The Numerical Calculation of Odd-degree Polynomial Splines with Equi-spaced Knots

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Abstract
Constructive equations for polynomial splines of odd-degree 2r+1 with knots x1 = x0+ih, i = 0(1)n are formulated in terms of even-order derivatives, odd-order derivatives being given by explicit formulae which are shown to be identical with truncated Taylor series expansions of the same form. The defining equations are written in a manner which reveals a strong connection with the well-known Numerov formula. Solution of the equations by block iterative methods is considered for the case when even derivatives are specified at x = x0 and xn. Block Jacobi and block Gauss—Seidel iteration are shown to be convergent for all positive r and optimum acceleration parameters for block S.O.R. are given for r = 2(1)6. It is shown that distinct computational advantages result from relaxing the condition for a true spline fit and considering instead a truncated spline of higher order.

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