In our earlier work, a computer code based on Method of Characteristics (MOC) was developed to solve the neutron transport equation for mainly assembly-level lattice calculations with reflective and periodic boundary conditions and to some extent core-level calculation with a vacuum boundary condition. Performance of the MOC code was also demonstrated with flat and linear flux approximations. Since neutron transport calculations involve extensive computation, an attempt is made to develop an efficient numerical recipe that will reduce the computation time. First, a conventional MOC solution of the neutron transport equation is transformed into a matrix equation to apply the Krylov subspace iteration method for accelerating the solution. It is found that even in the most sophisticated and compact formats, forming the matrix equation explicitly by storing its nonzero elements requires extremely large computer memory. Hence, an alternate way to apply the Krylov iteration is demonstrated by incorporating the effect of the matrix-based approach into the solution without storing the matrix elements. This computationally viable and novel acceleration technique is used in combination with the existing formalism of flat as well as linear flux approximation to solve a number of benchmark problems. Results show significant improvement in terms of faster convergence of the solution over the conventional inner-outer iteration without compromising accuracy.