Hindawi Publishing Corporation Advances in Numerical Analysis Volume 2012, Article ID 346420, 18 pages doi:10.1155/2012/346420 Research Article An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence Rajni Sharma 1 and Janak Raj Sharma 2 1 Department of Applied Sciences, DAV Institute of Engineering and Technology, Kabirnagar 144008, India 2 Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148106, India Correspondence should be addressed to Rajni Sharma, rajni gandher@yahoo.co.in Received 21 May 2012; Accepted 4 September 2012 Academic Editor: Nils Henrik Risebro Copyright q 2012 R. Sharma and J. R. Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub 1974 conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2 n-1 . Thus, the family agrees with Kung-Traub conjecture for the case n  4. Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods. 1. Introduction Solving nonlinear equations is one of the most important problems in science and engineering 1, 2. The boundary value problems arising in kinetic theory of gases, vibration analysis, design of electric circuits, and many applied fields are reduced to solving such equations. In the present era of advance computers, this problem has gained much importance than ever before. In this paper, we consider iterative methods to find a simple root r of the nonlinear equation f x 0, where f : R → R be the continuously differentiable real function. Newton’s method 1 is probably the most widely used algorithm for solving such equations, which starts with an initial approximation x 0 closer to the root r and generates a sequence of successive iterates {x i } ∞ 0 converging quadratically to the root. It is given by the following: x i1  x i - f x i  f ′ x i  , i  0, 1, 2, 3, .... 1.1