A fast and efficient composite Newton–Chebyshev method for systems of nonlinear equations

Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

“…We shall show the estimates (14)-(16) using mathematical induction. By hypothesis x 0 ∈ U(δ, r) − {δ}, (4), (9), and (10), we get that…”

“…for Banach space valued mappings with H : D ⊆ X → Y, and H is differentiable according to Fréchet [1,2]. Many authors have studied local and semilocal convergence criteria of iterative methods (see, for example [3][4][5][6][7][8][9][10][11][12][13][14]). The most well-known iterative method for approximating a solution δ of Equation 1is Newton's method, which is given by…”

“…, (2) which has a quadratic order of convergence. In order to achieve higher convergence order, a number of modified, multistep Newton's or Newton-type iterations have been developed in the literature; see [3,4,6,7,[9][10][11][12][15][16][17][18][19] and references cited therein. There is another important class of multistep methods based on Jarratt methods or Jarratt-type methods [20][21][22].…”

Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

“…for Banach space valued mappings with F : Ω ⊂ B → B, where F is differentiable in the sense of Fréchet [1,2]. For a good survey of literature on local and semilocal convergence criteria of iterative methods see [3][4][5][6][7][8][9][10][11][12][13].…”

“…In quest of efficient higher order method, a number of improved, multipoint Newton's or Newton-like iterative schemes have been proposed in literature; see, for example [3,5,[8][9][10][12][13][14][15][16][17][18][19] and references cited therein.…”