Haley's Methods for Solving Equations

On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation

“…The updating formula (15) in Lemma 4.2 can be expressed by using the general formula (17) with n = 2: 3 and…”

Some Derivatives of Newton's Method

“…for the function f. Not e that the equation f( x) = 0 has exact ly one (simple) real root, which is approximat ely -0.159704853 . (a) Nf( x) = ( What is the orbit of 1 und er iteration of (c) A blow-up of [1,2] x [1 ,2] in Figure 8 Next , we consider SN f (x ; 1), which has exac t ly one fixed point at the root of f (x ) = O. Figure 9(a) shows, however , that the orbit of 1 do es not converge to the fixed point ; rather , the orbit "escapes to infinity."…”