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6.4 Chebyshev Spaced PSOMs 83

each component d.
For m
the tensor splines are simple to extend in theory, but in prac-
tice they are not very useful (Friedman 1991). Here, the PSOM approach
can be easily extended as shown for the case m later in Sec. 8.2.

6.4 Chebyshev Spaced PSOMs

An alternative way to deal with the shortcomings addressed in the intro-
duction of this chapter, is the use of an improved scheme for constructing
the internal basis functions H a s . As described (in Sec. 4.5), they are con-
structed by products of Lagrange interpolation polynomials on a rectan-
gular grid of nodes in the parameter manifold. So far, a regular equidistant
division of an interval (e.g. [0,1] in Fig. 4.3) was suggested for placing the
nodes on each parameter manifold axis.
As is well-known from approximation theory, an equidistant grid point
spacing is usually not the optimal choice for minimizing the approxima-
tion error (Davis 1975; Press et al. 1988). For polynomial approximation
on an interval [-1,1], one should choose the support points a j along each
axis of A at locations given by the zeros a j of the Chebyshev polynomial
T n a
T n a cos n arccos a (6.2)

The (fixed location of the) n zeros of T n are given by

j

a j cos (6.3)
n
The Chebyshev polynomials are characterized by having all minima
and maxima of the same amplitude, which finally leads to a smaller bound
of the error polynomial describing the residual deviation. Fig. 6.5a dis-
plays for example the Chebyshev polynomial T . A characteristic feature
is the increasing density of zeros towards both ends of the interval.
It turns out that this choice of support points can be adopted for the
PSOM approach without any increase in computational costs. As we will
demonstrate shortly, the resulting Chebyshev spaced PSOM (“C-PSOM”)
tends to achieve considerably higher approximation accuracy as compared
to equidistant spaced PSOMs for the same number of node points. As a re-
sult, the use of Chebyshev PSOMs allows a desired accuracy to be achieved