Contemporary Mathematics A Superfast Algorithm for Confluent Rational Tangential Interpolation Problem via Matrix-vector Multiplication for Confluent Cauchy-like Matrices Vadim Olshevsky and Amin Shokrollahi Abstract. Various problems in pure and applied mathematics and engineer- ing can be reformulated as linear algebra problems involving dense structured matrices. The structure of these dense matrices is understood in the sense that their n 2 entries can be completeley described by a smaller number O(n) of parameters. Manipulating directly on these parameters allows us to de- sign efficient fast algorithms. One of the most fundamental matrix problems is that of multiplying a (structured) matrix with a vector. Many fundamen- tal algorithms such as convolution, Fast Fourier Transform, Fast Cosine/Sine Transform, and polynomial and rational multipoint evaluation and interpola- tion can be seen as superfast multiplication of a vector by structured matrices (e.g., Toeplitz, DFT, Vandermonde, Cauchy). In this paper, we study a gen- eral 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 prob- lems (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 multiply a con- fluent Cauchy-like matrix by a vector. This is precisely what has been done in this paper. 1991 Mathematics Subject Classification. Primary: 15A06 Secondary: 47N70, 42A70. Key words and phrases. Rational matrix tangential interpolation. Nevanlinna-Pick problem. Caratheodory-Fejer problem. Cauchy matrices. Superfast algorithms. This work was supported by NSF grant CCR 9732355. c 0000 (copyright holder) 1