John von Neumann's Analysis of Gaussian Elimination and the Origins of Modern Numerical Analysis

Abstract

Abstract. Just when modern computers (digital, electronic, and programmable) were being invented, John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did for the accuracy of mechanized Gaussian elimination-but their paper was about more than that. Von Neumann and Goldstine described what they surmized would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them.

Alan Turing and the origins of modern Gaussian elimination

“…After reading with detail Turing's paper (1948) and some recent works on the History of Numerical Analysis, I am convinced that Turing's paper deserves to be analysed in depth, both from the point of view of Turing's scientific biography and from the point of view of the History of Numerical Analysis. An extensive study in the spirit of the recent paper by Joseph Grcar (2011b) on von Neumann's contribution to GE is clearly necessary. However, this would lead to a very long paper or to a paper for specialists who already know the error analysis of GE and are interested in its origins and evolution.…”

“…It is a key paper that has been considered by several top numerical analysts as the first paper of modern Numerical Analysis, where "modern" has here the sense, already used before, of "analyzing methods to be used on digital, electronic, programmable computers". More information on von Neumann and Goldstine (1947) can be found in Grcar (2011b) and in Wilkinson (1971a).…”

“…A brief account of them may be found in Grcar (2011b) and a complete history in Rojas and Hashagen (2000). Here, I will say just a few words on the projects in which Alan Turing and John von Neumann were involved, because at that time they simultaneously became interested in the error analysis of GE.…”

“…Some modern texts (Higham, 2002, p. 185) and papers (Grcar, 2011b) indicate that von Neumann and Goldstine (1947) and Turing (1948) introduced "backward error analysis". In my opinion, this is not completely true.…”

“…Before Turing's paper, von Neumann and Goldstine used the condition number (in the 2-norm and with the name "figure of merit") in their error bounds, but they do not show any clear perturbation inequality involving the condition number. The first fully rigorous perturbation results on condition numbers were proved by Bauer in 1959 for matrix inverses and by Wilkinson in 1963 for linear systems (see Grcar, 2011b for more details).…”