rsform: rpform, rBform
Offhand, the two splines,
and
, in the rational spline
need not be
related to one another. They could even be of different forms. But,
in the context of this toolbox, it is convenient to restrict them
to be of the same form, and even of the same order and with the same
breaks or knots. For, under that assumption, we can (and do) represent
such a rational spline by the (vector-valued) spline function
whose values are vectors with one more entry than the values
of the rational spline
, and call this the rsform of the rational spline, or, more precisely,
the rpform or rBform,
depending on whether
and
are in ppform or in B-form. Internally, the only
thing that distinguishes these rational forms from their corresponding
ordinary spline forms, rpform and B-form, is their form part, i.e.,
the string obtained via fnbrk(r,'form'). This is
enough to alert the fn... commands to act appropriately
on a function in one of the rsforms.
For example, as is done in fnval, it is very
easy to obtain
from
. If v is the value of
at
, then v(1:end-1)/v(end) is
the value of
at
. If, in addition, dv is
, then (dv(1:end-1)-dv(end)*v(1:end-1))/v(end) is
. More generally,
by Leibniz's formula,
Therefore,
This shows that we can compute the derivatives of
inductively,
using the derivatives of
and
(i.e., the derivatives of
) along with
the derivatives of
of order less than
to compute the
th derivative
of
. This inductive scheme is used in fntlr to
provide the first so many derivatives of a rational spline. There
is a corresponding formula for partial and directional derivatives
for multivariate rational splines.
 | Example: Sphere | | Available Commands |  |
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