On the rate of convergence of the alternating projection method in finite dimensional spaces

Using the results of Smith, Solmon, and Wagner [K. Smith, D. Solomon, S. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977) 1227-1270] and Nelson and Neumann [S. Nelson, M. Neumann, Generalizations of the projection method with application to SOR theory for Hermitian positive semidefinite linear systems, Numer. Math. 51 (1987) 123-141] we derive new estimates for the speed of the alternating projection method and its relaxed version in Rm. These estimates can be computed in at most O(m3) arithmetic operations unlike the estimates in papers mentioned above that require spectral information. The new and old estimates are equivalent in many practical cases. In cases when the new estimates are weaker, the numerical testing indicates that they approximate the original bounds in papers mentioned above quite well.