The library QIBSH  (authors A. Iurino, F. Mazzia) is a collection of C procedures dealing with numerical computations for the Hermite Quasi-Interpolation of curves and surfaces. A theoretical description of the numerical methods and techniques implemented in the library is presented in the following references:

A. Iurino. BS hermite quasi-interpolation methods for curves and surfaces., 2014. Dipartimento di Matematica, Università degli Studi di Bari.

F. Mazzia and A. Sestini. The BS class of hermite spline quasi-interpolants on nonuniform knot distributions. Bit Numerical Mathematics, 49(3):pp. 611–628, 2009.

F. Mazzia, A. Sestini, and D. Trigiante. The continuous extension of the B-spline linear multistep methods forbvps on non-uniform meshes. Applied Numerical Mathematics, 59(3-4):pp. 723–738, 2009. 

F. Mazzia, A. Sestini, and D. Trigiante. BS linear multistep methods on non-uniform meshes. JNAIAM J. Numer. Anal. Indust. Appl. Math, 1(1):pp. 131–144, 2006.

F. Mazzia, A. Sestini, and D. Trigiante. BS linear multistep methods and their continuous extensions. SIAM Journal on Numerical Analysis, 44(5):pp. 1954–1973, 2006.


One and Two dimensional Quasi-Interpolation problems can be tackled using the procedures in this library. For example, say we have to interpolate some data from a one dimensional function. Providing a knot set, the desired degree of the spline approximation, the values of the function, together with its first derivatives on the knot set, the user can find the spline Quasi-Interpolating the data. The Quasi-Interpolating operator is of Hermite type, since the coefficients of the final spline depend on local values of the function and of its first derivative. The final representation is given in term of the B-spline coefficients of the curve. In cases when first derivative values are not available, the procedure approximating the first derivative values of the function is provided. It is based on a finite difference method for derivative approximations, by means of a generalized backward difference formula. In this way the user can apply the BSH Quasi Interpolant starting only from the function values.