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An Elementary Proof of the Exponential Conditioning of Real Vandermonde Matrices

Stefano Serra Capizzano — 2007

Bollettino dell'Unione Matematica Italiana

We provide and discuss an elementary proof of the exponential con- ditioning of real Vandermonde matrices which can be easily given in undergraduate courses: we exclusively use the definition of conditioning and the sup-norm formula on [ - 1 , 1 ] for Chebyshev polynomials of first kind. The same proof idea works virtually unchanged for the famous Hilbert matrix.

Multigrid Methods for (Multilevel) Structured Matrices Associated with a Symbol and Related Application

Marco DonatelliStefano Serra Capizzano — 2013

Bollettino dell'Unione Matematica Italiana

When dealing with large linear systems with a prescribed structure, two key ingredients are important for designing fast solvers: the first is the computational analysis of the structure which is usually inherited from an underlying infinite dimensional problem, the second is the spectral analysis which is often deeply related to a compact symbol, again depending on the infinite dimensional problem of which the linear system is a given approximation. When considering the computational view-point,...

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