Accurate Symmetric Rank Revealing and Eigendecompositions of Symmetric Structured Matrices

Abstract

We present new $O(n^3)$ algorithms that compute eigenvalues and eigenvectors to high relative accuracy in floating point arithmetic for the following types of matrices: symmetric Cauchy, symmetric diagonally scaled Cauchy, symmetric Vandermonde, and symmetric totally nonnegative matrices when they are given as products of nonnegative bidiagonal factors. The algorithms are divided into two stages: the first stage computes a symmetric rank revealing decomposition of the matrix to high relative accuracy, and the second stage applies previously existing algorithms to this decomposition to get the eigenvalues and eigenvectors. Rank revealing decompositions are also interesting in other problems, such as the numerical determination of the rank and the approximation of a matrix by a matrix with smaller rank.