The development of fast and efficient algorithms is crucial not only for computer scientists, but also for mathematicians and engineers as those algorithms lead to reduce complexity. Another common interest of these professionals is to construct models using existing data. This leads numerical analysts to explore interpolation techniques. One such technique is called cubic spline interpolation. In here, we will propose a cubic spline solver aiming to bridge the gap between numerical linear algebra, electrical engineering, systems engineering, sensor processing, and parallel processing. We will use quasiseparable structure to evaluate cubic splines by deriving a fast and stable algorithm. The derivation is carried through a specific factorization of the inverse of tridiagonal matrices. This factorization leads to an alternative method to solve the system of tridiagonal matrices as opposed to the existing methods. The proposed algorithm has the lowest computational complexity compared to existing algorithms.