On the discovery of the "good Broyden" method

This short note traces the events that led to the unsymmetric rank one formula known as the “good Broyden” update [5,6], which is widely used within derivative-free mathematical software for solving a system of nonlinear equations. I left University in 1956 with an indifferent degree in Physics and took up a post with the English Electric Company in Leicester. The company was involved in the design and construction of nuclear reactors and I was employed as a computer programmer. One of the problems with which I was concerned was the solution of systems of differential equations. These equations were used in the performance calculations of the reactors and modelled their behaviour as time passed, and some of the favoured methods of solution were the “predictor-corrector” methods. They worked roughly as follows. Given a temperature distribution at time t they would calculate the equivalent distribution at time t + δt where δt was a small time increment. This calculation would then be repeated many times in the hope that the computed temperature would subside before the Magnox cladding of the fuel rods melted. The new temperature distribution would be obtained by getting an approximate solution of the “corrector” equations, which were essentially of the form