Pivoting for structured matrices with applications

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 nite-precision-arithmetic, it often fails to compute the solution accurately because of the accumulation of small roundo s accompanying each elementary oating 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 di erent patterns of structure arise in the context of di erent 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 eÆciently obtain fast implementations of the Gaussian elimination procedure. There is a vast literature about such methods which are known under di erent