Collaborative Research on Quadrature and Orthogonal Polynomials in Large Scale Computation

对于大型的、可能是稀疏的、对称矩阵的函数的上界和下界的廉价计算在过去的几年里受到了很多的关注。本文提出了新的方法和新的应用，并讨论了允许矩阵非对称的扩展。矩阵泛函的上界和下界的计算是基于高斯型正交规则对的求值。本文提出了一种新的高斯型求积规则，该规则具有适合于估计非对称矩阵的矩阵泛函的性质。与这些正交规则相关的测度可以是不定值或复值。将这些正交规则应用于非对称矩阵线性方程组由迭代法确定的近似解的误差范数估计。此外，应用于非线性问题的迭代解也将被研究。科学计算的一个重要方面是解决结果的可靠性问题。特别重要的是，要知道计算结果的准确度，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，对这些问题来说，了解结果的数值准确性是非常重要的。这个项目通过发展计算一个方程组的某些测度的上界和下界的理论来解决这个问题。一个特殊的应用是求得大方程组近似解精度的上界和下界。

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.