In His Own Words

January 6, 2008

In the summer of 2005, Gene Golub visited the University of Manchester to attend a workshop, and Nick Higham took the opportunity to interview him. The following extracts are from an edited transcript of the interview, which will be posted on the Web site of the School of Mathematics at Manchester.

NJH: Did you have any particular mentors . . . up to, say, your PhD?
GHG:
[At the University of Illinois] I took a statistics course from a man called C.R. Rao. C.R. Rao was a great statistician and as it turns out he was visiting Illinois just that year. The course was probably beyond me, but he did lots of matrix manipulations: for instance, he didn't call it this, but block Gaussian elimination and so forth. So that really expanded my vision. Also at Illinois there was a large group of people who did psychometrics and there was really a well established programme. And of course they had the Illiac, so it was one of the first psychometrics groups to have a computer available for it. . . . The thing we were able to do was compute the eigenvalues and eigenvectors of a 23 x 23 matrix in 15 minutes. Well, that for then was a big breakthrough because there were some classical cases in psychometrics that had never been done so extensively. I learnt a lot about factor analysis and in a way it's just one step away from the singular value decomposition. Rao devised a method called maximum likelihood factor analysis, or canonical factor analysis, which was a new form of factor analysis. The big question in factor analysis at that time was, how many factors should there be? And it's equivalent to saying, how many singular values are there? . . . So I was just very fortunate: this man J.P. Nash brought me to the computing laboratory, through that laboratory I met Charles Wrigley, I met C.R. Rao, and I met many other people. My first lecture was at the International Congress of Psychologists.

NJH: "Some Modified Matrix Eigenvalue Problems" in SIAM Review in 1973 . . . was one of your earlier papers on this topic and has led to a lot of other work, hasn't it?
GHG:
Well, I'm pleased that you say that because the origins are as follows. I was on sabbatical one year and I had been collecting results. For instance, I worked on eigenvalue problems with homogeneous constraints; that was at the suggestion of Henrici, as a matter of fact, and I devised a very simple algorithm for doing a computation, and then I discovered that this problem, minimizing quadratic forms subject to linear constraints, comes up in very many different places. So I had been working on a lot of small results, and then the time came I had to show I had done something on my sabbatical. So I wrote down this review paper. I love that paper, in part because it has a lot of nice useful tools for solving matrix problems.

NJH: I'd like to ask you about the background to . . . your 1969 paper "Calculation of Gauss Quadrature Rules" with Welsch in Mathematics of Computation. How did you get interested in that particular topic?
GHG:
Well, I'm not sure how I became particularly interested in quadrature. Paul Concus was a dear friend of mine and asked me some things about quadrature, and then I realised that all that's really necessary to do is compute eigenvalues of matrices. I was seated in Peter Lax's office. I was on leave at the Courant Institute and I was sort of bored by what was taking place, so I happened to see a book by Wilf. There he had some of the relationships about computing the quadrature weights, and then I worked out the fact that all you need is the square of the first element of the eigenvector. Actually, other people had figured this out too and known this before. What I was able to do is to recognise that when you mix that with the QR method you can organise the QR method just to compute the first component of an eigenvector. So I was really pleased when that all fell into place. . . .

NJH: I must ask you about your book Matrix Computations with Charlie Van Loan, originally published in 1983, now in its third edition (1996). When did you actually start writing the book?
GHG:
I gave some lectures at Johns Hopkins University. What happened is, they formed a Math Sciences Department at Johns Hopkins and the head was Roger Horn. I had known Roger Horn because he was in the first course of advanced numerical analysis I taught. I taught three quarters of the course; the first two quarters were out of Varga's book. Roger Horn is a wonderful fellow, and I hadn't realised what impact that had on him because, for instance, he told me he learned about the Perron–Frobenius theory from me and also the singular value decomposition, and those have both been of some importance to him. . . . The class consisted of a whole bunch of people; I don't know specifically who now, but Charlie Van Loan was there and also Richard Bartels. Originally the idea was for the three of us to write a book, but then Richard's interests were developing otherwise; it was going to include a large component of math programming. So then it was just Charlie and I. He did most of the writing, as you probably know, but it was really a great stimulant for both of us. For instance, although I had done some work on total least squares, we developed a paper on total least squares with a lot of analysis and showing how the computations will go, and so forth. So that was really a wonderful effort, and he would send me some section and I would say, "Oh, well, we can do better like this". There is the Sylvester equation that we did work on; again I looked at it and realised we could do one eigenvalue computation: we did one upper Hessenberg form and one Schur decomposition. So that was really a great period for both of us because we were incorporating new material in the book and we were also writing the book.


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