Convergence of Multipoint Padé-type Approximants

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...

Multipoint Padé approximants to complex Cauchy transforms with polar singularities

We study diagonal multipoint Padé approximants to functions of the form F (z) = Z dλ(t) z − t +R(z), where R is a rational function and λ is a complex measure with compact regular support included in R, whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some c...

Shanks' convergence acceleration transform, Padé approximants and partitions

Padé Approximants in Complex Points Revisited

In 1976, Chisholm et al. 1 published a paper concerning the location of poles and zeros of Padé approximants of ln 1 − z developed at the complex point ζ : ln 1 − z ln 1 − ζ − ∑∞ n 1 1/n z − ζ/1 − ζ . They claimed that all poles and zeros of diagonal Padé approximants n/n interlace on the cut z ζ t 1 − ζ , t ∈ 1,∞ . Unfortunately, this result is only partially true, for poles. Klarsfeld remarke...

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...