Fast Bit Complexity in Symbolic Computation Algorithms

埃里希·卡尔托芬正在研究数字系数上的算术比特成本与符号计算算法的整体效率之间的关系。他设计并实现了精确多项式和线性代数中基本问题的新算法，以及通过控制中间计算的有理数的长度来实现加速的多项式结果。通过固定或可变精度的浮点运算也可以获得更快的算术成本，这种近似的输入和输出数据是我们研究用于多项式因式分解和结构化系统求解的混合符号/数字算法的主题。稀疏内插问题的随机化算法被作为良好的启发式算法执行，只需有限次数的掷硬币，以保持中间系数较小。LinBox程序库中用于稀疏、结构化和黑盒矩阵的算法通过其通用的、可重复使用的设计，用专门的算法编译，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。像Wolfram Research Inc.的MATHEMICAL A和Maplesoft的Maple这样的程序已经接触到了数百万用户，他们使用这些程序来自动、无错误地执行数学处理的机制。因此，用户可以集中精力解释数学结果，同样重要的是，可以操纵更接近现实的大型数学模型。卡尔托芬的研究为计算机基础数学引擎的基础设施做出了贡献。所调查的加速比使执行速度显著加快，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。在LinBox集团(www.linalg.org)旗下的Kaltofen正在免费提供开发的软件。用户可以下载并运行该算法，该学科的专家可以仔细查看细微之处。

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.