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In this paper, we show that the integro quintic splines can locally be constructed without solving any systems of equations. The new construction does not require any additional end conditions. By virtue of these advantages the proposed algorithm is easy to implement and effective. At the same time, the local integro quintic splines possess as good approximation properties as the integro quintic splines. In this paper, we have proved that our local integro quintic spline has superconvergence properties at the knots for the first and third derivatives. The orders of convergence at the knots are six (not five) for the first derivative and four (not three) for the third derivative.

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This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.

In this work, we first develop a new family of three-step seventh and eighth-order Newton-type iterative methods for solving systems of nonlinear equations. We also propose some different choices of parameter matrices that ensure the convergence order. The proposed family includes some known methods of special cases. The computational cost and efficiency index of our methods are discussed. Numerical experiments give to support the theoretical results.