IJST (2015) 39A1: 91-99 Iranian Journal of Science & Technology http://ijsts.shirazu.ac.ir Modified chain least squares method and some numerical results F. Goharee 1 , E. Babolian 1 * and A. Abdollahi 2 1 Department of Mathematics, College of Basic Sciences, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran 2 Department of Mathematics, Maragheh Branch, Islamic Azad University, Maragheh, Iran E-mail: fgoharee@srbiau.ac.ir, babolian@khu.ac.ir & a.abdollahi@iau-maragheh.ac.ir Abstract Recently, in order to increase the efficiency of least squares method in numerical solution of ill-posed problems, the chain least squares method is presented in a recurrent process by Babolian et al. Despite the fact that the given method has many advantages in terms of accuracy and stability, it does not have any stopping criterion and has high computational cost. In this article, the attempt is to decrease the computational cost of chain least squares method by introducing the modified least squares method based on stopping criterion. Numerical results show that the modified method has high accuracy and stability and because of its low computational cost, it can be considered as an efficient numerical method. Keywords: Chain least squares; Lagrange multipliers method; Ill- posed problem; Integral equations; Singular second order initial value differential equations 1. Introduction Least squares method is one of the efficient methods in the numerical solution of many engineering and physics problems (Aksan et al. 2006; Alexander and George, 1990; Ching and Suh- Yuh, 2002; Jagalur-Mohan et al. 2013; Jannike and Hugo, 2012; Jinming, 2012; King and Krueger, 2003; Laeli and Maalek, 2012). In order to increase the efficiency of this method in numerical solution of some ill-posed problems, the chain least squares method is presented in recurrent form by Babolian et al. (2014). In this approach, by reducing an - term least squares problem to the ( − 1)-term ones and continuation of this trend up to the last stage (1-term problem), the efficiency of the least squares method in numerical solution of ill-posed problems has been significantly increased (Babolian et al. 2014). Thus, for solving an -term problem by chain least squares method, we have to continue the recurrent process up to the last stage. In this article, the attempt is to prevent the continuation of the recurrent process up to the last stage by providing a logical and experimental stopping criterion. Besides decreasing the computational cost of chain least squares method, the definition of the stopping criterion maintains the stability and accuracy of this method. This stopping *Corresponding author Received: 12 April 2014 / Accepted: 8 October 2014 criterion is based on the convergence of intermediate matrix elements of least squares method to zero. This is inspired by the convergence of the Galerkin method in numerical solution of Fredholm integral equation of the second kind (Delves and Mohamed, 1985). It should be mentioned that by intermediate matrices, we mean the coefficient matrix of system of equation corresponding to the chain least squares method in turning -term problem ( = , … ,2) to ( − 1)- term one. In the second step, considering the main role of the artificial trajectories in the definition of chain least squares method (Babolian et al. 2014), in order to decrease the computational cost of this method, a new process is introduced in defining of artificial trajectories. According to the kinds of problems solved by the chain least squares method, at least one of the artificial trajectories is decreased. In the new trend, instead of reducing -term problem to ( − 1)-term one, the attempt is to change -term problem to ( − )-term one (≥2) in such a way that the computational cost of this method is decreased. By presenting numerical examples in each section, the stability and accuracy of the new method will be shown. Firstly, a review of chain least squares method has been given, then the modified chain least squares method is presented. Finally, the efficiency of the modified methods is investigated by solving several ill-posed functional equations.