Accurate eigenvalues of certain sign regular matrices

We present a new O(n3) algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic. The accuracy and cost are unaffected by the conventional eigenvalue condition numbers.A matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors. The sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign (-1)k(k-1)/2 for all k. This class of matrices can also be characterized as “nonsingular totally nonnegative matrices with columns in reverse order”.We exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity. We arrange the computations in such a way that no subtractive cancellation is encountered, thus guaranteeing high relative forward accuracy.