On Calculating with Mplines

In computational dealings with splines, the question of representation is of primary importance. For splines of fixed order on a fixed partition, this is a question of choice of basis, since such splines form a linear space. Only three kinds of bases for spline spaces have actually been given serious attention; those consisting of truncated power functions, of cardinal splines, and of B-splines. Truncated power bases are known to be open to severe illconditioning, while cardinal splines are difficult to calculate. By contrast, bases consisting of B-splines are well-conditioned, at least for orders < 20. Such bases are also local in the sense that at every point only a fixed number (equal to the order) of B-splines is nonzero. B-splines are also evaluated quite easily, using their definition as a divided difference of the truncated power function. Unfortunately, such calculations are ill-conditioned, particularly for partitions of widely varying interval lengths, as is indicated by the fact that special measures have to be taken in case of coincident knots. In this note, a different way of evaluating B-splines is discussed which is very well conditioned yet efficient, and which needs no special adjustments in case of coincident knots. It is also shown that the condition of the B-spline basis increases exponentially with the order.