In a recent paper, Trefethen and al. [1] have proposed a method for computing a robust Padé approximant based on Singular Value Decomposition techniques. They observe numerically that these approximants are insensitive to perturbations in the data, and do not have so-called spurious poles, that is, poles with close-by zero or poles with small residues. A black box procedure for eliminating spurious poles would have a major impact on the convergence theory of Padé approximants since it is known that convergence in capacity plus absence of poles in some domain D implies locally uniform convergence in D.
In this talk we will propose a mathematical analysis of these numerical phenomena. We will study forward and backward conditioning of the application going from the Taylor coefficients (ci) of the function to the vector of coefficients of the numerator and denominator of the Padé approximant and provide a proof for robustness. We will show that the conditioning of underlying rectangular Toeplitz and Sylvester like matrices plays an important role and and will prove the absence of spurious poles for the subclass of so-called well conditioned Padé approximants.

References
[1] P. Gonnet, S. Guttel, N.L. Trefethen, Robust Padé approximation via SVD, SIAM Review 55(1), 101117. (2013)