The Akima's fitting method for quartic splines

For the Hermite type quartic spline interpolating on the partition knots and at the midpoint of each subinterval, we consider the estimation of the derivatives on the knots, and the values of these derivatives are obtained by constructing an algorithm of Akima's type. For computing the derivatives on endpoints are also considered alternatives that request optimal properties near the endpoints. The error estimate in the interpolation with this quartic spline is generally obtained in terms of the modulus of continuity. In the case of interpolating smooth functions, the corresponding error estimate reveal the maximal order of approximation O(h^3). A numerical experiment is presented for making the comparison between the Akima's cubic spline and the Akima's variant quartic spline havingdeficiency 2 and natural endpoint conditions.