An iterative algorithm for solving linear least squares problems has been developed and tested on an IBM 1620 computer. This article traces the development of the algorithm from that described by Björck and Golub [1] to the present, and shows how, with a slight change in algebra, the Householder triangularization may be replaced equally successfully by the simpler method of Cholesky factorization. The algorithm appears accurate and efficient even for highly ill-conditioned problems. Use of the residual vector in the iteration process is the main source of the algorithm's success.