Application of Gauss quadrature rule in finding bounds for solution of linear systems of equations

In this paper, application of the quadrature rules such as Gauss, Gauss–Radau and Gauss–Lobatto, for finding a bound for f(A) where f is a smooth function of the matrix A  , will be discussed. We also use this approach for the case where f(x)=1x or f(A) = A−1. Employing this technique, we can find bounds for A−1 and consequently bounds for x = A−1b will be obtained. An example is given to show some applications of the presented method.