Collaborative Research on Quadrature and Orthogonal Polynomials in Large Scale Computation

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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.