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

9707359理查德·S·瓦尔加这项建议的目标是通过潜在的理论技术，在复杂平面上用变权多项式和有理逼近来解决基本问题。最近，提出者在这一领域进行了卓有成效的初步调查，产生了一批新的研究手稿，我们认为这些手稿具有突破性。我们清楚地感到，这些初步调查是如此有希望，以至于有理由将这项研究建议提交给国家科学基金会。但是，这一建议的一个重要和新颖的特点是，通过迭代方法有效地求解大型非奇异线性方程组的联系，通过势能理论方法直接与复平面上的这一理论工作相联系。因此，所提出的研究将在复函数理论和数值分析的研究之间架起桥梁，并将具有很强的数值成分。在大规模电路设计、油藏机械采油和核反应堆发电设计中出现的问题都有以下共同之处：对这些物理现象的建模导致了大量的非奇异线性方程组，其未知数可以超过一百万。为了有效地求解这些方程组，通常采用迭代方法。这一建议的目的之一是使用位势理论和复变函数论领域的工具，从理论上攻击一个特定的问题，即如何最好地迭代如此大的方程组。

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.