Semilocal convergence analysis of an efficient Steffensen-type fourth order method

From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

“…It may be noted that the method (4), denoted by M1, requires only two function evaluations per iteration (𝑓 (x n ) and 𝑓 ′ (x n )) and re-uses the evaluations of the previous iteration (𝑓 (x n−1 ) and 𝑓 ′ (x n−1 )) to converge with an approximate order of convergence of 1 + √ 3 ≈ 2.73205. The second memory-based one-step method, denoted by M2, requires only one function evaluation per iteration (𝑓 (x n )) to converge with an approximate order of convergence of 1 2 (1 + √ 5) ≈ 1.61803 (golden ratio). The algorithm works as follows:…”

“…where 𝑓 is a differentiable operator defined on a non-empty, open convex subset 𝛶 of the real line R with values in itself [1]. In numerical analysis, methods for locating roots are crucial.…”

“…In numerical analysis, methods for locating roots are crucial. There are several algorithms for finding the places where a function in (1) has a value of zero, also known as its roots. At first appearance, this may seem theoretical, yet it is actually a fundamental principle with many real-world implications.…”

“…A nonlinear function in its scalar form is written as $$$$ f(x)=0, $$$$ where $$$ f $$$ is a differentiable operator defined on a non‐empty, open convex subset $$$ Y $$$ of the real line $$$ \mathrm{\mathbb{R}} $$$ with values in itself [1]. In numerical analysis, methods for locating roots are crucial.…”