A Comparative Analysis of Sixth-Order Algebraic-Hyperbolic Natural Splines and Integral Quasi-Interpolating Splines

Algebraic-hyperbolic splines are important in numerical analysis because they combine smoothness and approximation accuracy. In a number of applications, pointwise function values may be noisy or unavailable, while subinterval integral values constitute the available data. Existing integro-spline constructions address this situation, but comparison with sixth-order algebraic-hyperbolic natural splines constructed from point values remains relevant. The purpose of this work is to construct a sixth-order algebraic-hyperbolic natural interpolation spline and to compare its approximation accuracy with several spline methods constructed from integral values. The interpolating natural spline is constructed in a functional space associated with a third-order differential operator containing a hyperbolic parameter. A discrete analogue of the corresponding operator is used to derive the computational algorithm and explicit systems for the spline coefficients. An important reproduction property of the proposed spline is established: it exactly reproduces constant and hyperbolic functions. Numerical comparisons are carried out for two test functions and several numbers of knots. The approximation errors of the proposed algebraic-hyperbolic natural spline are compared with those of an algebraic-hyperbolic integro quasi-interpolant, local integrosplines, an integro-quadratic spline, and an algebraic-trigonometric integro-spline. For the reported tests, the proposed method yields smaller maximum approximation errors. The results indicate that the constructed algebraic-hyperbolic natural spline is a useful high-accuracy approximation tool and provides a basis for comparison between point-interpolation and integral-data spline techniques.