A Class of Steffensen-Type Iterative Methods for Nonlinear Systems

Abstract

A class of iterative methods without restriction on the computation of Fréchet derivatives including multisteps for solving systems of nonlinear equations is presented. By considering a frozen Jacobian, we provide a class ofm-step methods with order of convergencem+1. A new method named as Steffensen-Schulz scheme is also contributed. Numerical tests and comparisons with the existing methods are included.

Multipoint Iterative Methods for Finding All the Simple Zeros in an Interval

“…We note that keeping tighter conditions will produce fractals with much more quality and reliability. For further application, refer to [4,23,24].…”

“…If ( ) ̸ = 0, we categorize the solution as a simple one, while if ( ) = ( ) = ⋅ ⋅ ⋅ = ( −1) ( ) = 0 and ( ) ( ) ̸ = 0, then the solution is multiple. The application of such schemes, from finding Moore-Penrose inverse to calculating the -adic inverse of a number to the module , has attracted the researchers for constructing multipoint schemes possessing a much more reliable computational efficiency; see, for example, [1][2][3][4].…”

An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs

“…For the purpose of comparison we write the multi-step extensions of iterative schemes (13) and (14) as follows: x ðkþ1Þ ¼ w ðkÞ À 5 2 I À 3 2 t ðkÞ À Á F 0 ðx ðkÞ Þ À1 Fðw ðkÞ Þ:…”

“…Multi-step quasi-linearization methods are constructed in [5,6], but both methods face low convergence-order. Many authors [7][8][9][10][11][12][13][14] has investigated the iterative methods for nonlinear equations with higher convergence-order. For the systems of nonlinear equation recent advancements are addressed in [15,[17][18][19][20][21][22][23][24][25][26][27].…”

A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems

“…It involved predictor-corrector methods with a high order of convergence, with the corrector step coming from a Gaussian quadrature. Moreover, other techniques have been used: Adomian decomposition [5][6][7], multipoint methods free from second derivative [8,9], multidimensional Steffensen-type schemes [10][11][12][13], and even derivative-free methods with memory [14][15][16].…”