By definition 3.1 and the algorithm 3.1,
the hybrid rational function, *p*(*x*)/*q*(*x*), with the accuracy level
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.

for s.t. .

The relaxed termination and implies and respectively. Thus, rhs of (10) is estimated by (9), for s.t. .

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:

- 1.
- Compute rational interpolation
*P*(*x*)/*Q*(*x*) to interpolate*D*. - 2.
- Let
*f*_{1}=*P*(*x*),*f*_{2}=*Q*(*x*) input polynomials of algorithm 3.1. Compute 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

where positive constant

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