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

在过去的几年里，大的，可能是稀疏的，对称矩阵的泛函的上界和下界的廉价计算受到了很多关注。这个建议关注新的方法和新的应用，并讨论了扩展，允许矩阵是非对称的。矩阵泛函的上界和下界的计算是基于对高斯型求积规则的评估。概述的工作提出了研究新的高斯型求积规则的性质，使它们适合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定值或复值的。这些正交规则的应用程序，以估计的误差在近似解的线性方程组与非对称矩阵的迭代方法确定的范数将被追求。此外，还将研究非线性问题的迭代解的应用。科学计算的一个重要方面是结果的可靠性。特别是，重要的是要知道计算结果的精度，由误差来衡量。科学计算中普遍存在的一类问题是大型代数方程组的求解。由于这类方程的解是如此广泛，它们代表了一类问题，对于这类问题，结果的数值精度的知识是非常重要的。这个项目通过发展计算方程系统的某些测量的上界和下界的理论来解决这个问题。一个特殊的应用是得到上界和下界的精度近似解的大型系统的方程。

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.