On the Convergence of Cyclic Jacobi Methods

In a cyclic Jacobi method for calculating the eigenvalues and eigenvectors of a symmetric matrix, the pivots are chosen in any fixed cyclic order. It is not known in theory whether convergence to the solution is always obtained, although convergence has been proved subject to a restriction on the angle of rotation about each pivot (Henrici, 1958). Now we report an actual computer calculation where a cyclic Jacobi method failed, due to computer rounding errors, so in practice the angle restriction may be needed. A new bound for the angle restriction is given that is less severe than the one proposed originally.