Polynomials vs. Splines
Polynomials are the approximating functions of choice when a
smooth function is to be approximated locally. For example, the truncated Taylor series
provides a satisfactory approximation for
if
is
sufficiently smooth and
is sufficiently close to
. But if a function
is to be approximated on a larger interval, the degree,
, of the approximating
polynomial may have to be chosen unacceptably large. The alternative
is to subdivide the interval
of approximation into sufficiently
small intervals
, with
,
so that, on each such interval, a polynomial
of
relatively low degree can provide a good approximation to
. This can even
be done in such a way that the polynomial pieces blend smoothly, i.e.,
so that the resulting patched or composite function
that equals
for
, all
, has several
continuous derivatives. Any such smooth piecewise polynomial function
is called a spline. I.J. Schoenberg coined this term since a twice continuously
differentiable cubic spline with sufficiently small first derivative
approximates the shape of a draftsman's spline.
There are two commonly used ways to represent a polynomial spline,
the ppform and the B-form. In this toolbox, a spline in ppform
is often referred to as a piecewise polynomial, while a piecewise
polynomial in B-form is often referred to as a spline. This reflects
the fact that piecewise polynomials and (polynomial) splines are just
two different views of the same thing.
 | Introduction | | ppform |  |
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