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Published in:
2018 | OriginalPaper | Chapter
Exact Solutions to the Spline Equations
Authors : Anthony A. Ruffa, Bourama Toni
Published in: Advanced Research in Naval Engineering
Publisher: Springer International Publishing
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Abstract
The exact solution to the cubic spline equations is developed for the case of equal knot spacing. It exhibits an oscillatory response in the region of a discontinuity, which is a consequence of the row structure of the resulting tridiagonal Toeplitz system. The oscillations cancel in the absence of discontinuities. Splines under tension exhibit a similar oscillatory response; however, increasing the tension attenuates the oscillations over a shorter length scale. The use of imaginary tension removes the large amplitude oscillations in the region of a discontinuity at the expense of introducing a low-amplitude oscillation throughout the entire curve fit. A composite spline (i.e., a spline under tension in the region surrounding a discontinuity, and a cubic spline elsewhere) can confine such oscillations to an arbitrarily defined region surrounding each discontinuity.