Conclusion

In the present work we have presented rational interpolants with guaranteed interpolation, no poles in the interval of interpolation, and an error which usually decreases -- and never increases -- with the degree of the denominator. The error is consistently smaller than that of classical rational interpolation with the same denominator degree.

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 $T'\in\Rscr_{m',n'}$, $m'\le m$, $n'\le n$, is to consider interpolation nodes $s_0,\ldots,s_{m'}$ (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 $p\in\,\Pscr_{m'}$ 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 $\Vert Lr-h\Vert$ (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 $\Rscr_{N,P}$ 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 $\Rscr_{N,N}$ 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.