Metode numerik

 4.10 TENSION SPLINES

Splines are a wonderful tool for approximation, but they can still exhibit some poor behavior. Consider the data set plotted in Fig. 4.25. Obviously, this represents a function with a severe jump near x — 0.5, but there is no sign of oscillatory behavior. However, a B-spline representation of this data (Fig. 4.26) shows small wiggles on either side of a sharp front. This is fundamentally an artifact of the steep gradient in the data, but in other contexts a spline fit can display behavior that does not match the "sense" of the data. One way to avoid the problem is the notion of a taut spline or tension spline, an idea that appears to have been first published by Schweikert [17], but which also owes a lot to the work of A. K. Cline [4]; we relied heavily on a short paper of Marusic and Rogina [12] in our presentation here.

Imagine that the curve in Fig 4.26 is a piece of string that is constrained to pass through small loops at the data points. If we were to pull the string taut, we would smooth out the spurious oscillations in the curve. This amounts to studying the mechanical properties of a cable hanging between two supports. More prosaically, we construct our spline from the new basis set {1, a;, coshpx, sinhpa;}, where p > 0 is the tension parameter: p = 0 means no tension, and it can be shown that this corresponds to the pure spline approximation; p —» oo gives us a piecewise linear approximation. (We will not attempt to justify either of these statements other than by examples and exercises.)

The reader may well be wondering how practical this scheme might be. After all, we have traded a set of polynomial basis functions for a set of transcendental basis functions. Not only is this going to make execution of any program more expensive, but it leaves open the entire question of how to construct the approximation. The basic idea is the same as in §4.8: First, we construct a primary basis function, as in (4.34):

τ(x)=2/α {█(sinh⁡〖 px cosh⁡〖 2p+cosh⁡〖 px sinh⁡〖 2p-px-2p,〗                x∈[-2,-1]〗 〗 〗@-(sinh⁡〖 px-px)β-2cosh px sinh⁡ p+γ                         x ∈[-1,0]    〗@(sinh⁡〖 px-px)β-2cosh⁡ px sinh⁡p+γ                              x∈[0,1]〗     @-sinh⁡〖 px cosh⁡〖 2p+cosh⁡〖 px sinh⁡〖 2p+px-2p,           x∈[1,2]〗 〗 〗 〗 )┤ (4.56)

where α=p cosh⁡〖p-sinh⁡〖p,β=(1+2 cosh⁡〖p),and γ=2p cosh⁡〖p.〗 〗 〗 〗 Confirmation of this formula is deferred to the exercises. A plot of  τ is given in Fig. 4.27, for p = 4; note that it does not look very different from Fig. 4.13.