A new approach to the rational interpolation problem: The vector case (Q805947)

Let \({\mathbb{K}}\) be a field, let m and p be positive integers, and let both \(z_ j\in {\mathbb{K}}\) and \(f_ j\in {\mathbb{K}}^ m\), \(1\leq j\leq p\). The ``vector rational interpolation problem'' (VRIP) is the problem of finding a ``vector rational function'' \[ n(z)/d(z)=[n_ 1(z)/d(z),n_ 2(z)/d(z),...,n_ p(z)/d(z)], \] where \(n_ 1,n_ 2,...,n_ p\), and d are all polynomials with coefficients in \({\mathbb{K}}\), such that both \(n(z_ j)/d(z_ j)=f_ j\) and \(d(z_ j)\neq 0\), \(1\leq j\leq p\). The authors produce a specific recursive algorithm to find all solutions to this VRIP, and similar algorithms which give all solutions to some variations of this VRIP, for example, if the \(d(z_ j)'s\) are not all required to be zero. The algorithm given has the advantage over other known algorithms that it does not require an reordering of the data in the problem. The basic ideas of this paper are modifications of techniques from the author's paper [J. Comput. Appl. Math. 32, No.1/2, 281-289 (1990; Zbl 0715.41018)] in which the seemingly simpler case of scalar rational interpolation is considered.