Simple comparison is not fair, however: our *r* has total degree *N*+*P*,
as opposed to *N* for the
classical rational interpolant. More importantly, it requires
knowledge of the interpolated
function in the entire interpolation interval, so that it is more an
``interpolative approximant''
than an interpolant in the classical sense.
We do think, however, that it will have
interesting applications.

One of them is *model reduction* in control system design. Indeed, it often
happens there that a rational transfer function, *T*(*s*) say, has too large
numerator and/or denominator degrees *m*, resp. *n*, and that one wants to decrease
these without loosing the main features of *T*. Several methods exist
for that purpose, and their results can vary enormously [15]. The approach we
suggest for replacing *T* with
,
,
,
is to
consider interpolation nodes
(and corresponding interpolated
values *T*(*s*_{i})) well chosen to maintain the main features of *T*(e.g., its extrema). Then our method can be used to construct *T*' by optimally
attaching *n*' poles to the polynomial
interpolating between the
*s*_{i}.

Another application is the *numerical solution of two-point boundary value
problems **Lu* = *h* [7]. We suggest to iteratively improve upon the
classical polynomial pseudospectral (collocation) method: once approximate values
*u*_{k} of the solution at the *N*+1 collocation points *x*_{k} have been found, we
optimally attach poles by minimizing the residuum
(with respect to
*P* poles *z*_{i}) among all rational interpolants *r* of the *u*_{k}'s as given in
(3). The optimal *z*_{i} determine a denominator as in (3), and the set of all
rational functions in
interpolating between the *x*_{k}'s and sharing this
common denominator form a linear space. The second step of our iteration procedure
consists in solving the original problem *Lu* = *h* in this space by the linear rational
collocation method [3]. We then
simply repeat the two steps described above until convergence, see [7]
for details. Numerical tests are encouraging.

*A final remark: *the alert reader will wonder why we did not simply
minimize the approximation error
with respect to all of the *w*_{k}'s in (1) -- after all, as mentioned in §2,
every rational interpolant in
can be written as such a
barycentric expression.
In fact, this is the way
we started, but we encountered
difficulties, both on the theoretical side (existence of an optimum,
of an alternating
sequence, etc.) as on the practical side (too many parameters to optimize).
That led us to
the present compromise of merely optimizing low degree denominators,
which gives
satisfactory results in many cases, as demonstrated in our experiments.

2000-05-30