Convergence rates for inverse-free rational approximation of matrix functions

Superlinear convergence of the rational Arnoldi method for the approximation of matrix functions

A superlinear convergence bound for rational Arnoldi approximations to functions of matrices is derived. This bound generalizes the well-known superlinear convergence bound for the CG method to more general functions with finite singularities and to rational Krylov spaces. A constrained equilibrium problem from potential theory is used to characterize a max-min quotient of a nodal rational func...

On the Convergence of Polynomial Approximation of Rational Functions

This paper investigates the convergence condition for the polynomial approximation of rational functions and rational curves. The main result, based on a hybrid expression of rational functions (or curves), is that two-point Hermite interpolation converges if all eigenvalue moduli of a certain r_r matrix are less than 2, where r is the degree of the rational function (or curve), and where the e...

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class Rn,m of all rational functions of order (n,m) are considered. Let ρn,m = ρn,m( f ; E) be the distance of f in the uniform metric on E from the class Rn,m . We obtain results characterizing the rate of convergence to zero of the sequen...

Approximation by Rational Functions

Approximation by Rational Functions

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If f' is in L log L on a finite interval, then f can be approximated in the uniform norm by rational functions of degree n to an error 0(1/n) on that interval. It is well known that approximation by rational functions of degree n can produce a dramatically smaller error than that for p...