Convergence of Padé approximants in the general case

Convergence of Multipoint Padé-type Approximants

Let µ be a finite positive Borel measure whose support is a compact subset K of the real line and let I be the convex hull of K. Let r denote a rational function with real coefficients whose poles lie in C \ I and r(∞) = 0. We consider multipoint rational interpolants of the function f (z) = dµ(x) z − x + r(z), where some poles are fixed and others are left free. We show that if the interpolati...

Shanks' convergence acceleration transform, Padé approximants and partitions

General order multivariate Padé approximants for Pseudo-multivariate functions. II

Explicit formulas for general order multivariate Padé approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function E (x) = ∞ ∑ j1,j2,...,jm=0 x1 1 x j2 2 · · ·x jm m (j1 + j2 + · · ·+ jm)! , the logarithm function L(x) = ∑ j1+j2+···+jm≥1 x1 1 x j2 2 · · ·x jm m j1 + j2 + · · ·+ jm , the Lauricella funct...

General order multivariate Padé approximants for pseudo-multivariate functions

Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inheri...

Evaluating Padé Approximants of the Matrix Logarithm

The inverse scaling and squaring method for evaluating the logarithm of a matrix takes repeated square roots to bring the matrix close to the identity, computes a Padé approximant, and then scales back. We analyze several methods for evaluating the Padé approximant, including Horner’s method (used in some existing codes), suitably customized versions of the Paterson– Stockmeyer method and Van L...