Let
be *m*+*n*+1 distinct points,
and let
,
where *f*(*x*) is an unknown function.
They are changed to a set of data

The classical algebraic problem of rational interpolation is
to compute a pair of polynomials *P*(*x*) and *Q*(*x*) satisfying
the relations
,
,

for a given pair of nonnegative integers

If we put and , then we can compute the values

- There may arise the rational interpolation of the form
at
*x*_{i}, i.e.*unattainable point*. - There may be poles between the points of interpolation.

To avoid the difficulties, several rational interpolations
have been proposed.
One of them is called linear rational interpolation by
Berrut and Mittelmann [2].
This interpolation has
no unattainable points and no poles in the interval of the interpolation
[*x*_{0},*x*_{m+n}].
However, the degree of the approximation will become very high to obtain
better approximation of *f*(*x*). Thus it is difficult to use the approximation
in other hybrid applications, e.g. hybrid integral.

If *Q*(*x*) has zeros in
[*x*_{0},*x*_{m+n}], i.e.
a rational interpolation has poles, and *P*(*x*) has close zeros to the
zeros of *Q*(*x*),
hybrid rational function approximation (HRFA) [4] based
on symbolic-numeric hybrid computation
is available to remove the poles in
[*x*_{0},*x*_{m+n}].
However, if unattainable points occur in the data set *D*,
HRFA becomes ill-conditioned.
Hereafter is a detailed discussion of HRFA.

In HRFA, the approximate-GCD by Sasaki and Noda [11] is used
to remove the undesired poles in
[*x*_{0},*x*_{m+n}].
For the computation of approximate-GCD, the quantity
in general is different from zero
(even if it may be closed to zero), where *p*(*x*) and *q*(*x*) are the
numerator and denominator polynomials of rational function obtained by HRFA.
We must estimate the accuracy
of the hybrid rational interpolation
*r*_{k,l}(*x*) as

where degrees of

In this paper, we propose a method of the error estimation of HRFA.
For this purpose, the approximate-GCD proposed by Hribernig and
Stetter is used.
HRFA and
the approximate-GCD are briefly summarized in **2** and **3**,
respectively.
In **4**, We show a theorem of the accuracy of HRFA using
the approximate-GCD.
A symbolic-numeric hybrid example is shown in **5**.
The result satisfies the theorem.