In this paper, we present the cross rules of some extrapolation algorithms. These new cross rules are similar in their spirit to the cross rule for the scalar e-algorithm and the -algorithm. The first interest of such rules is that, in the algorithms for implementing sequence transformations for the acceleration of the convergence, they allow us to compute only the quantities which have a signification for extrapolation and to jump over the intermediate ones. The second interest comes out from the link between convergence acceleration methods, soliton theory and discrete integrable systems. In some cases, cross rules directly lead to physically significant integrable systems. For example, the cross rule of the confluent qd-algorithm leads to the celebrated Toda equation, which is known to have physical applications.