Computational Optimization and Applications

, Volume 54, Issue 3, pp 579–593

Variable projection for nonlinear least squares problems


DOI: 10.1007/s10589-012-9492-9

Cite this article as:
O’Leary, D.P. & Rust, B.W. Comput Optim Appl (2013) 54: 579. doi:10.1007/s10589-012-9492-9


The variable projection algorithm of Golub and Pereyra (SIAM J. Numer. Anal. 10:413–432, 1973) has proven to be quite valuable in the solution of nonlinear least squares problems in which a substantial number of the parameters are linear. Its advantages are efficiency and, more importantly, a better likelihood of finding a global minimizer rather than a local one. The purpose of our work is to provide a more robust implementation of this algorithm, include constraints on the parameters, more clearly identify key ingredients so that improvements can be made, compute the Jacobian matrix more accurately, and make future implementations in other languages easy.


Data fittingModel fittingVariable projection methodNonlinear least squares problemsJacobian approximationLeast squares approximationStatistical softwareMathematical software design and analysis

Supplementary material

Copyright information

© US National Institute of Standards and Technology 2012

Authors and Affiliations

  1. 1.National Institute of Standards and TechnologyGaithersburgUSA
  2. 2.Computer Science Department and Institute for Advanced Computer StudiesUniversity of MarylandCollege ParkUSA
  3. 3.Applied and Computational Mathematics DivisionNational Institute of Standards and TechnologyGaithersburgUSA