Jacobi's method is more accurate than QR

We show that Jacobi's method (with a proper stopping criterion) computes small eigenvalues of symmetric positive de nite matrices with a uniformly better relative accuracy bound than QR, divide and conquer, traditional bisection, or any algorithm which rst involves tridiagonalizing the matrix. In fact, modulo an assumption based on extensive numerical tests, we show that Jacobi's method is optimally accurate in the following sense: if the matrix is such that small relative errors in its entries cause small relative errors in its eigenvalues, Jacobi will compute them with nearly this accuracy. In other words, as long as the initial matrix has small relative errors in each component, even using in nite precision will not improve on Jacobi (modulo factors of dimensionality). We also show the eigenvectors are computed more accurately by Jacobi than previously thought possible. We prove similar results for using one-sided Jacobi for the singular value decomposition of a general matrix. 1The rst author would like to acknowledge the nancial support of the NSF via grants DCR-8552474 and ASC-8715728, and the support of DARPA via grant F49620-87-C-0065. Part of this work was done while the rst author was visiting the Fernuniversitat Hagen, and he acknowledges their support as well. He is also a Presidential Young Investigator.