The Dirac equation is the basic equation for relativistic spin-1/2 particles. Here we consider it in the Feshbach-Villars formalism with linear time derivative and quadratic spatial derivative. In this picture, the relativistic equations appear in Hamiltonian form, similar to the Schrodinger equation, and the relativistic features and the spin structures are manifested in a multi-component wave function. In previous studies, the relativistic spin-0 Feshbach-Villars equation was solved in a discrete basis representation with the help of matrix continued fractions. In this work we extended this solution to the spin-1/2 Feshbach-Villars equation. The extension amounts to replacing the basis with integer angular momentum to non-integer angular momentum. The validity of the numerical procedure was tested with comparison to the exact Dirac hydrogen results and perfect agreement was found.