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 $n\times n$ , 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

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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[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.