4,826 research outputs found
A polynomial-division-based algorithm for computing linear recurrence relations
International audienceSparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidi-mensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp– Massey–Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp–Massey–Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. Finally, we give a partial solution to the transformation of this algorithm into an adaptive one
Automated Generation of Non-Linear Loop Invariants Utilizing Hypergeometric Sequences
Analyzing and reasoning about safety properties of software systems becomes
an especially challenging task for programs with complex flow and, in
particular, with loops or recursion. For such programs one needs additional
information, for example in the form of loop invariants, expressing properties
to hold at intermediate program points. In this paper we study program loops
with non-trivial arithmetic, implementing addition and multiplication among
numeric program variables. We present a new approach for automatically
generating all polynomial invariants of a class of such programs. Our approach
turns programs into linear ordinary recurrence equations and computes closed
form solutions of these equations. These closed forms express the most precise
inductive property, and hence invariant. We apply Gr\"obner basis computation
to obtain a basis of the polynomial invariant ideal, yielding thus a finite
representation of all polynomial invariants. Our work significantly extends the
class of so-called P-solvable loops by handling multiplication with the loop
counter variable. We implemented our method in the Mathematica package Aligator
and showcase the practical use of our approach.Comment: A revised version of this paper is published in the proceedings of
ISSAC 201
Fast algorithm for border bases of Artinian Gorenstein algebras
Given a multi-index sequence , we present a new efficient algorithm
to compute generators of the linear recurrence relations between the terms of
. We transform this problem into an algebraic one, by identifying
multi-index sequences, multivariate formal power series and linear functionals
on the ring of multivariate polynomials. In this setting, the recurrence
relations are the elements of the kerne l\sigma of the Hankel operator
$H$\sigma associated to . We describe the correspondence between
multi-index sequences with a Hankel operator of finite rank and Artinian
Gorenstein Algebras. We show how the algebraic structure of the Artinian
Gorenstein algebra \sigma\sigma yields the
structure of the terms $\sigma\alpha N nAK[x 1 ,. .. , xnIHIA$ and the tables of multiplication by the variables in these
bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with
improved complexity bounds. We present applications of the method to different
problems such as the decomposition of functions into weighted sums of
exponential functions, sparse interpolation, fast decoding of algebraic codes,
computing the vanishing ideal of points, and tensor decomposition. Some
benchmarks illustrate the practical behavior of the algorithm
Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus
We present a probabilistic Las Vegas algorithm for computing the local zeta
function of a genus- hyperelliptic curve defined over with
explicit real multiplication (RM) by an order in a degree-
totally real number field.
It is based on the approaches by Schoof and Pila in a more favorable case
where we can split the -torsion into kernels of endomorphisms, as
introduced by Gaudry, Kohel, and Smith in genus 2. To deal with these kernels
in any genus, we adapt a technique that the author, Gaudry, and Spaenlehauer
introduced to model the -torsion by structured polynomial systems.
Applying this technique to the kernels, the systems we obtain are much smaller
and so is the complexity of solving them.
Our main result is that there exists a constant such that, for any
fixed , this algorithm has expected time and space complexity as grows and the characteristic is large enough. We prove that
and we also conjecture that the result still holds for .Comment: To appear in Journal of Complexity. arXiv admin note: text overlap
with arXiv:1710.0344
Invariant Generation for Multi-Path Loops with Polynomial Assignments
Program analysis requires the generation of program properties expressing
conditions to hold at intermediate program locations. When it comes to programs
with loops, these properties are typically expressed as loop invariants. In
this paper we study a class of multi-path program loops with numeric variables,
in particular nested loops with conditionals, where assignments to program
variables are polynomial expressions over program variables. We call this class
of loops extended P-solvable and introduce an algorithm for generating all
polynomial invariants of such loops. By an iterative procedure employing
Gr\"obner basis computation, our approach computes the polynomial ideal of the
polynomial invariants of each program path and combines these ideals
sequentially until a fixed point is reached. This fixed point represents the
polynomial ideal of all polynomial invariants of the given extended P-solvable
loop. We prove termination of our method and show that the maximal number of
iterations for reaching the fixed point depends linearly on the number of
program variables and the number of inner loops. In particular, for a loop with
m program variables and r conditional branches we prove an upper bound of m*r
iterations. We implemented our approach in the Aligator software package.
Furthermore, we evaluated it on 18 programs with polynomial arithmetic and
compared it to existing methods in invariant generation. The results show the
efficiency of our approach
Fast algorithms for computing isogenies between elliptic curves
We survey algorithms for computing isogenies between elliptic curves defined
over a field of characteristic either 0 or a large prime. We introduce a new
algorithm that computes an isogeny of degree ( different from the
characteristic) in time quasi-linear with respect to . This is based in
particular on fast algorithms for power series expansion of the Weierstrass
-function and related functions
Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K
We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of
higher order q-recurrence equations with rational coefficients. We extend a
method for finding a bound on the maximal power of t in the denominator of
arbitrary rational solutions y(t) as well as a method for bounding the degree
of polynomial solutions from the scalar case to the systems case. The approach
is direct and does not rely on uncoupling or reduction to a first order system.
Unlike in the scalar case this usually requires an initial transformation of
the system.Comment: 8 page
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
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