Hybrid Symbolic-Numeric Computing in Distributed LSC Environments

希兹，马库斯·a .北乔治亚大学在实验中收集的数据，或者物理和其他科学中的常数，精度有限，在计算中引入了一定程度的不确定性。传统的精确方法不能揭示具有不确定参数的代数问题的全部解集。通常，这些问题本质上是病态的，因此数值方法对输入参数的微小扰动变得敏感。符号-数值混合算法已被证明能够成功地找到不能精确解决的问题的“最近”解。有效的（多项式时间）算法已经开发用于常见问题，例如计算单变量多项式的近似gcd。对于其他问题，例如寻找最近的奇异Hankel或Toeplitz矩阵，目前尚不知道是否存在有效的算法。本项目将继续研究这一领域，特别是关于近似二元和多元分解的问题。对于符号和数值计算这两个领域，大量的软件库和编程环境都是现成的。对于混合计算，这些库要么具有计算开销大的特性（符号），要么根本缺乏符号支持（数字）。一类新的系统将有限符号能力（LSC）添加到已有的优化了有理数和浮点运算实现的库中。我们将对LSC系统进行调查并作出贡献。这是一个RUI项目。本科生将从事与他们的专业水平相适应的研究。在此过程中，他们还将获得配置局域网和操作在服务器应用中越来越重要的计算机集群的宝贵技能。

Proposal #0098175Hitz, Markus A.North Georgia CollegeData collected in experiments, or constants in physics and other sciences are of limited precision, introducing some degree of uncertainty in computations. Traditional exact methods fail to reveal the entire set of solutions to algebraic problems that have uncertain parameters. Often, these problems are inherently ill-conditioned, such that numerical methods become sensitive to small perturbations of the input parameters.Hybrid symbolic-numeric algorithms have proven to be successful for finding "nearest" solutions of problems that cannot be solved exactly. Efficient (polynomial time) algorithms have been developed for common problems, such as computing approximate GCDs of univariate polynomials. For other problems, e.g., finding the nearest singular Hankel or Toeplitz matrix, it is currently unknown whether there exist efficient algorithms. This project will continue research in this area, in particular on the problem of approximate bivariate and multivariate factorization.For both areas, symbolic and numeric computing, vast software libraries and programming environments are readily available. For hybrid computing those libraries either have features that are computationally expensive (symbolic), or lack symbolic support at all (numeric). A new class of systems adds Limited Symbolic Capabilities (LSC) to existing libraries that have optimized implementations of rational and floating point arithmetic. We will investigate, and contribute to, LSC systems.This is an RUI project. Undergraduate students will engage in research that is appropriate for their level of expertise. In the process they will also gain valuable skills in configuring local area networks, and in operating clusters of computers which become more and more important in server applications.