chain least squares method and ill-posed problems

the main purpose of this article is to increase the efficiency of the least squares method in numerical solution of ill-posed functional and physical equations. determining the least squares of a given function in an arbitrary set is often an ill-posed problem. in this article, by defining artificial constraint and using lagrange multipliers method, the attempt is to turn -dimensional least squares problems into  ones, in a way that the condition number of the corresponding system with -dimensional problem will be low. at first, the new method is introduced for  and -term basis, then the presented method is generalized for -term basis. finally, the numerical solution of some ill-posed problems like fredholm integral equations of the first kind and singularly perturbed linear fredholm integral equations of the second kind are approximated by chain least squares method. numerical comparisons indicate that the chain least squares method yields accurate and stable approximations in many cases.