Iterative methods for the solution of linear systems of equations produce a sequence of approximate solutions. In many applications it is desirable to be able to compute estimates of the norm of the error in the approximate solutions generated and terminate the iterations when the estimates are sufficiently small. This research focuses on new iterative methods based on the Lanczos process for the solution of linear systems of equations as well as on the computation of bounds and estimates of the norm of the error in the approximate solutions. These estimates are determined by evaluating certain Gauss, anti-Gauss, or Gauss-Radau quadrature rules.