Given a sequence of vectors $\{ x^{(0)} ,x^{(1)} ,x^{(2)} , \cdots \} $, which may be convergent or divergent and which is governed by $x^{(i + 1)} = Mx^{(i)} + g$, where M and g may be unknown, an extrapolation method is derived for finding either the limit or antilimit of the sequence. The extrapolation method is intended for those problems where the numerical rank of M is small, or where M has only a few dominant eigenvalues. An error analysis is given; in particular, it is shown that the extrapolation method is stable provided that the eigenvalues of M are bounded away from 1.

The method is compared with the epsilon algorithm and the vector-epsilon algorithm. Numerical evidence is provided showing that the method obtains the same accuracy as the epsilon algorithms but with approximately half the cost.

It is also shown that the extrapolation method is competitive with the Woodbury method for solving the system $(I - M)x = g$, where M is known.