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Algebraic properties of robust Padé approximants. (English) Zbl 1311.41009

The \([m/n//]\) Padé approximant of a function \(f(z)=\sum_{j=0}^{\infty}\,c_j z^j\) is the rational function \(p(z)/q(z)\) with \(\operatorname{deg}p\leq m\), \(\operatorname{deg}q\leq n\) and
\[ f(z)q(z)-p(z)={\mathcal O}(z^{m+n+1}),\quad z\rightarrow 0. \]
Although the polynomials found might be non-unique, the rational function is.
Using the recently introduced concept of robust Padé approximant [P. Gonnet et al., SIAM Rev. 55, No. 1, 101–117 (2013; Zbl 1266.41009)], the authors provide a proof for forward stability for the procedures (a.o. eliminating spurious poles) and even show absence of these spurious poles for well-conditioned approximants.
The work by H. Stahl [J. Comput. Appl. Math. 86, No. 1, 287–296 (1997; Zbl 0888.41008)] and [J. Approx. Theory 91, No. 2, 139–204, Art. No. AT973141 (1997; Zbl 0896.41009)] plays an important role in the paper.
The paper also contains several numerical examples.

MSC:

41A21 Padé approximation
65F22 Ill-posedness and regularization problems in numerical linear algebra
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References:

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