Quasi-Newton inverse least squares method

In numerical analysis, the quasi-Newton inverse least squares method is a quasi-Newton method for finding roots of functions of several variables. It was originally described by Degroote et al. in 2009.[1]

Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian is a difficult (sometimes even impossible) and expensive operation. The idea behind the quasi-Newton inverse least squares method is to build up an approximate Jacobian based on known input–output pairs of the function f.

Haelterman et al. also showed that when the quasi-Newton inverse least squares method is applied to a linear system of size n × n, it converges in at most n + 1 steps, although like all quasi-Newton methods, it may not converge for nonlinear systems.[2]

The method is closely related to the quasi-Newton least squares method.

References
  1. J. Degroote; R. Haelterman; S. Annerel; A. Swillens; P. Segers; J. Vierendeels (2008). "An interface quasi-Newton algorithm for partitioned simulation of fluid-structure interaction". Proceedings of the International Workshop on Fluid–Structure Interaction. Theory, Numerics and Applications. S. Hartmann, A. Meister, M. Schfer, S. Turek (Eds.), Kassel University Press, Germany.
  2. R. Haelterman; J. Petit; B. Lauwens; H. Bruyninckx; J. Vierendeels (2014). "On the Non-Singularity of the Quasi-Newton-Least Squares Method". Journal of Computational and Applied Mathematics. 257: 129–131. doi:10.1016/j.cam.2013.08.020.
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