Weighted Approximation in the Complex Plane and Iterative Methods, via Potential Theory and Function Theory

9707359理查德S.瓦尔加 该提案的目标是解决多项式中的基本问题， 有理逼近，具有不同的权重，在复平面上，通过潜在的理论技术。近年来，提出者们在这一领域开始了卓有成效的初步调查，产生了一些新的研究成果 手稿，我们认为这是突破性的。我们清楚地感到，这些初步调查是如此有希望，以保证 将该研究计划提交给国家科学基金会。 但是，该提案的一个重要且新颖的特征是，通过迭代方法有效求解大型非奇异线性方程组的联系与这项理论工作直接相关。 用位势理论方法研究复平面。因此，本文的研究是复变函数理论与数值分析研究的桥梁，具有很强的数值性。 在大规模电路设计中出现的问题，通过石油储层力学采油，以及用于发电的核反应堆的设计 电力，都有以下共同点：建模 这些物理现象导致大量的非奇异线性系统， 方程，其未知数可以超过一百万。为了有效地求解这些方程组，通常使用迭代法。这个建议的一个目的是使用工具，从潜在的理论和dd复函数理论领域，从理论上攻击一个具体的问题，如何最好地迭代 如此庞大的方程组

9707359 Richard S. Varga The goal of this proposal is to attack basic problems in polynomial and rational approximation, with varying weights, in the complex plane, by means of potential theoretic techniques. Recently, the proposers have begun fruitful initial investigations in this area, resulting in a number of new research manuscripts, which we consider to be of break-through character. We clearly feel that these initial investigations are so promising as to warrant the submission of this research proposal to the National Science Foundation. But, an important and novel feature of this proposal is that connections, with the efficientsolution of large nonsingular systems of linear equations by iterative methods, are directly associated with this theoretical work in the complex plane by potential theoretic methods. Thus, the proposed research bridges research in complex function theory and numerical analysis, and will have a strong numerical component. Problems arising in large-scale circuit design, oil recovery via petroleum reservoir machanics, and the design of nuclear reactors for generating electrical power, all have the following item in common: the modeling of these physical phenomena leads to massive nonsingular systems of linear equations, whose unknowns can exceed one million. To efficently solve these systems, iterative methods are commonly used. One aim of this proposal is to use tools, from the fields of potential theory anddd complex function theory, to theoretically attack a specific question on how best to iteratively such large systems of equations.