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IMA Journal of Applied Mathematics 1971 7(3):384-397; doi:10.1093/imamat/7.3.384
© 1971 by Institute of Mathematics and its Applications
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The Numerical Calculation of Odd-degree Polynomial Splines with Equi-spaced Knots

E. L. ALBASINY1 and W. D. HOSKINS2

1National Physical Laboratory Teddington, Middlesex, England
2Mathematics Department, Brunel University Uxbridge, Middlesex, England

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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