For extrapolation and interpolation with complete or quasi complete Chebychev systems, several methods for computing a table of approximants by recurrence are well-know, e.g., the E-algorithm for extrapolation, the Mühlbach-Neville-Aitken algorithm and the recurrence relation for Generalized Divided Differences together with a corresponding Newton Interpolation Formula for interpolation. There exist applications ---as for example the problem of determining a table of multivariate rational interpolants--- where it is quite natural that the system of interpolating functions does not form a Chebychev system.

If the system of interpolating functions is non-regular, then an application of the algorithms mentioned above might fail. It is shown that the technique of Neville pivoting still allows to determine all regular entries of the considered table by recurrence. Replacing the main rule of the above algorithms by new two-step formulas yields methods for extrapolation and interpolation having the same complexity where no conditions on the interpolating system are required.