The Computation of all the Derivatives of a B-spline Basis

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Abstract
Splines are currently much used in the field of interpolation to functions and their derivatives. In this context for a given argument two relationships between derivatives of B-spline bases of consecutive orders are derived. Using these relationships it is shown there are (K—1)!((K—m1)!m!) schemes for the evaluation of the mth derivative of a B-spline basis of order k. Analyses of error growth in terms of a matrix notation are carried out in order to see which of the schemes is the most numerically stable, for uniform or highly non-uniform knot sets. The computation of the B-spline basis of order K and its (K—1)th derivative are shown to have small a priori relative error bounds.

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