When s = 1 and N = ‖n‖ − 1 this is the classical Hermite Padé approximation problem. Here we allow N to be arbitrary. We describe algorithms for computing an order N genset of type n: a matrix V ∈ k[x]∗×m such that every row of V is a solution to (1) and every solution P of (1) can be expressed as a k[x]-linear combination of the rows of V . Ideally, V will be a minbasis of solutions: V has ful...

In this article, we design fast algorithms for the computation of approximant bases in shifted Popov normal form. We first recall the algorithm known as PM-Basis, which will be our second fundamental engine after polynomial matrix multiplication: most other fast approximant basis algorithms basically aim at efficiently reducing the input instance to instances for which PMBasis is fast. Such red...

Within the context of massive N -component φ scalar field theory, we use asymptotic Padéapproximant methods to estimate from prior orders of perturbation theory the five-loop contributions to the coupling-constant β-function βg, the anomalous mass dimension γm, the vacuum-energy βfunction βv, and the anomalous dimension γ2 of the scalar field propagator. These estimates are then compared with e...

Asymptotic Padé-approximant methods are utilized to estimate the O(α5s) contribution to the H → gg rate and the O(α4s) contribution to the H → bb̄ rate. The former process is of particular interest because of the slow convergence evident from the three known terms of its QCD series, which begins with an O(α2s) leading-order term. The O(α5s) contribution to the H → gg rate is expressed as a degre...

The structure of a complicated quasicrystal approximant epsilon(16) was predicted from a known and related quasicrystal approximant epsilon(6) by the strong-reflections approach. Electron-diffraction studies show that in reciprocal space, the positions of the strongest reflections and their intensity distributions are similar for both approximants. By applying the strong-reflections approach, t...

The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in MATLAB’s expm function. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Padé approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error...

The scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ‖A‖ 1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the...

Problems concerning complete structural characterization of quasicrystals involve locating the atomic positions as well as determining the distribution of elements at each site. Quasicrystalline approximants provide models for potential building units of quasicrystals, but a clear determination of the elemental decorations in such approximants also remains incomplete. We report experimental and...

The Padé-Legendre method has been introduced as an effective approach to characterize uncertainties in the presence of strongly non-linear or discontinuous system responses—thus, it supports forward propagation. The method is based on the construction of a ratio of polynomials that approximate the available data. Two criteria for the choice of the best approximant are considered and an optimiza...