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 $\{\langle x, \phi _{n} \rangle \}$. 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 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 almost effective. We also obtain some partial results characterizing when the sequence of approximations from $\{\langle x, \phi _{n} \rangle \}$ using $\{\psi _{n}\}$ converges to $x$, in which case $\{(\phi _{n}, \psi _{n})\}$ is called an effective pair.