Kantorovich-ostrowski convergence theorems and optimal error bounds for jarratt's iterative method

Abstract

In the first part of this paper, we consider the Kantorovich-Ostrowski convergence theorem of a fourth order method which requires two evaluations of f′ and one of f for solving nonlinear equations. The convergence of the iteration is established under the following assumptions:

Let f:D⊂K→K be C ⁴ on D with |f″(x)|≦M,D be a convex open domain |f″(x)−f″(y)|≦L|x−y| and |f ⁽⁴⁾(x)|≦N for all x,y∊D. Assume that x ₀∊D y ₀ = u(x ₀): = x ₀−f(x ₀)/f′(x ₀), |f′(x ₀)^−l|≦ β, |y _o−x _o|≦μ,

Here x _n and y _n are generated by the following schemes:

In the second part, we show that in fact Jarratt's method is equivalent to Newton's method under optimal operator. Also we give an exact error constant and show that it cannot be improved.

Keywords:

• Nonlinear equation
• Kantorovich theorem
• multipoint iterative method