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 entriesâ€¦Â CONTINUE READING