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An Example of HRFA

We show how the error of HRFA satisfies the error estimate (11). As a practical example, a data set


is considered and approximated by HRFA with the accuracy level $\alpha=10^{-12}$.

The data set D is obtained by discretization of a function f(x)=1/(1+25x2). Because of the presence of a roundoff error, rational interpolation of D is obtained by linearized equations such as


The leading and the second coefficients of P(x) is smaller than $\alpha$. Thus deg (P(x))=3. Since $\min_i\vert Q(x_i)\vert=0.751654$, we put c=1.33041. Thus, the accuracy of symbolic-numeric hybrid rational interpolation is

\begin{displaymath}\left\vert f_i-\frac{p(x_i)}{q(x_i)}\right\vert\leq 2\times

near-GCD(P,Q) with $\alpha=10^{-12}$ is obtained as


Thus rational function is obtained by (8) as

\begin{displaymath}r(x)=\frac{p(x)}{q(x)}=\frac{1}{1+5.34132\cdot 10^{-15}x+25x^2}.\end{displaymath}

The expression can be replaced by


The result satisfies the theorem 4.1.

IMACS ACA'98 Electronic Proceedings