In this work, a solution strategy based on the use of Padé approximants is investigated for efficient solution of parametric finite element problems such as, for example, frequency sweep analyses. An improvement to the Padé-based expansion of the solution vector components is proposed, suggesting the advantageous a priori estimate of the poles of the solution. This allows for the intervals of a...

Several existing algorithms for computing the matrix cosine employ polynomial or rational approximations combined with scaling and use of a double angle formula. Their derivations are based on forward error bounds. We derive new algorithms for computing the matrix cosine, the matrix sine, and both simultaneously, that are backward stable in exact arithmetic and behave in a forward stable manner...

The usual way to compute a low-rank approximant of a matrix H is to take its truncated SVD. However, the SVD is computationally expensive. This paper describes a much simpler generalized Schur-type algorithm to compute similar low-rank approximants. For a given matrix H which has d singular values larger than ε, we find all rank d approximants Ĥ such that H − Ĥ has 2-norm less than ε. The set o...

It is well known that outer functions are zero-free on the unit disk. If an outer function, f , is given as an infinite series and a finite (polynomial) approximation is chosen, then it is desirable that the approximants retain the zero-free property of f . We observe for outer functions that the standard Taylor approximants do not, in general, retain the zero-free property – even when fairly r...

This manuscript reviews the study of the asymptotic behavior of meromorphic approximants to classes of functions holomorphic at infinity. The asymptotic theory of meromorphic approximation is primarily concerned with establishing the types of convergence, describing the domains where this convergence takes place, and identifying its exact rates. As the first question is classical, it is the lat...