11 research outputs found
Quasi-optimal multiplication of linear differential operators
We show that linear differential operators with polynomial coefficients over
a field of characteristic zero can be multiplied in quasi-optimal time. This
answers an open question raised by van der Hoeven.Comment: To appear in the Proceedings of the 53rd Annual IEEE Symposium on
Foundations of Computer Science (FOCS'12
Evaluating parametric holonomic sequences using rectangular splitting
We adapt the rectangular splitting technique of Paterson and Stockmeyer to
the problem of evaluating terms in holonomic sequences that depend on a
parameter. This approach allows computing the -th term in a recurrent
sequence of suitable type using "expensive" operations at the cost
of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either
naive evaluation or asymptotically faster algorithms for ranges of
encountered in applications. As an example, fast numerical evaluation of the
gamma function is investigated. Our work generalizes two previous algorithms of
Smith.Comment: 8 pages, 2 figure
Fast multidimensional Bernstein-Lagrange algorithms
In this paper we present two fast algorithms for the BĂ©zier curves and surfaces of an arbitrary dimension. The first algorithm evaluates the Bernstein-BĂ©zier curves and surfaces at a set of specific points by using the fast Bernstein-Lagrange transformation. The second algorithm is an inversion of the first one. Both algorithms reduce the initial problem to computation of some discrete Fourier transformations in the case of geometrical subdivisions of the d-dimensional cube. Their orders of computational complexity are proportional to those of corresponding d-dimensional FFT-algorithm, i.e. to O (N logN) + O (dN), where N denotes the order of the Bernstein-BĂ©zier curves
Fast systematic encoding of multiplicity codes
We present quasi-linear time systematic encoding algorithms for multiplicity
codes. The algorithms have their origins in the fast multivariate interpolation
and evaluation algorithms of van der Hoeven and Schost (2013), which we
generalise to address certain Hermite-type interpolation and evaluation
problems. By providing fast encoding algorithms for multiplicity codes, we
remove an obstruction on the road to the practical application of the private
information retrieval protocol of Augot, Levy-dit-Vehel and Shikfa (2014)
Nearly Optimal Computations with Structured Matrices
We estimate the Boolean complexity of multiplication of structured matrices
by a vector and the solution of nonsingular linear systems of equations with
these matrices. We study four basic most popular classes, that is, Toeplitz,
Hankel, Cauchy and Van-der-monde matrices, for which the cited computational
problems are equivalent to the task of polynomial multiplication and division
and polynomial and rational multipoint evaluation and interpolation. The
Boolean cost estimates for the latter problems have been obtained by Kirrinnis
in \cite{kirrinnis-joc-1998}, except for rational interpolation, which we
supply now. All known Boolean cost estimates for these problems rely on using
Kronecker product. This implies the -fold precision increase for the -th
degree output, but we avoid such an increase by relying on distinct techniques
based on employing FFT. Furthermore we simplify the analysis and make it more
transparent by combining the representation of our tasks and algorithms in
terms of both structured matrices and polynomials and rational functions. This
also enables further extensions of our estimates to cover Trummer's important
problem and computations with the popular classes of structured matrices that
generalize the four cited basic matrix classes.Comment: (2014-04-10
Power Series Composition and Change of Basis
Efficient algorithms are known for many operations on truncated power series
(multiplication, powering, exponential, ...). Composition is a more complex
task. We isolate a large class of power series for which composition can be
performed efficiently. We deduce fast algorithms for converting polynomials
between various bases, including Euler, Bernoulli, Fibonacci, and the
orthogonal Laguerre, Hermite, Jacobi, Krawtchouk, Meixner and
Meixner-Pollaczek
Nearly optimal computations with structured matrices
International audienceWe estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic and most popular classes, that is, Toeplitz, Hankel, Cauchy and Vandermonde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in [10], except for rational interpolation. We supply them now as well as the Boolean complexity estimates for the important problems of multiplication of transposed Vandermonde matrix and its inverse by a vector. All known Boolean cost estimates from [10] for such problems rely on using Kronecker product. This implies the d-fold precision increase for the d-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions. This also enables further extensions of our estimates to cover Trummerâs important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes, as well as the transposed Vandermonde matrices. It is known that the solution of Toeplitz, Hankel, Cauchy, Vandermonde, and transposed Vandermonde linear systems of equations is generally prone to numerical stability problems, and numerical problems arise even for multiplication of Cauchy, Vandermonde, and transposed Vandermonde matrices by a vector. Thus our FFT-based results on the Boolean complexity of these important computations could be quite interesting because our estimates are reasonable even for more general classes of structured matrices, showing rather moderate growth of the complexity as the input size increases
Algorithmes rapides pour les polynÎmes, séries formelles et matrices
Notes d'un cours dispensĂ© aux JournĂ©es Nationales du Calcul Formel 2010International audienceLe calcul formel calcule des objets mathĂ©matiques exacts. Ce cours explore deux directions : la calculabilitĂ© et la complexitĂ©. La calculabilitĂ© Ă©tudie les classes d'objets mathĂ©matiques sur lesquelles des rĂ©ponses peuvent ĂȘtre obtenues algorithmiquement. La complexitĂ© donne ensuite des outils pour comparer des algorithmes du point de vue de leur efficacitĂ©. Ce cours passe en revue l'algorithmique efficace sur les objets fondamentaux que sont les entiers, les polynĂŽmes, les matrices, les sĂ©ries et les solutions d'Ă©quations diffĂ©rentielles ou de rĂ©currences linĂ©aires. On y montre que de nombreuses questions portant sur ces objets admettent une rĂ©ponse en complexitĂ© (quasi-)optimale, en insistant sur les principes gĂ©nĂ©raux de conception d'algorithmes efficaces. Ces notes sont dĂ©rivĂ©es du cours " Algorithmes efficaces en calcul formel " du Master Parisien de Recherche en Informatique (2004-2010), co-Ă©crit avec FrĂ©dĂ©ric Chyzak, Marc Giusti, Romain Lebreton, Bruno Salvy et Ăric Schost. Le support de cours complet est disponible Ă l'url https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-2