AN APPLICATION OF SCHUR’S ALGORITHM TO VARIABILITY REGIONS OF CERTAIN ANALYTIC FUNCTIONS II

We extend our study of variability regions, Ali et al. [‘An application of Schur algorithm to variability regions of certain analytic functions–I’, Comput. Methods Funct. Theory, to appear] from convex domains to starlike domains. Let $\mathcal {CV}(\Omega )$ be the class of analytic functions f in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in {\Omega }$ . As an application of the main result, we determine the variability region of $\log f'(z_0)$ when f ranges over $\mathcal {CV}(\Omega )$ . By choosing a particular $\Omega $ , we obtain the precise variability regions of $\log f'(z_0)$ for some well-known subclasses of analytic and univalent functions.

A general approach to multivariable recursive interpolation

We consider here the problem of constructing a general recursive algorithm to interpolate a given set of data with a rational function. While many algorithms of this kind already exist, they are either providing non-minimal degree solutions (like the Schur algorithm) or exhibit jumps in the degree of the interpolants (or of the partial realization, as the problem is called when the interpolation is at infinity, see Rissanen (SIAM J Control 9(3):420–430, 1971) and Gragg and Lindquist (in: Linear systems and control (special issue), linear algebra and its applications, vol 50. pp 277–319, 1983)). By imbedding the solution into a larger set of interpolants, we show that the increase in the degree of this representation is proportional to the increase in the length of the data. We provide an algorithm to interpolate multivariable tangential sets of data with arbitrary nodes, generalizing in a fundamental manner the results of Kuijper (Syst Control Lett 31:225–233, 1997). We use this new approach to discuss a special scalar case in detail. When the interpolation data are obtained from the Taylor-series expansion of a given function, then the Euclidean-type algorithm plays an important role.

The Generalized Schur Algorithm and Some Applications

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the Q R and L U factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the Q R factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices.