Quasi-Newton Least Squares Method

In numerical analysis, The Quasi-Newton Least Squares Method is a quasi-Newton method for finding roots in n variables. It was originally described by Rob Haelterman 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 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 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.

The method is closely related to the Quasi-Newton Inverse Least Squares Method.

References

  1. ↑ R. Haelterman; J. Degroote; D. Van Heule; J. Vierendeels (2009). "The quasi-Newton Least Squares method: a new and fast secant method analyzed for linear systems". SIAM J. Numer. Anal. 47 (3): 2347–2368. doi:10.1137/070710469.


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