On certain (block) Toeplitz matrices related to radial functions

Spectral analysis and preconditioning techniques for radial basis function collocation matrices

“…From the numerical experiments performed in 9, the above asymptotic bounds are very strict even for small values of g when ρ( g ) is small, while the quantity ρ( g ) becomes an extremely pessimistic upper bound when ρ( g ) is moderate (say e.g. g = 3, 4).…”

“…In 9, the authors provided explicit asymptotic estimates, as function of c/h , c being the shape parameter, h being the step size, to the condition number µ( T n ) of the Toeplitz matrix T n related to the approximated one‐dimensional model problem with the collocation technique over a grid of equally spaced points and based on the MQ, IMQ, and Gaussian radial functions, respectively.…”

“…According to Bini et al 9, there exists a function ρ( g ) of g such that for any n we observe µ( T n )<ρ( g ), where equality is reached only for n →∞. Furthermore, an interesting asymptotical estimate γ( g ) of ρ( g ), for g →∞, was proved: Here, ϕ( x )≈ψ( x ) if lim x →∞ ϕ( x )/ψ( x ) = 1, while ϕ( x )∼ψ( x ) means asymptotical equivalence that is the existence of two positive constants r and R independent of x , such that r ϕ( x )ψ( x ) R ϕ( x ) for every x large enough.…”

“…More in detail, concerning the one‐dimensional setting, we give a more precise analysis than in 9, by showing that for some choices of RBF, e.g. IMQ and Gaussian, and for some values of the parameter c / h , the conditioning is not exponential, but grows mildly as n 2 .…”

“…In Section 3 we state in some detail the one‐dimensional and two‐dimensional problems, respectively, by emphasizing the linear algebra viewpoint and by reporting the known results. In Section 4 we analyze both the one‐level and two‐level cases giving a rigorous explanation of some numerics reported in 9 and we study the problem from the point of view of extremal spectral values (conditioning). In Section 5 we study the global spectral distribution of the complete matrix‐sequence where also the boundary conditions are considered.…”