Optimization, Randomization, and Generalization in Symbolic Computation

Erich Kaltofen建议研究因数字数据而不精确的问题公式的数学优化技术；作为设计快速算法的强大工具的随机化技术，以及通过通用、可重复使用的编程和接口协议等方法将这些算法应用到多个领域的计算机程序。符号/数字领域的一个困难是，由于物理测量或数值计算而具有不精确系数的输入可能缺少感兴趣的已知属性。例如，多项式不是双实根，而是两个相距很近的共轭复根。优化问题是高效地计算具有期望奇异性的最近输入，我们的焦点将集中在系数方向变化(无穷大范数)下具有公根的最近多项式对、最近的奇异Toeplitz矩阵和因数在复数上的最近的二元复多项式，后两个在任何合理范数下。黑盒矩阵被存储为执行矩阵乘以向量积的函数。基于Lanczos/Wiedemann方法的精确域算法是基于Lanczos/Wiedemann方法进行线性代数计算的有效算法，矩阵预条件和随机投影的Krylov子间隔等随机化技术避免了当系数来自有限域时的自正交性，以及一般算法由于矩阵的某些不变因素而崩溃。我们建议研究不同的线性代数问题是如何相互关联的，例如，如果非齐次黑箱线性系统的解的计算可以有效地归结为计算黑箱矩阵的列相关性。泛型编程是一种软件设计技术，它概括了一个程序，以便它可以与多个基础领域一起使用。我们的黑盒算法的拟议实现不仅允许在本地系数域上插入黑盒矩阵，而且代码还从跨通用接口的专业微调的现有包中导入内部需要的运算，如多项式和向量运算。我们的库的应用程序接口为黑盒对象提供了一种序列化机制，该机制符合MathML/OpenMath的原理，并允许互联网通信。最后，我们提出了对两个经典问题的研究。首先，给出了高次稀疏多元有理系数多项式的稠密低次因子的多项式时间计算，推广了Hendrik W.Lenstra，Jr.第二，将矩阵乘向量函数变换为转置向量乘矩阵运算的转置原理的时间和空间效率的实现。

Erich Kaltofen proposes to study mathematical optimization techniques on problem formulations that are imprecise due to numerical data; randomization techniques that are the power tools in the design of fast algorithms with exact arithmetic on black box matrices; and generalization of the computer programs that implement these algorithms to multiple domains with methodologies such as generic, reusable programming and interface protocols. A difficulty in the symbolic/numeric area is that inputs with imprecise coefficients as a result of a physical measurement or a numerical computation can lack a known property that is of interest. For example, instead of a double real root, a polynomial has two conjugate complex roots that are very close together. The optimization problem is to compute efficiently the nearest input that has the desired singularity.Our focus will be on the nearest pair of polynomials with a common root under coefficient-wise change (infinity norm), the nearest singular Toeplitz matrix, and the nearest bi-variate complex polynomial that factors over the complex numbers, the latter two under any reasonable norm. A black box matrix is stored as a function that performs the matrix-times vector product. Efficient algorithms for performing linear algebra computations with exact field arithmetic, such as solving a linearsystem with a black box coefficient matrix, are based on the Lanczos/Wiedemann approach.Randomization techniques, for instance, matrix pre-conditioning and Krylov sub-spacing with random projections, avoid self-orthogonality when the coefficients are from a finite field and general algorithmicbreak-down due to certain invariant factors of the matrix. We propose to investigate how the different linear algebra problems are interelated, for example if the computation of a solution of an inhomogeneous black box linear system can be efficiently reduced to the computation of a column dependency of a black box matrix. Generic programming is a software design technique that generalizes a program so that it can be used with more than one underlying domain. The proposed implementation of our black box algorithms notonly permits the plug-in of a black box matrix over a native coefficient field, but the code also imports internally needed operations like polynomial and vector operations from expertly fine-tuned existing packages across generic interfaces. Our library's application program interface provides a serialization mechanism for black box objects that is compliant with the MathML/OpenMath philosophy and that allowsInternet communication. Finally, we propose research on two classical problems. First, the polynomial-time computation of dense, low-degree factors of high-degree sparse multi-variate polynomials with rational coefficients, which would generalize a recent related result by Hendrik W. Lenstra, Jr. Second, a time- and space-efficient realization of the transposition principle for transforming a matrix-times-vector function to the transpose vector-times-matrix operation.