Rational functions like for instance Padé approximants play an important role in signal processing, sparse interpolation and exponential analysis. However, for a successful modeling with help of rational functions we want to make sure that there is no ”similar” rational function being degenerate, i.e., having strictly smaller degree of both degrees of numerator and denominator. In particular, we prefer having rational functions without Froissart doublets (i.e., roots close to a pole) because their presence induce numerical instabilities: small variations in the argument of the function give rise to large variations in the function values.
In this talk I’ll present recent work in controlling the presence of Froissart doublets and/or small residuals. More precisely, we will give lower bounds on the distance of a zero and a pole of a rational function r = p/q. We will show that three quantities play an important role:
- the condition number of the Sylvester matrix built with the coefficients of the numerator p and denominator q polynomials;
- a measure for numerical coprimeness of p and q representing the minimal distance in the coefficient vector metric to a couple of degenerate polynomials (with a joint root allowing for cancelling the fraction);
- the spherical derivative of the rational function r.
Based on these quantities we will give sufficient conditions for the absence of Froissart doublets. We also obtain relations between these three quantities and define classes of rational functions with ”good” properties (no Froissart doublets).
As we are interested in numerical computations, we want our inequalities to be valid in a neighborhood of the rational function. The distance between two rational functions will be measured in two different metrics, the chordal distance of the values on the unit disk, and the distance of normalized coefficient vectors. We will compare these two metrics.
The previous quantities can be used as indicators of the presence of Froissart doublets. This may be interesting in approximation theory: construct a rational approximant solution of an optimization problem with a penalization term, obliging this indicator to be not too small.