International Symposium on Symbolic and Algebraic Computation最新文献_第4页

Solving higher order linear differential equations having elliptic function coefficients 求解具有椭圆函数系数的高阶线性微分方程

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608675

Reinhold Burger

{"title":"Solving higher order linear differential equations having elliptic function coefficients","authors":"Reinhold Burger","doi":"10.1145/2608628.2608675","DOIUrl":"https://doi.org/10.1145/2608628.2608675","url":null,"abstract":"We consider the problem of finding closed form solutions of a linear homogeneous ordinary differential equation having coefficients which are elliptic functions. In particular, the input coefficients are assumed to be represented as elements of C(p,p'), where C is the complex number field, and p(x) and p' (x) are the Weierstrass p function and its first derivative, respectively. The specific closed form solutions y(x) which we seek are hyperexponential over C(p,p'), i.e., solutions y(x) such that y' (x)/y(x) is in C(p,p'). Such solutions correspond to first order right-hand factors of the associated linear differential operator, and are analogous to hyperexponential solutions over C(x), in the more well-known case where the coefficients of the ode are in C(x). A previous paper [4] gave an algorithm for equations of second order. The algorithm presented here works for equations of arbitrary order, and will find all such hyperexponential solutions that may exist. It relies on determining the structure of such first order factors to construct an ansatz of a solution, which can then be completely determined by solving a system of multivariate polynomial equations. The algorithm works well for solutions having few singularities and hidden poles, but can slow as the number of such points increases.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128844737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Reduction among bracket polynomials 括号多项式之间的约简

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608630

Hongbo Li, Changpeng Shao, Lei Huang, Yue Liu

引用次数: 4

Online order basis algorithm and its impact on the block Wiedemann algorithm 在线订单基算法及其对块Wiedemann算法的影响

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608647

Pascal Giorgi, R. Lebreton

引用次数: 20

Covering of surfaces parametrized without projective base points 无投影基点的曲面参数化覆盖

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608635

J. Sendra, David Sevilla, Carlos Villarino

引用次数: 13

Linear independence oracles and applications to rectangular and low rank linear systems 线性无关指令及其在矩形和低阶线性系统中的应用

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608673

A. Storjohann, Shiyun Yang

{"title":"Linear independence oracles and applications to rectangular and low rank linear systems","authors":"A. Storjohann, Shiyun Yang","doi":"10.1145/2608628.2608673","DOIUrl":"https://doi.org/10.1145/2608628.2608673","url":null,"abstract":"Randomized algorithms are given for linear algebra problems on an input matrix A ∈ Knxm over a field K. We give an algorithm that simultaneously computes the row and column rank profiles of A in 2r3 + (r2 + n + m + |A|)1+o(1) field operations from K, where r is the rank of A and |A| denotes the number of nonzero entries of A. Here, the +o(1) in cost estimates captures some missing log n and log m factors. The rank profiles algorithm is randomized of the Monte Carlo type: the correct answer will be returned with probability at least 1/2. Given a b ∈ Knx1, we give an algorithm that either computes a particular solution vector x ∈ Kmx1 to the system Ax = b, or produces an inconsistency certificate vector u ∈ K1xn such that uA = 0 and ub ≠ 0. The linear solver examines at most r + 1 rows and r columns of A and has running time 2r3 + (r2 + n + m + |R| + |C|)1+o(1) field operations from K, where |R| and |C| are the number of nonzero entries in the rows and columns, respectively, that are examined. The solver is randomized of the Las Vegas type: an incorrect result is never returned but the algorithm may report FAIL with probability at most 1/2. These cost estimates are achieved by making use of a novel randomized online data structure for the detection of linearly independent rows and columns.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125560719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 6

LLL reducing with the most significant bits 用最有效位降低LLL

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608645

Goel Sarushi, I. Morel, D. Stehlé, G. Villard

引用次数: 14

Fast arithmetic for the algebraic closure of finite fields 有限域代数闭包的快速算法

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608672

L. Feo, Javad Doliskani, É. Schost

引用次数: 6

The asymptotic analysis of some interpolated nonlinear recurrence relations 一些插值非线性递归关系的渐近分析

