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We describe explicitly each stage of a numerically stable algorithm for calculating with exponential tension B-splines with non-uniform choice of tension parameters. These splines are piecewisely in the kernel of D 2 (D 2 − p 2 ), where D stands for ordinary derivative, defined on arbitrary meshes, with a different choice of the tension parameter p on each interval. The algorithm provides values of the associated B-splines and their generalized and ordinary derivatives by performing positive linear combinations of positive quantities, described as lower-order exponential tension splines. We show that nothing else but the knot insertion algorithm and good approximation of a few elementary functions is needed to achieve machine accuracy. The underlying theory is that of splines based on Chebyshev canonical systems which are not smooth enough to be ECC-systems. First, by de Boor algorithm we construct exponential tension spline of class C 1 , and then we use quasiOslo type algorithms to evaluate classical non-uniform C 2 tension exponential splines.

We propose a knot insertion algorithm for splines that are piecewisely in L{1, x, sin x, cos x}. Since an ECC-system on [0, 2π ] in this case does not exist, we construct a CCC-system by choosing the appropriate measures in the canonical representation. In this way, a B-basis can be constructed in much the same way as for weighted and tension splines. Thus we develop a corner cutting algorithm for lower order cycloidal curves, though a straightforward generalization to higher order curves, where ECC-systems exist, is more complex. The important feature of the algorithm is high numerical stability and simple implementation.

Splines determined by the kernel of the differential operator D k (D √x D) are known to be useful to solve the singular boundary value problems of the form D √x Du = f (x, u). One of the most successful methods is the collocation method based on special Chebyshev splines. We investigate the construction of the associated B-splines based on knot-insertion algorithms for their evaluation, and their application in collocation at generalized Gaussian points. Specially, we show how to obtain these points as eigenvalues of a symmetric tridiagonal matrix of order k.