Untere Fehlerschranken für Regula-falsi-Verfahren

A Unified Convergence Analysis for Secant-Type Methods

“…For example: Case 1 (secant method) l 0 = l, λ = 1 and µ = 1 has been studied in [5,7,10,11,13,14,18,20] and for l 0 ≤ l in [2,3]. Case 2 (Newton's method) l 0 = l, λ = 0, c = 0 and µ = 2 has been studied in [1, 7-9, 11, 13, 14, 16, 17, 19-21] and for l 0 ≤ l in [2][3][4].…”

“…Using the Banach lemma on invertible operators [8], [10], [14], [17], [18] and (3.4), we deduce that M is invertible and…”

“…Schmidt [18], Potra-Ptáck [14], Dennis [7], Ezquerro el at. [10], used the majorizing sequence {α n } for θ n ∈ [0, 1] and l 0 = l. That is, they used the sequence {t n } given by (3.27) t −1 = 0, t 0 = c, t 1 = c + ν…”

“…There is a plethora of sufficient convergence criteria for special cases of secant-type methods (1.3)-(1.5) under Lipschitz-type conditions (1.2) (see [1][2][3][4][5][6][7][8][9][10][11][13][14][15][16][17][18][19][20][21] and the references there in). Therefore, it is important to study the convergence of the secant-type method in a unified way.…”

Convergence conditions for secant-type methods

“…Bosarge and Falb [7], Dennis [9], Potra [16], Argyros [1]- [5], Hernández et al [10] and others [11], [15], [18] have provided sufficient convergence conditions for the Secant method based on Lipschitz-type conditions on δF (see also relevant results in [6]- [9], [12], [14], [16], [17], [19]). The conditions usually associated with the semilocal convergence of the Secant method (1.2) are:…”