In this thesis, a brief introduction to the mathematical background of the problems under consideration is given in Chapter 1. It is well known that the conjugate gradient method is an efficient method for solving linear system. In Chapter 2, we apply the conjugate gradient method to solve the inverse eigenvalue problems, which is a nonlinear problem. We prove that the method converges locally. Numerical results show that the method converges even for some initial guesses that are not close to the true solution. In Chapter 3, the superconvergent points for the evaluation of Hadamard finite part integrals by quadrature rules are studied. People may be interested in estimating the number of superconvergent points. It has been proved that there are at most k-(-1)k superconvergent points if some conditions are satisfied. Some researchers conjectured that the conditions hold in general. In Chapter 3, the conditions are proved to be true in some special cases.