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 Author Goldstine, H. H. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1960 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Abstract In a recent paper1 I stated that von Neumann had originated the suggestion for the use of Schur's canonical form for arbitrary matrices. I have since learned that the suggestion actually is due in the first instance to John Greenstadt, who brought it to von Neumann's attention. The history of this is rather interesting and was communicated to me in a letter from John Greenstadt, which I quote below.“The full story is, that the triangularization occurred to me early in 1953, after trying in vain to find a general iterative diagonalization procedure, even where one knew that it was possible to diagonalize (defective degeneracy being the impossible case). It seemed to me that one thing that made for the stability of the Jacobi method was the fact that all the elements in the transformation matrix were less than 1. A natural generalization embodying this requirement was to consider unitary transformations. Then, a quick check of Murnaghan's book showed that one could hope only to triangularize, but that this was always possible.“I did some hand calculations on this, and lo and behold! it converged in the few cases I tried. I then programmed it for the CPC and tried many other cases. For several months thereafter, Kogbetliantz, John Sheldon, and I tried to prove convergence, when the algorithm involved the sequential annihilation of off-diagonal elements. We (particularly Sheldon) tried many approaches, but with no hint of success. Finally, in the latter part of 1953, we decided to ask von Neumann, who was then a consultant for IBM, when he was in New York at our offices. “I had prepared a writeup describing the procedure, but von Neumann (rightly) didn't want to bother reading it, so I explained it to him in about two minutes. He spent the next 15 minutes thinking up $\textit{all}$ the approaches we had thought of in three or four months, plus a few ones—all, however, without promise.“At this point he decided that it was a nontrivial problem, and perhaps not worth it anyway, and immediately suggested minimizing the sum of squares of subdiagonal elements, which is, of course, the truly natural generalization of the Jacobi method. For the next 15 minutes he investigated the case when it would be impossible to make an improvement for a particular pivotal element and found that these cases were of measure zero.“I recoded my procedure for the 701 and tried many other matrices of various sizes. I myself never had a failure, but it has since been demonstrated that the method will indeed fail for a class of matrices. Hence, a proof is clearly impossible. However, I think a $\textit{statistical}$ proof is possible, along lines suggested by Kogbetliantz, which, however, I have not been able to find. I do not think von Neumann's variation of the method would fail. (However, it is more complicated and time consuming.)” ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1960-01-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 7 Issue Number 1 Page Count 2 Starting Page 78 Ending Page 79

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Source: ACM Digital Library