The Dual Kaczmarz Algorithm

  • Anna Aboud
  • Emelie Curl
  • Steven N. Harding
  • Mary Vaughan
  • Eric S. WeberEmail author


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.


Kaczmarz algorithm Effective sequence Frame Gram matrix Hilbert space 

Mathematics Subject Classification (2010)

41A65 65D15 42C15 65F10 



Anna Aboud and Eric Weber were supported in part by the National Science Foundation and the National Geospatial-Intelligence Agency under NSF award #1832054.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA

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