THE INSTABILITY OF PARALLEL PREFIX MATRIX MULTIPLICATION

It is shown that when parallel prefix is used to compute the leading principal miners of a tridiagonal matrix T within a bisection algorithm to compute the eigenvalues of T the relative error in the computed ei,aenvalues can be as great as epsilon kappa(3), where epsilon is machine precision and kappa is the condition number for the problem of computing the eigenvalues of T. An ideal algorithm, like serial bisection, would have forward error epsilon kappa. Forward and backward error bounds for the computed leading principal miners are given. Also, error bounds for the parallel prefix computation of the partial products of a sequence of matrices are given and some applications to other related problems in numerical Linear algebra, including the parallel implementation of the differential qd algorithm, are presented.