An Example of HRFA

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

$$\begin{eqnarray*}D&=&\{(-1.0,0.0384615),(-0.8,0.588235),(-0.6,0.1),\\ &&(-0.4,0... ...5),(0.4,0.2),\\ &&(0.6,0.1),(0.8,0.0588235),(1.0,0.00384615)\}, \end{eqnarray*}$$

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

$$\begin{eqnarray*}R_{5,5}&=&\frac{P(x)}{Q(x)},\\ P(x)&=&-1.58989\cdot10^{-14}x^5... ...(x)&=&-532.11x^5+138.4x^4+199.658x^3+30.536x^2\\ &&+8.83771x+1. \end{eqnarray*}$$

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

$$\left\vert f_i-\frac{p(x_i)}{q(x_i)}\right\vert\leq 2\times \... ...041\cdot10^{-12}}{1-1.33041\cdot10^{-12}}=2.66082\cdot10^{-12}.$$

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

$$10^{-12}\mbox{-GCD}(P,Q)=-21.2844x^3+5.53599x^2+8.83771x+1.00000.$$

Thus rational function is obtained by (8) as

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

The expression can be replaced by

$$r(x)=\frac{p(x)}{q(x)}=\frac{1}{1+25x^2}.$$

The result satisfies the theorem 4.1.