the existence of recurrences of
order and degree close to optimal. We study the complexity of computing these
differential equations and recurrences. We deduce a fast algorithm for the
expansion of algebraic series
Algorithms Seminar, 2000-2001
These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, asymptotic analysis, computational biology, and average-case analysis of algorithms and data structures
Algorithms Seminar, 2002-2004
These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, and the asymptotic analysis of algorithms, data structures, and network protocols
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