Contemporary Algorithms: Theory and Applications Volume II

Extended Semilocal Convergence for Chebyshev-Halley-Type Schemes for Solving Nonlinear Equations under Weak Conditions

“…Notice also that using (A3) the following estimate was used in the earlier studies [3][4][5][7][8][9][10][11][12][13][14][15][16]:…”

“…by the right hand side of ( 12), thus 3 The condition ( 12) is the sufficient convergence criterion for the sequence {t k }. Such a criterion is standard in this type of study.…”

“…Many applications in computational Sciences require finding a solution x * of the nonlinear equation (1) ( ) 0, x =  where the operator  : D ⊂ X → Y is acting between Banach spaces X and Y. Higher convergence order schemes have been used extensively to generate a sequence approximating x * under certain conditions [1][2][3][4]. In particular, the third order scheme [5] has been used and is defined ∀k = 0, 1, 2, … by…”

“…Using conditions (A1)-(A3) and (C4) the R−order was increased. In particular, if the second derivative satisfies (A4) the R−order of the scheme (3) is at least five which is higher than Chebyshev's, Halley's, and Super-Halley's [1][2][3][4]17].…”

“…and (Q6) Provide the results in affne invariant form. The advantages of (Q6) are well-known [3,[17][18]. Denote this set of questions by (Q).…”