A Unified Superfast Algorithm for Boundary Rational Tangential Interpolation Problems and for Inversion and Factorization of Dense Structured Matrices

• Authors: Vadim Olshevsky;Victor Pan
• Venue: FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
• Year: 1998

The classical scalar Nevanlinna-Pick interpolation problem has a long and distinguished history, appearing in a variety of applications in mathematics and electrical engineering. There is a vast literature on this problem and on its various far reaching generalizations. It is widely known that the now classical algorithm for solving this problem proposed by Nevanlinna in 1929 can be seen as a way of computing the Cholesky factorization for the corresponding Pick matrix. Moreover, the classical Nevanlinna algorithm takes advantage of the special structure of the Pick matrix to compute this triangular factorization in only $O(n^2)$ arithmetic operations, where $n$ is the number of interpolation points, or, equivalently, the size of the Pick matrix. Since the structure-ignoring standard Cholesky algorithm [though applicable to the wider class of general matrices] has much higher complexity $O(n^3)$, the Nevanlinna algorithm is an example of what is now called fast algorithms. In this paper we use a divide-and-conquer approach to propose a new superfast $O(n \log^3 n)$ algorithm to construct solutions for the more general boundary tangential Nevanlinna-Pick problem. This dramatic speed-up is achieved via a new divide-and-conquer algorithm for factorization of rational matrix functions; this superfast algorithm seems to have a practical and theoretical significance itself. It can be used to solve similar rational interpolation problems [e.g., the matrix Nehari problem], and a variety of engineering problems. It can also be used for inversion and triangular factorization of matrices with displacement structure, including Hankel-like, Vandermonde-like, and Cauchy-like matrices.