Attaching poles to the interpolating polynomial

Let $\Pscr_m$ and $\Rscr_{m,n}$ respectively denote the linear space of all polynomials of degree $\le m$ and the set of all rational functions with numerator degree $\le m$ and denominator degree $\le n$; furthermore, denote by f_k the interpolated values f(x_k), k=0(1)N, of f. Then the unique polynomial $p\in\,\Pscr_N$ that interpolates f between the x_k's can be written in its barycentric form [12]

$$p(x) = {\sum_{k=0}^N {w_k\over x-x_k}f_k}\Bigg/% {\sum_{k=0}^N {w_k\over x-x_k}},$$ (1)

where the so-called weight w_k corresponding to the point x_kis given by

$$w_k := 1\Bigg/\kern-5pt\prod_{i=0,\ i\ne k}^N(x_k-x_i).$$

The barycentric formula has several advantages [18], [6, p. 357]. One of them is the fact that the weights appear in the numerator and in the denominator, so that they can be divided by any common factor. For example, simplified weights for equidistant points are given by

$$w_k^* = (-1)^k {N \choose k}$$

[8], while for the Cebysevpoints of the second kind $\cos \phi_k$, $\phi_k := {k}{\pi\over N}$, one simply has [17]

$$w_k^* = (-1)^k \delta_k,\qquad \delta_k=\left\{\begin{array}{... ...mbox{ or }k=N,\\ 1,&\quad\mbox{otherwise}. \end{array}\right.$$

As explained in the introduction, we now want to improve the quality of approximation of the interpolant, for instance for functions with very large derivatives. For that purpose, we will divide the interpolant by an optimized denominator, while maintaining interpolation.

Let P, $P\le N$, be the number of poles z_i, i=1(1)P, we want to attach to the polynomial. If some rational interpolant $r\in \Rscr_{N,P}$ exists with poles at the z_i's and only there, then its denominator takes the values

$$d_k := a\prod_{i=1}^P (x_k - z_i),\quad a\ne 0\in\hbox{\rm C... ...em\hbox{\vrule width 0.015em height .6em}}\\ mbox{ arbitrary,}$$ (2)

at the interpolation points x_k. (The fact that it does not exist may mean that attaching the poles is not advisable from an approximation point of view, see [4].) To insure interpolation, the values of the numerator at the same points will be f_kd_k. Writing the numerator and the denominator as interpolating polynomials in their barycentric form (1) and simplifying, one gets

$$r := {\sum_{k=0}^N {w_k {\prod\limits_{i=1}^P}(x_k- z_i) \ov... ...N {w_k {\prod\limits_{i=1}^P}(x_k- z_i) \over x-x_k}}\right. .$$ (3)

(3) is the barycentric representation of rwith weights $v_k := w_k\prod_{i=1}^P (x_k - z_i)$. In the present case, with all poles prescribed, the weights are unique up to a constant [4]. Barycentric representations exist for every rational interpolant in $\Rscr_{N,N}$[5], and computing them is a way of solving the classical rational interpolation problem also when only a subset of the poles are prescribed [4].

In order to stay with real interpolants, we will assume here that the poles z_i with $\Im z_i\ne 0$ arise in complex conjugate pairs.