Pivoting for Structured Matrices and Rational Tangential Interpolation

Gaussian elimination is a standard tool for computing triangular factorizations for general matrices, and thereby solving associated linear systems of equations. As is well-known, when this classical method is implemented in finite-precision-arithmetic, it often fails to compute the solution accurately because of the accumulation of small roundoffs accompanying each elementary floating point operation. This problem motivated a number of interesting and important studies in modern numerical linear algebra; for our purposes in this paper we only mention that starting with the breakthrough work of Wilkinson, several pivoting techniques have been proposed to stabilize the numerical behavior of Gaussian elimination. Interestingly, matrix interpretations of many known and new algorithms for various applied problems can be seen as a way of computing triangular factorizations for the associated structured matrices, where different patterns of structure arise in the context of different physical problems. The special structure of such matrices [e.g., Toeplitz, Hankel, Cauchy, Vandermonde, etc.] often allows one to speed-up the computation of its triangular factorization, i.e., to efficiently obtain fast implementations of the Gaussian elimination procedure. There is a vast literature about such methods which are known under different names, e.g., fast Cholesky, fast Gaussian elimination, generalized Schur, or Schur-type algorithms. However, without further improvements they are efficient fast implementations of a numerically inaccurate [for indefinite matrices] method. In this paper we survey recent results on the fast implementation of various pivoting techniques which allowed us to improve numerical accuracy for a variety of fast algorithms. This approach led us to formulate new more accurate numerical methods for factorization of general and of J-unitary rational matrix functions, for solving various tangential interpolation problems, new Toeplitz-like and Toeplitz-plus-Hankel-like solvers, and new divided differences schemes. We beleive that similar methods can be used to design accurate fast algorithm for the other applied problems and a recent work of our colleagues supports this anticipation. 1991 Mathematics Subject Classification. Primary 65F30, 15A57; Secondary 65Y20.