A family of fourth order iterative methods for solving nonlinear operator equations (Q762888)

For the solution of an equation \(P(x)=0\) involving a nonlinear operator P between Banach spaces, the author considers the family of iterative methods \(x_{n+1}=w_ n-[3I-\Gamma_ n(P(w_ n,x_ n)+P(v_ n,w_ n))]\Gamma_ nP(w_ n)\) where \(w_ n=x_ n-\Gamma_ nP(x_ n);\quad \Gamma_ n=[P(x_ n,v_ n)]^{-1},\quad v_ n=x_ n-cP(x_ n),\) P(x,y) is a divided difference operator of P(x), and c a real parameter. For \(c=0\) this represents the method of \textit{R. F. King} [SIAM J. Num. Anal. 10, 876-879 (1973; Zbl 0235.65036)] in a Banach space setting. A semi-local convergence theorem is proved which also shows that the method has R-order four. A trivial example for a one-dimensional cubic equation is given.