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In this article, a two-parameter class of hybrid block methods for integrating first-order initial value ordinary differential systems is proposed. The methods exhibit hybrid nature which helps in bypassing the first Dahlquist barrier existing for linear multistep methods. The approach used in the development of a class of methods is purely interpolation and collocation technique. The class of methods is based on four intra-step points from which two intra-step points have been optimized by using an optimization strategy. In this optimization strategy, the values of two intra-step points are obtained by minimizing the local truncation errors of the formulas at the points x n+1∕2 and x n+1 .The order of accuracy of the proposed methods is six. A method as a special case of this class of methods is considered and developed into a block form which produces approximate numerical solutions at several points simultaneously. Further, the method is formulated into an adaptive step-size algorithm using an embedded type procedure. This method which is a special case of this class of methods has been tested on six well-known first-order differential systems. K E Y W O R D Sblock methods, embedded-type procedure, hybrid methods, ordinary differential equations INTRODUCTIONNumerous numerical integrators are available for solving first-order initial value ordinary differential systems numerically. Runge-Kutta and linear multistep methods are the usual classes of methods that are being used for solving these systems numerically. These days many built-in computer codes based on these classes of methods are available in software like Matlab, Mathematica, and so forth. To develop more accurate and efficient codes for solving the differential systems is an ongoing process of research. This article is concerned with the development of a class of optimized codes for solving the following system:whereTo proceed further, we assume that the system (1) has a unique continuously differentiable solution that we denote by Y(x). As a first step, the interval of interest [x 0 , x N ] is discretized as follows:x n = x 0 + nh, n = 0, 1, 2, 3, … , N; h = x n+1 − x n and an approximate numerical solution of the system (1) at x n is denoted as Y n ≈ Y(x n ).

In this paper, we derive new one-parameter family of Newton's method, Schröder's method, super-Halley method and Halley's method respectively for finding simple zeros of nonlinear functions, permitting f (x n ) = 0 at some points in the vicinity of required root. Using the newly derived family of super-Halley method, we further obtain new interesting families of famous quartically convergent Traub-Ostrowski's and Jarratt's methods respectively. Further, the approach has been extended to solve a system of nonlinear equations. It is found by way of illustration that the proposed methods are very useful in high-precision computing environment and non-convergent cases.