Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications. Volume IV

Extended three step sixth order Jarratt-like methods under generalized conditions for nonlinear equations

“…Estimate (2.15) and the lemma by Banach on linear invertible operators [3,7] imply F (v) −1 exists with…”

“…Additionally, a lot of researchers have been designing higher order modifications of conventional procedures like Newton's, Chebyshev's, Jarratt's, etc. [1][2][3][5][6][7]9,[11][12][13][14][15][16][17]21,23,[25][26][27]31]. Kou and Li [18] presented a sixth order variant of Jarratt's algorithm to address nonlinear equations in R. They added Newton's iterate as the third step in Jarratt's iterate and used linear interpolation formula to eliminate the additional evaluation of the first derivative.…”

Extending the Convergence of Two Similar Sixth Order Schemes for Solving Equations under Generalized Conditions

“…Some standard definitions and results are restated in order to make the paper as self-contained as possible. More on this topic can be found in [4,[9][10][11]17]. The set L(X, Y) denotes the space of bounded linear operators from X into Y.…”

“…This way, we extend the applicability of these schemes. We have used, the Computational Order of Convergence (COC) and Approximate Computational Order of Convergence (ACOC) to determine the convergence order which does not require the usage of higher-order derivatives or divided differences (see Remark 2.2) [8][9][10][11][12]. This is done in Section 2 and Section 3.…”

An extended radius of convergence comparison between two sixth order methods under general continuity for solving equations

“…The convergence order of iterative methods, in general, was obtained using Taylor expansions and conditions on high order derivatives not appearing on the method. These conditions limit the applicability of the methods [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. For example: Let X = Y = R, Ω = [− 1 2 , 3 2 ].…”