Certain simple non-linear algorithms which have useful application in numerical analysis have recently been introduced. When using the formulae of these algorithms it may sometimes occur that a quantity is numerically ill-determined. As a result of this and because of the way in which the algorithmic formulae are used, all quantities lying in a certain sector become ill-determined. By considering theε-,ϱ-, andq —d-algorithms in detail it is shown how this misfortune may be overcome. ALGOL programmes are developed to show how the derived formulae may be mechanized.