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Inversion of generalized Cauchy matrices and other classes of structured matrices. (English) Zbl 0823.65020
Bojanczyk, Adam (ed.) et al., Linear algebra for signal processing. Papers based on lectures presented at the IMA workshop, held at IMA, University of Minnesota, Minneapolis, MN, USA, April 6-10, 1992. New York, NY: Springer-Verlag. IMA Vol. Math. Appl. 69, 63-81 (1995).
Summary: Fast inversion algorithms for strongly nonsingular matrices of the form $$C = \left[ {z^ T_ i y_ j \over c_ i - d_ j} \right]$$ (generalized Cauchy matrices), where $$z_ i$$, $$y_ j$$ are column vectors and $$c_ i$$, $$d_ j$$ are complex numbers, are presented. The approach is based on the interpretation of equations $$C\xi = \eta$$ as tangential interpolation problems. Furthermore, it is described how other types of structured matrices like Toeplitz matrices and their generalizations can be transformed into generalized Cauchy matrices. This transformation can be utilized in order to get stable algorithms.
For the entire collection see [Zbl 0819.00018].

##### MSC:
 65F05 Direct numerical methods for linear systems and matrix inversion 65D05 Numerical interpolation 15B57 Hermitian, skew-Hermitian, and related matrices 41A20 Approximation by rational functions