Chapter 8.1 Least squares

Mathematical, physical and chemical tables

First Online Edition (2006)

Part 8. Refinement of structural parameters

  1. E. Prince1,
  2. P. T. Boggs2

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000609

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Prince, E. and Boggs, P. T. 2006. Least squares. International Tables for Crystallography. C:8:8.1:678–688.

Author Information

  1. 1

    NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

  2. 2

    Scientific Computing Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

Publication History

  1. Published Online: 1 JAN 2006


Chapter 8.1 summarizes the mathematics of both the statistical and numerical analytical aspects of data fitting by linear and nonlinear least squares. It includes discussions of various algorithms for the solution of the least squares problem and, for the nonlinear case, of the stability and convergence properties of different procedures. Finally, it includes information on the availability of existing high-quality computer software.


  • conditioning;
  • conjugate-gradient methods;
  • Gauss–Newton algorithm;
  • large-scale problems;
  • least-squares calculations;
  • Levenberg–Marquardt algorithm;
  • linear algebra;
  • non-linear least squares;
  • normal equations;
  • numerical methods;
  • quasi-Newton methods;
  • secant methods;
  • sparse matrices;
  • statistics;
  • stopping rules;
  • trust-region methods