71 research outputs found

    Prediction based task scheduling in distributed computing

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    In Memory of Vladimir Gerdt

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    Center for Computational Methods in Applied Mathematics of RUDN, Professor V.P. Gerdt, whose passing was a great loss to the scientific center and the computer algebra community. The article provides biographical information about V.P. Gerdt, talks about his contribution to the development of computer algebra in Russia and the world. At the end there are the author’s personal memories of V.P. Gerdt.Настоящая статья - мемориальная, она посвящена памяти руководителя научного центра вычислительных методов в прикладной математике РУДН, профессора В.П. Гердта, чей уход стал невосполнимой потерей для научного центра и всего сообщества компьютерной алгебры. В статье приведены биографические сведения о В.П. Гердте, рассказано о его вкладе в развитие компьютерной алгебры в России и мире. В конце приведены личные воспоминания автора о В.П. Гердте

    Fast in-place accumulated bilinear formulae

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    Bilinear operations are ubiquitous in computer science and in particular in computer algebra and symbolic computation. One of the most fundamental arithmetic operation is the multiplication, and when applied to, e.g., polynomials or matrices, its result is a bilinear function of its inputs. In terms of arithmetic operations, many sub-quadratic (resp. sub-cubic) algorithms were developed for these tasks. But these fast algorithms come at the expense of (potentially large) extra temporary space to perform the computation. On the contrary, classical, quadratic (resp. cubic) algorithms, when computed sequentially, quite often require very few (constant) extra registers. Further work then proposed simultaneously ``fast'' and ``in-place'' algorithms, for both matrix and polynomial operations We here propose algorithms to extend the latter line of work for accumulated algorithms arising from a bilinear formula. Indeed one of the main ingredient of the latter line of work is to use the (free) space of the output as intermediate storage. When the result has to be accumulated, i.e., if the output is also part of the input, this free space thus does not even exist. To be able to design accumulated in-place algorithm we thus relax the in-place model to allow algorithms to also modify their input, therefore to use them as intermediate storage for instance, provided that they are restored to their initial state after completion of the procedure. This is in fact a natural possibility in many programming environments. Furthermore, this restoration allows for recursive combinations of such procedures, as the (non concurrent) recursive calls will not mess-up the state of their callers. We propose here a generic technique transforming any bilinear algorithm into an in-place algorithm under this model. This then directly applies to polynomial and matrix multiplication algorithms, including fast ones

    Implementing the Thull-Yap algorithm for computing Euclidean remainder sequences

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    International audienceThere are two types of integer gcd algorithms: those which compute the sequence of remainders of Euclid's algorithm and those which build different sequences. The former are more difficult to validate and analyse, whereas the latter are simpler and more efficient. When one wants the euclidean remainders (for instance if one wants to compute continued fractions), only the former can be used. Our main focus is the subquadratic time Thull-Yap GCD algorithm, and in fact on its core computing a half gcd (TYHGCD). This algorithm is tricky due to the difficulty in correcting the remainder sequence that comes back from a recursive call. The aim of this work is to revise TYHGCD in order to implement it using GMP. We clarify some points of the algorithm, in particular the stopping conditions that are always difficult to set correctly. We add a base case to speed up the whole algorithm, using Jebelean's quadratic algorithm with a stopping condition. We give our own modified version and add the proofs where needed. We insist on the test phase for this algorithm, giving families of hard cases for all branches, some of which are rarely activated. We give some details on our implementation in GMP using low-level functions, adding some remarks on the use of fast multiplications techniques. We pay attention to the data structure needed to store partial quotients, enabling to navigate rapidly back and forth in the sequence of Euclidean remainders. Benchmarks are provided. Some comments are made on Lichtblau's algorithm, which is close in spirit to the Thull-Yap algorithm

    Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

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    The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables.We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (2017). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus logarithm of the degree) for not only on the average, but all higher moments.[This is the journal version of the conference paper with the same title.
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