ABSTRACT. It is the purpose of this note to give a description of a simpleminded “alternating projections” algorithm and to very briefly mention some of its many and diverse applications. (A more comprehensive and detailed survey — with a lengthier list of reference — is planned for a later date.)

In its simplest (abstract) form, the algorithm goes back to von Neumann [20] and can be described as follows. Suppose M_{1} and M_{2} are two closed subspaces of a Hilbert space X and x is any element of X. The vector x is first (orthogonally) projected onto M_{1} to obtain a new vector x_{1}; then x_{1} is projected onto M_{2} to obtain x_{2}; then x_{2} is projected back onto M_{1} to obtain x_{3}, and we continue to alternately project back and forth between the two subspaces. The sequence of vectors (x_{n}) thus generated converges to the projection of x onto the intersection M_{1} ∩M_{2} of M_{1} and M_{2} (see Figure 1 below).