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...

We show that a simple and straightforward rational approximation to the Thomas– Fermi equation provides the slope at origin with unprecedented accuracy and that relatively small Padé approximants are far more accurate than more elaborate approaches proposed recently by other authors. We consider both the Thomas–Fermi equation for isolated atoms and for atoms in strong magnetic fields.

We calculate the low-temperature series of the free energy in absolutevalue solid-on-solid (ASOS) model to order u23 using finite-lattice method. The property of the obtained series and the behavior of their Padé approximants confirms us that the roughening transition in ASOS model is of Kosterlitz-Thouless type.

We classify all integer squares (and, more generally, q-th powers for certain values of q) whose ternary expansions contain at most three digits. Our results follow from Padé approximants to the binomial function, considered 3-adically. –Dedicated to the memory of John Selfridge.

Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, f ∈ A (C \A), ]A < ∞. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Padé approximants for f . The Padé approxi...