Diagonal pivoting for partially reconstructible Cauchy-like Matrices

Abstract In an earlier paper we exploited the displacement structure of Cauchy-like matrices to derive for them a fast O ( n 2 ) implementation of Gaussian elimination with partial pivoting. One application is to the rapid and numerically accurate solution of linear systems with Toeplitz-like coefficient matrices, based on the fact that the latter can be transformed into Cauchy-like matrices by using the fast Fourier, sine, or cosine transform. However, symmetry is lost in the process, and the algorithm given is not optimal for Hermitian coefficient matrices. In this paper we present a new fast O ( n 2 ) implementation of symmetric Gaussian elimination with partial diagonal pivoting for Hermitian Cauchy-like matrices, and show how to transform Hermitian Toeplitz-like matrices to Hermitian Cauchy-like matrices, obtaining algorithms that are twice as fast as those in the earlier work. Numerical experiments indicate that in order to obtain not only fast but also numerically accurate methods, it is advantageous to explore the important case in which the corresponding displacement operators have nontrivial kernels; this situation gives rise to what we call partially reconstructible matrices, which are introduced and studied in the present paper. We extend the transformation technique and the generalized Schur algorithms (i.e., fast displacement-based implementations of Gaussian elimination) to partially reconstructible matrices. We show by a variety of computed examples that the incorporation of diagonal pivoting methods leads to high accuracy. We focused in this paper on the design of new numerically reliable algorithms for Hermitian Toeplitz-like matrices. However, the proposed algorithms have other important applications; in particular, we briefly describe how they recursively solve a boundary interpolation problem for J -unitary rational matrix functions.