Fast and Accurate Con-Eigenvalue Algorithm for Optimal Rational Approximations

Abstra t. The need to ompute small on-eigenvalues and the asso iated on-eigenve tors of positive-de nite Cau hy matri es naturally arises when onstru ting rational approximations with a (near) optimally small L error. Spe i ally, given a rational fun tion with n poles in the unit disk, a rational approximation with m ≪ n poles in the unit disk may be obtained from the mth on-eigenve tor of an n × n Cau hy matrix, where the asso iated on-eigenvalue λm > 0 gives the approximation error in the L norm. Unfortunately, standard algorithms do not a urately ompute small on-eigenvalues (and the asso iated on-eigenve tors) and, in parti ular, yield few or no orre t digits for on-eigenvalues smaller than the ma hine roundo . We develop a fast and a urate algorithm for omputing on-eigenvalues and on-eigenve tors of positive-de nite