A Superfast Algorithm for Confluent Rational Tangential Interpolation Problem via Matrix-vector Multiplication for Confluent Cauchy-like Matrices∗

Various problems in pure and applied mathematics and engineering can be reformulated as linear algebra problems involving dense structured matrices. The structure of these dense matrices is understood in the sense that their n2 entries can be completeley described by a smaller number O(n) of parameters. Manipulating directly on these parameters allows us to design efficient fast algorithms. One of the most fundamental matrix problems is that of multiplying a (structured) matrix with a vector. Many fundamental algorithms such as convolution, Fast Fourier Transform, Fast Cosine/Sine Transform, and polynomial and rational multipoint evaluation and interpolation can be seen as superfast multiplication of a vector by structured matrices (e.g., Toeplitz, DFT, Vandermonde, Cauchy). In this paper, we study a general class of structured matrices, which we suggest to call confluent Cauchy-like matrices, that contains all the above classes as a special case. We design a new superfast algorithm for multiplication of matrices from our class with vectors. Our algorithm can be regarded as a generalization of all the above mentioned fast matrix-vector multiplication algorithms. Though this result is of interest by itself, its study was motivated by the following application. In a recent paper [18] the authors derived a superfast algorithm for solving the classical tangential Nevanlinna-Pick problem (rational matrix interpolation with norm constrains). Interpolation problems of Nevanlinna-Pick type appear in several important applications (see, e.g., [4]), and it is desirable to derive efficient algorithms for several similar problems. Though the method of [18] can be applied to compute solutions for certain other important interpolation problems (e.g., of Caratheodory-Fejer), the solution for the most general confluent tangential interpolation problems cannot be easily derived from [18]. Deriving new algorithms requires to design a special fast algorithm to ∗This work was supported by NSF grant CCR 9732355. multiply a confluent Cauchy-like matrix by a vector. This is precisely what has been done in this paper.