The Dual Kaczmarz Algorithm

Abstract

The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector x in a (separable) Hilbert space from the inner-products { x, φn }. The Kaczmarz algorithms defines a sequence of approximations from the sequence { x, φn }; these approximations only converge to x when {φn} is effective. We dualize the Kaczmarz algorithm so that x can be obtained from { x, φn } by using a second sequence {ψn} in the reconstruction. This allows for the recovery of x even when the sequence {φn} is not effective; in particular, our dualization yields a reconstruction when the sequence {φn} is almost effective. We also obtain some partial results characterizing when the sequence of approximations from { x, φn } using {ψn} converges to x, in which case {(φn, ψn)} is called an effective pair.2010 Mathematics Subject Classification. Primary: 41A65, 65D15. Secondary: 42C15, 65F10.