In this thesis, we consider two interesting problems on computational mathematics, named inverse eigenvalue problems and Ostrowski-like type integral inequalities. We first give an introduction in Chapter 1. In Chapter 2, we propose a Gauss-Newtonlike method for finding least squares solutions to inverse eigenvalue problems. We show that the proposed method converges under some mild conditions. In particular, if the method converges to the exact solution, the convergence rate is at least quadratic in the root sense. Numerical examples are given to justify the theoretical result. In Chapter 3, we consider some Ostrowski-like type integral inequalities proposed by Huy and Ngˆo. We give some improvements of these inequalities which are inspired by a result established recently. An example is also given to verify our improvements. This thesis ends with a chapter of a brief conclusion.