Quasi-Newton least squares method
 | It has been suggested that this article be merged into quasi-Newton method. (Discuss) Proposed since July 2020. |
In numerical analysis, the quasi-Newton least squares method is a quasi-Newton method for finding roots in [math]\displaystyle{ n }[/math] variables. It was originally described by Rob Haelterman et al. in 2009.[1]
Newton's method for solving [math]\displaystyle{ f(x) = 0 }[/math] uses the Jacobian matrix, [math]\displaystyle{ J }[/math], 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 [math]\displaystyle{ f }[/math].
Haelterman et al. also showed that when the quasi-Newton least squares method is applied to a linear system of size [math]\displaystyle{ n \times n }[/math], it converges in at most [math]\displaystyle{ n + 1 }[/math] 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
- ↑ 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.
 |
