Accuracy of HRFA

By definition 3.1 and the algorithm 3.1, the hybrid rational function, p(x)/q(x), with the accuracy level $\alpha$ is obtained. We consider, here, the error between p(x)/q(x)and the interpolated rational function, P(x)/Q(x). The following theorem is established.

Theorem 4.1   Let P(x) and Q(x) be polynomials in C[x] and let p(x) and q(x) be polynomials satisfying $\Vert \delta P(x)\Vert\leq\alpha<1$ and $\Vert \delta Q(x)\Vert\leq\alpha<1$in (8). Then

$$\left\vert\frac{P(z)}{Q(z)}-\frac{p(z)}{q(z)}\right\vert\leq\... ...rac{P(z)}{Q(z)}\right\vert\right)\cdot\frac{\alpha}{1-\alpha},$$ (9)

for $z\in[-1,1]$ s.t. $\vert Q(z)\vert\geq1$.  

proof

 

$$\left\vert\frac{P(z)}{Q(z)}-\frac{p(z)}{q(z)}\right\vert \left\vert\frac{\delta P(z)}{Q(z)}-\frac{P(z)}{Q(z)}\cdot \frac{\delta Q(z)}{Q(z)}\right\vert\cdot\frac{1}{1-\frac{\delta Q(z)}{Q(z)}}.$$ = (10)

The relaxed termination $\Vert \delta P(z)\Vert\leq\alpha<1$ and $\Vert \delta Q(z)\Vert\leq\alpha<1$ implies $\max_{-1\leq z\leq 1}\vert\delta P(z)\vert\leq\alpha<1$ and $\max_{-1\leq z\leq 1}\vert\delta Q(z)\vert\leq\alpha<1$ respectively. Thus, rhs of (10) is estimated by (9), for $z\in[-1,1]$ s.t. $\vert Q(z)\vert\geq1$. $\Box$

This theorem 4.1 gives the error of HRFA and is the main result of this paper. To show how apparent the error is, we rewrite HRFA algorithm, Algorithm 2.1, as follows:

Algorithm 4.1 (HRFA by near-GCD)  
Input: data set D s.t. $x_i\in [-1,1]$ and parameter $\alpha$
Output: numerical rational interpolation p(x)/q(x)
Method:

1.

Compute rational interpolation P(x)/Q(x) to interpolate D.

2.

Let f₁=P(x),f₂=Q(x) input polynomials of algorithm 3.1. Compute $\alpha-GCD(f_1,f_2)$ and obtain

p(x)/q(x)=s⁽¹⁾_j/s⁽²⁾_j.

 

After the computation of the procedure 1. of the algorithm 4.1, accuracy of numerical rational interpolation p(x)/q(x) of the data set D is estimated by Theorem 4.1 as

$$\vert f(x_i)-p(x_i)/q_(x_i)\vert\leq (1+\vert f(x_i)\vert)\cdot\frac{c\alpha}{1-c\alpha},$$ (11)

where positive constant c satisfies the condition $c\cdot \vert Q(x_i)\vert\geq 1$ for $i=0,\cdots,m+n$.

The result (11) shows that our a posteriori error estimate is dominated by the parameter, the accuracy level, $\alpha$. Similar results may obtained by using other approximate-GCDs but with different accuracy level parameters.