Wide-angle scalar beam propagation methods (BPMs) for effective modelling of optical propagation in three-dimensional (3D) waveguides are usually limited to the low-order-accurate Padé(1,1) approximant. Presented is a 3D wide-angle scalar BPM based on higher-order Padé approximant operators which is factored into a series of simpler first-order Padé(1,1) approximants. This results in a multiste...

A new complex Jacobi iterative technique adapted for the solution of three-dimensional (3D) wide-angle (WA) beam propagation is presented. The beam propagation equation for analysis of optical propagation in waveguide structures is based on a novel modified Padé(1,1) approximant operator, which gives evanescent waves the desired damping. The resulting approach allows more accurate approximation...

A more interesting use of Padé approximation is “two-point Padé approximation” or Padé interpolation in which a Padé approximant for f(x) is constructed from two truncated Taylor series: one for when x is very small and one is for when x is very large. The Padé approximant in this method interpolates between the values at which f(x) is known. This method is directly applicable to perturbation t...

The matrix variational Padé approximant and its generalization to elastic wave scattering are discussed. Predictions of the method for the scattering of a longitudinal plane wave are compared with the exact scattering from spherical voids and inclusions. Its predictions are also compared to those of the first and second Born approximations and to the standard matrix Padé approximant based on th...

A new algorithm is developed for computing arbitrary real powers Ap of a matrix A ∈ Cn×n. The algorithm starts with a Schur decomposition, takes k square roots of the triangular factor T , evaluates an [m/m] Padé approximant of (1 − x)p at I − T 1/2 , and squares the result k times. The parameters k and m are chosen to minimize the cost subject to achieving double precision accuracy in the eval...

Let m, n be nonnegative integers and a<m+n) be a set of m + n + 1 real in~lation points (not necessarily distinct). Let Rm.n = P m.n / Qm.n be the unique rational function with degPm.n ::; m, deg Qm,n :$n, that in~lates e'" in the points of a<nr+n). Ifm = mv, n = nv with mv + nv --+ 00, and mv /nv --+ A as v --+ 00, and the sets a<nr+n) are uniformly bounded, we show that Pm.n(z) --+ ~/(l+)'), ...

Padé approximant methods are applied to the known terms of the DRED β-function for N = 1 supersymmetric SU(3) Yang-Mills theory. Each of the [N |M ] approximants with N + M ≤ 4, M 6= 0 constructed from this series exhibits a positive pole which precedes any zeros of the approximant, consistent with the same infrared-attractor pole behaviour known to characterize the exact NSVZ β-function. A sim...

UNLABELLED This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value proble...

In this paper, free convection flow and heat transfer of nanofluids of differently-shaped nano-sized particles over a vertical plate at very low and high Prandtl numbers are analyzed.  The governing systems of nonlinear partial differential equations of the flow and heat transfer processes are converted to systems of nonlinear ordinary differential equation through similarity transformations. T...