Computations of linearized equations (3)
in the presence of roundoff errors may lead to the
following interesting phenomenon.
Suppose that the unknown function *f*(*x*) is a rational function
which is continuous in
[*x*_{0},*x*_{m+n}].
Let the actual degrees of the numerator and the denominator polynomials
of *f*(*x*) be *m*-*d* and *n*-*d*, respectively, for a positive *d*. That is
.
Here, we consider rational function
interpolating *D*such as
.
There are a lot of rational functions passing the data set *D* but have
different degrees.
There are any polynomials
whose degree is *d* such as
.
Thus, linearized equations (3) have no unique solution.
However, in the floating number computations, because of the presence of
roundoff errors the numerical solution will produce the pair of
polynomials *P*(*x*) and *Q*(*x*) of degrees *m* and *n*, respectively.
The numerical rational interpolation *P*(*x*)/*Q*(*x*) should approximate one of
.
That is
*P*(*x*)/*Q*(*x*)=(*g*_{p}(*x*)*p*_{m-d}(*x*))/(*g*_{q}(*x*)*q*_{n-d}(*x*)).
In algebraic computations we can remove
from rational
function
by GCD computations. However
in floating computations the coefficients of *P*(*x*)/*Q*(*x*)are slightly different from
.
Thus, polynomials *g*_{p}(*x*) and *g*_{q}(*x*) remain in the numerical rational
interpolation.
Furthermore, it may occur that for some *z*we have simultaneously
,
and
then the computed ratio *P*(*x*)/*Q*(*x*)will be a very poor approximation to *f*(*x*) at *z* and near *z*.
In this case, a natural way to improve the
approximation is to compute
*g*(*x*), an approximate-GCD of *P*(*x*)and *Q*(*x*), then compute the symbolic quotients
*p*(*x*)=*P*(*x*)/*g*(*x*) and
*q*(*x*)=*Q*(*x*)/*g*(*x*), and finally
output *p*(*x*)/*q*(*x*) as an improved hybrid
rational interpolation to *f*(*x*). We call these procedures *Hybrid
Rational Function Approximation* and show the algorithm as follows:

- 1.
- Compute rational interpolation
*P*(*x*)/*Q*(*x*) to interpolate*D*. - 2.
- Obtain the approximate-GCD
*g*(*x*) of*P*(*x*) and*Q*(*x*) using the algorithm by Sasaki and Noda. - 3.
- and , where is a quotient obtained by polynomial division.

Any algorithms for (non-unique) approximate-GCD and for
polynomial division are available.
Several methods of calculating approximate-GCDs have been proposed other than
Schönhage [7], Sasaki and Noda [11].
They use different norms for the definition of approximate-GCDs,
i.e.
*L*_{1} norm (c.f. Pan[9], Hribernig and Stetter[8]),
*L*_{2} norm (c.f. Corless, Gianni, Trager and Watt[6], Karmarkar and
Lakshman[5])
or norms estimated in a domain (c.f. Sederberg and Chang[10]).
For further discussions to establish the error estimate of HRFA,
we use the approximate-GCD proposed by Hribernig and Stetter.