The stability of Newton’s methods for function minimization, or variants of it such as quasi-Newton or Newton–Raphson methods, can be improved by modifying the acceleration matrix by adding a scalar parameter to the diagonal elements. It can be further improved by optimizing the function in the direction of the search vector. This paper derives an algorithm for determining when to add these parameters, and how to determine their magnitude, by considering the function as a function of a single scalar parameter. Computational results are included to show the stability of the derived algorithm.