Search for: All records

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 $$\{ \langle x, \phi_{n} \rangle \}$$. The Kaczmarz algorithm defines a sequence of approximations from the sequence $$\{ \langle x, \phi_{n} \rangle \}$$; these approximations only converge to $$x$$ when $$\{ \phi_{n} \}$$ is \emph{effective}. We dualize the Kaczmarz algorithm so that $$x$$ can be obtained from $$\{\langle x, \phi_{n} \rangle\}$$ by using a second sequence $$\{\psi_{n}\}$$ in the reconstruction. This allows for the recovery of $$x$$ even when the sequence $$\{\phi_{n}\}$$ is not effective; in particular, our dualization yields a reconstruction when the sequence $$\{\phi_{n}\}$$ is \emph{almost effective}. We also obtain some partial results characterizing when the sequence of approximations from $$\{\langle \vec{x}, \phi_{n} \rangle\}$$ converges to $$x$$, in which case $$\{ (\phi_{n}, \psi_{n}) \}$$ is called an effective pair.