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
[x0,xm+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 Dsuch 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)=(gp(x)pm-d(x))/(gq(x)qn-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 gp(x) and gq(x) remain in the numerical rational
interpolation.
Furthermore, it may occur that for some zwe 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:
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. L1 norm (c.f. Pan[9], Hribernig and Stetter[8]), L2 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.