Collaborative Research on Quadrature and Orthogonal Polynomials in Large-Scale Computation

在过去的几年里，计算大型的，可能是稀疏的对称矩阵的泛函的上下界的费用很低，受到了很大的关注。这项建议涉及新的方法和新的应用，并讨论了允许矩阵非对称的扩展。矩阵泛函的上下界的计算是基于对高斯型求积规则对的计算。本文的主要工作是研究新的高斯型求积规则，这些求积规则的性质使其适用于估计非对称矩阵的矩阵泛函。与这些求积规则相关联的度量可以是不确定的或复值的。这些求积规则将应用于由迭代方法确定的具有非对称矩阵的线性方程组的近似解的误差范数的估计。此外，还将研究其在非线性问题迭代求解中的应用。科学计算的一个重要方面是解决结果的可靠性。特别是，重要的是要知道计算结果的精度，用误差来衡量。科学计算中普遍存在的一类问题是求解大型代数方程组。由于这类方程的解是如此广泛，它们代表了一类问题，对其结果的数值精度的了解是非常重要的。这个项目通过发展计算方程组某些措施的上下限的理论来解决这个问题。一个特殊的应用是得到大型方程组近似解的精度的上下界。

The inexpensive computation of upper and lower bounds for functionals of large, possibly sparse, symmetric matrices has received a lot of attention in the last few years. This proposal is concerned with new methods and new applications, and discusses extensions that allow the matrices to be nonsymmetric. The computation of upper and lower bounds for matrix functionals is based on the evaluation of pairs of Gauss-type quadrature rules. The outlined work proposes to study new quadrature rules of Gauss-type with properties which make them suitable for estimating matrix functional of nonsymmetric matrices. The measure associated with these quadrature rules may be indefinite or complex valued. Applications of these quadrature rules to the estimation of the norm of the error in the approximate solutions determined by iterative methods for linear systems of equations with nonsymmetric matrices will be pursued. Furthermore, applications to the iterative solutions of nonlinear problems will also be studied. An important aspect of scientific computations addresses the reliability of the results. In particular, it is important to know the accuracy, measured by the error, of a computed result. One class of problems ubiquitous in scientific computing is the solution of large systems of algebraic equations. Since the solution of these kinds are equations is so widespread, they represent a class of problems for which knowledge of the numerical accuracy of the results is of great importance. This project addresses the issue by developing theory for computing the upper and lower bounds for certain measures of a system of equations. One particular application is to get the upper and lower bounds on the accuracy of approximate solutions of large systems of equations.