Convergence of ray sequences of Frobenius-Padé 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...

Szegő-type Asymptotics for Ray Sequences of Frobenius-padé Approximants

Let σ̂ be a Cauchy transform of a possibly complex-valued Borel measure σ and {pn} be a system of orthonormal polynomials with respect to a measure μ, supp(μ)∩ supp(σ) = ∅. An (m,n)-th Frobenius-Padé approximant to σ̂ is a rational function P/Q, deg(P) 6m, deg(Q) 6 n, such that the first m+n+ 1 Fourier coefficients of the linear form Qσ̂−P vanish when the form is developed into a series with respe...

Convergence of ray sequences of Padé approximants for 2 F 1 ( a , 1 ; c ; z ) , for c > a > 0 .

The Padé table of 2F1(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n−1 and c / ∈ Z − , the denominator polynomial Qmn(z) in the [m/n] Padé approximant Pmn(z)/Qmn(z) for 2F1(a, 1; c; z) and the remainder term Qmn(z)2F1(a, 1; c; z)−Pmn(z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n−1, the poles of Pmn(z)/Qmn(z) lie on the cut (1,∞). We d...

Shanks' convergence acceleration transform, Padé approximants and partitions

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...