A numerically stable form of the simplex algorithm

Standard implementations of the Simplex method have been shown to be subject to computational instabilities, which in practice often result in failure to achieve a solution to a basically well-determined problem. A numerically stable form of the Simplex method is presented with storage requirements and computational efficiency comparable with those of the standard form. The method admits non-Simplex steps and this feature enables it to be readily generalized to quadratic and nonlinear programming. Although the principal concern in this paper is not with constraints having a large number of zero elements, all necessary modification formulae are given for the extension to these cases.