The aim of this Book was to present some new derivations and applications and uses of the numerical analysis in interpolation of the spline functions. The development of new algorithms by interpolated spline functions with convergence; existence; uniqueness and stability analysis were investigated. Firstly the lacunary interpolation problem consisted of, finding the sixth degree spline of deficiency four, interpolating the data given on the function value with third and fifth order in the interval[0, 1]. Under suitable assumptions, the existences; uniqueness and the error bounds investigated. Secondly a quintic-spline interpolation algorithm presented for the solution of second order initial value problems with a new class of interpolations based on quintic -splines as an approximation to the exact solution of such problems. Finally, a quartic interpolated spline function algorithm were developed to find an approximation solutions to error estimate for -spline which interpolate the first & third derivatives of a given smooth functions. The results indicated that the presented spline function algorithm proved their effectiveness in solving the second order initial value problems.