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608677

Robert M Corless, D. J. Jeffrey, Fei Wang

{"title":"The asymptotic analysis of some interpolated nonlinear recurrence relations","authors":"Robert M Corless, D. J. Jeffrey, Fei Wang","doi":"10.1145/2608628.2608677","DOIUrl":"https://doi.org/10.1145/2608628.2608677","url":null,"abstract":"We study discrete dynamical systems, or recurrence relations, of the general form\u0000 [EQUATION]\u0000 with explicitly known series coefficients αk and α1 ≠ 0. We associate with the discrete system an interpolating continuous system Y (t), such that Y (n) = yn. The asymptotic behaviour of yn can then be investigated through Y (t). The corresponding continuous system is\u0000 [EQUATION]\u0000 where G is called the generator (following Labelle's terminology), and is given by an explicit formula in terms of the recurrence relation. This continuous system may fail to be smooth everywhere but nonetheless may be useful. Analytic solution is only rarely possible.\u0000 We analyze the equation for Y under assumptions of an asymptotic limit, and show that the asymptotic behaviour can be obtained by reverting a series containing logarithms and powers. We introduce a novel reversion based on the Wright ω function.\u0000 An application of the theory is made to functional iteration of the Lambert W function and the asymptotic behaviour of the iteration is obtained.\u0000 The iteration of functions is a central topic in the theory of complex dynamical system, and a sophisticated use of conjugation is only one key tool used there. We show here that Labelle's theory and generator can be used to compute the conjugated mapping of functional iterations to simple non-iterative functions in general. We use the Lambert W function again as an example to illustrate this. We also discuss the curious asymptotic series ln z ~ Σk ≥ 1 W<k>(z).\u0000 This study uses the truncated generalized series tools available in Maple, particularly the logarithmic-and-power series that is usual in Maple. We also use Levin's u-transform as a key piece in interpolating the discrete dynamical system.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129020670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

How to develop a mobile computer algebra system 如何开发一个移动计算机代数系统

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2627492

M. Fujimoto

引用次数: 0

Synthesis of optimal numerical algorithms using real quantifier elimination (case study: square root computation) 利用实量词消去法综合最优数值算法(案例研究:平方根计算)

International Symposium on Symbolic and Algebraic Computation Pub Date : 2014-07-23 DOI: 10.1145/2608628.2608654

Madalina Erascu, H. Hong

{"title":"Synthesis of optimal numerical algorithms using real quantifier elimination (case study: square root computation)","authors":"Madalina Erascu, H. Hong","doi":"10.1145/2608628.2608654","DOIUrl":"https://doi.org/10.1145/2608628.2608654","url":null,"abstract":"We report on on-going efforts to apply real quantifier elimination to the synthesis of optimal numerical algorithms. In particular, we describe a case study on the square root problem: given a real number x and an error bound ε, find a real interval such that it contains [EQUATION] and its width is less than or equal to ε.\u0000 A typical numerical algorithm starts with an initial interval and repeatedly updates it by applying a \"refinement map\" on it until it becomes narrow enough. Thus the synthesis amounts to finding a refinement map that ensures the correctness and optimality of the resulting algorithm.\u0000 This problem can be formulated as a real quantifier elimination. Hence, in principle, the synthesis can be carried out automatically. However, the computational requirement is huge, making the automatic synthesis practically impossible with the current general real quantifier elimination software.\u0000 We overcame the difficulty by (1) carefully reducing a complicated quantified formula into several simpler ones and (2) automatically eliminating the quantifiers from the resulting ones using the state of the art quantifier elimination software.\u0000 As the result, we were able to synthesize semi-automatically, under mild assumptions, a class of optimal maps, which are significantly better than the well known hand-crafted Secant-Newton map. Interestingly, the optimal synthesized maps are not contracting as one would naturally expect.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116770206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 11