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Many applications from a wide variety of disciplines in the natural sciences and also in engineering are reduced to solving of an equation or a system of equations in a correspondingly chosen abstract area. For most of these problems, the solutions are found iterative, because their analytic versions are difficult to find or impossible. This article encompasses efficient, derivatives-free, high-convergence iterative methods. Convergence of two types: Local and Semi-local areas will be investigated under the conditions of the ϕ, ψ-continuity utilizing operators on the method. The new method can also be applied to other methods, using inverses of the linear operator or the matrix.