Fast Bit Complexity in Symbolic Computation Algorithms

Erich Kaltofen正在研究数字系数运算的位成本与符号计算算法的整体效率之间的联系。他设计并实现了新的算法的基本问题，在确切的多项式和线性代数等多项式resultants，实现加速通过控制长度的中间计算的有理数。更快的算术成本也实现了固定或可变精度浮点运算，这样的近似输入和输出数据是我们的调查的主题混合符号/数值算法多项式因式分解和结构化系统求解。稀疏插值问题的随机算法被执行为具有有限数量的硬币翻转以保持中间系数小的良好算法。LinBox程序库中的稀疏矩阵、结构矩阵和黑盒矩阵的算法通过其通用的、可重用的设计，用专门的算法进行编译，例如，特别有效的有限域运算。符号计算领域的首要目标是在计算机的帮助下进行数学运算。 Wolfram Research Inc.的Mathematica等程序。Maplesoft的Maple和Maple已经拥有数百万用户，他们使用它们来自动和无错误地执行数学操作的机制。 因此，用户可以集中精力解释数学结果，同样重要的是，操纵更接近现实的大型数学模型。Kaltofen的研究有助于计算机上基础数学引擎的基础设施。研究的加速使执行速度明显加快，从而允许更好的模型，并为从实践科学家到高中生的更多用户提供数学服务器。Kaltofen在LinBox集团（www.linalg.org）的保护下免费提供所开发的软件。用户可以下载并运行算法，该学科的专家可以仔细检查细节。

Erich Kaltofen is studying the connection of the bit cost of arithmetic on the numeric coefficients to the overall efficiency of symbolic computation algorithms. He designs and implements new algorithms for fundamental problems in exact polynomial and linear algebra and such polynomial resultants that achieve speedup through controlling the lengths of the intermediately computed rational numbers. Faster arithmetic cost is also achieved by fixed or variable precision floating point operations, and such approximate input and output data is the subject of our investigations into hybrid symbolic/numeric algorithms for polynomial factorization and structured system solving. Randomized algorithms for sparse interpolation problems are being executed as good heuristics with a limited number of coin flips in order to keep intermediate coefficients small. The algorithms in the LinBox program library for sparse, structured and black box matrices through its generic, reusable design are compiled with arithmetic that is specialized, for example, for particularly efficient finite field operations.The overarching goal of the field of symbolic computation is doing mathematics with the aid of a computer. Programs such as Mathematica by Wolfram Research Inc. and Maple by Maplesoft have already reached millions of users, who use them to automatically and error-free perform the mechanics of mathematical manipulation. Thus, the users can concentrate on the interpretation of the mathematical results and, equally important, manipulate large mathematical models that are closer to reality. Kaltofen's research contributes to the infrastructure of the underlying mathematics engine on the computer. The investigated speedups make the execution significantly faster, thus allowing even better models and providing mathematics servers to even more users ranging from practicing scientists to high school students. Kaltofen under the umbrella of the LinBox group (www.linalg.org) is making the developed software freely available. Users can download and run the algorithms and experts in the discipline can scrutinize the fine points.