On the Convergence of Broyden-Like Methods

On Newton-like methods of “bounded deterioration” using recurrent functions

“…Using our new idea of recurrent functions, a combination of Lipschitz and center-Lipschitz conditions on the first Fréchet-derivative, instead of only Lipschitz conditions [10], we provided an analysis with the following advantages over the work in [4,10,11,13,16,19,20]: weaker sufficient convergence conditions, and larger convergence domain. Note that these advantages are obtained under the same computational cost as in [10], since in practice the computation of the Lipschitz constant K requires the computation of K 0 (see Sect.…”

“…We provide a convergence analysis based on our new idea of recurrent functions. In particular, using the majorant method [1][2][3][4][5][6], and more precise majorizing sequences than before [4,10,11,13,16,19,20], we provide under the same hypotheses and computational cost: in the semilocal case finer error bounds on the distances x n+1 − x n , x n − x , (n ≥ 0), and more precise information on the location of the solution x .…”

“…The conditions in [4,11,13,16,19,20] are also weakened, since the methods there are special cases of the ones in [10]. Then, with these changes in Proposition 2.7, and Theorem 2.3, the conclusions will still hold in this setting.…”

“…(d) We showed that our semilocal convergence analysis is finer at the general setting of [10]. All references in the literature [4,11,13,16,19,20] reduce to (2.79), which can always be replaced by (2.77). (e) A popular choice (but not the only one) for the triplet (b n , Δ, K 1 ) is given by…”

A Kantorovich-type analysis of Broyden’s method using recurrent functions

“…In such cases, the iteration sequences converge to an optimal solution of the problem at hand. Since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”