## An Interview with Peter Lancaster

**July 6, 2005**

Peter Lancaster, of the University of Calgary, March 2005.

*Peter Lancaster, interviewed by Nicholas J. Higham at the University of Manchester on March 15, described a career that began in England, with an undergraduate degree in mathematics (University of Liverpool, 1952) and a few post-World War II years in industry. In "the time-honored British tradition," he then moved to the University of Singapore, "one of the few places that would offer me a teaching and research position," working on iterative methods for eigenvalue problems and receiving a PhD in 1962. The next major move, in 1962, was to Canada---the University of Calgary---where the mathematics department was then a work in progress. He has been at Calgary ever since, retiring from teaching in 1994 but continuing to do research and supervise students and postdocs.The interview is part of a project, endorsed by the SIAM Activity Group on Linear Algebra (of which both Higham and Lancaster are active members), to interview significant senior people in the field. Readers are encouraged to seek out the complete transcript. Meanwhile, SIAM News is happy to publish the following extracts, chosen in part for the good fit with Philip Davis's book review on the facing page. The transcript was posted in June at http://www.ma.man .ac.uk/~nareports *

*(Numerical Analysis Report 468, Manchester*Centre for Computational Mathematics).

*NJH: I know that after your undergraduate degree in mathematics at the University of Liverpool you worked in the aircraft industry in the northwest of England. What sort of problems were you working on, and how did this influence your mathematical career?*PL: I was in the aero-structures group at what was then called Warton Aerodrome and was the research arm of the English Electric Company, which later became British Aerospace. I was involved in setting up mathematical models of aircraft structures for analysis of the "flutter" problem, first as one of a group led by Ivan Yates, but then being able to take more initiative as time went on (I was with the English Electric for five years). A number of methods were available. The book of Frazer, Duncan, and Collar was our standard reference at that time, and it had some leads on iterative methods, for example. In this connection, the work of A.C. Aitken was quite influential. It was part of my job to work with people who were generating the structural data and then to put it into the mathematical model and do the computing. We did both analogue and digital computing to produce results which, hopefully, would guarantee the safety of the aircraft.

There was one interesting occasion when, in a sense, we didn't guarantee the safety of the aircraft. Naturally this was on a Friday. There was an incident with the aircraft being flight-tested at the time: a P1 prototype, which later became the Lightning (see http://www.thunder-and-lightnings.co.uk). They were flying through the sound barrier, so operating at about Mach number 0.95, and a flutter incident occurred involving the rudder. This scared the living daylights out of the pilot, Roly Beamont, but he managed to keep things under control and land the aircraft. Of course we were all called in to work flat-out over the weekend, to try to understand why it happened, and to recommend changes to the structure. I remember we recommended subtle changes involving mass-balances which we were confident would be effective. But in the end they didn't accept our proposals. They made an engineering change: They just beefed up the structure instead, and took the expense of extra weight.

*NJH: So you were solving eigenvalue problems?*PL: Yes, quadratic eigenvalue problems, in particular, for vibrating systems, the elastic system being the aircraft itself, and of course the tricky part was modelling the velocity-dependent aerodynamics. It was the interaction of the aerodynamics and the structures which was our concern, and we felt, rightly or wrongly, that we had a pretty firm grip on the structure, but the aerodynamics was considerably more difficult. The theory of transonic flight was in its infancy then. There was a strong group here in Manchester at that time, including James Lighthill and Fritz Ursell. They were the people who, in the 1950s, were developing the theory of shock waves.

*NJH: What size were these eigenvalue problems?*PL: For computing vibration modes and frequencies, the matrices would be approximately 20-by-20. But for flutter itself they were small. In retrospect, it is amazing what we could do with just two degrees of freedom (one bending and one torsion mode). We thought if we could get up to four or five (for control surface problems) we were being quite ambitious. But those degrees of freedom were the so-called modal coordinates, which describe the structure quite effectively and implicitly contain a vast amount of information. Near the end of my time there, I could see the beginnings of the movement toward finite element methods, where, instead of looking at modal coordinates, you'd look at displacements at discrete points on the structure as your degrees of freedom and link them with the so-called influence coefficients, and this would provide a representation of the structure. So these matrices were considerably bigger. I think this probably gave the finite element method some impetus, but of course finite elements developed as an effective tool quite a bit later than that.

*NJH: What computing facilities were available?*PL: Desk-top calculators (Monroes and one or two Brunsvigas) were the standard equipment to begin with. However, the English Electric Company had its aircraft division in Warton (near Preston) and a digital computer division in Stafford. So we would occasionally travel down to Stafford to do our calculations on budding high-speed digital computers (the DEUCE). Of course, at that time, we could use only machine language. There was nothing else--no high-level language. This probably turned me away from writing programmes for life. Nowadays I am happy to use a high-level language, MATLAB in particular, but I lost my taste for programming in those early difficult years. We also had a fascinating analogue capability which included a life-size cockpit [see photo on this page]: By turning a key we could vary one of the coefficients in a quadratic matrix function and watch the effect on the eigenvalues on a screen.

* * *

*NJH: How did your 1966 book,* Lambda Matrices and Vibrating Systems*, come about?*PL: That began life as my PhD dissertation. In those days--and it may still be possible in British universities, or universities in that tradition--one could register for a degree while teaching. My first appointment in Singapore was as an assistant lecturer, but after a couple of years it was clear that my research was going quite well and I was managing to publish some papers. But I didn't have a doctorate and I was wondering about registering for such a degree.

After talking to Richard Guy and Alexander Oppenheim, who was the Vice Chancellor at the time (another mathematician), I decided to register for a PhD in Singapore.

I completed that dissertation before leaving Singapore in '62, and the two external examiners were analyst Ian Sneddon from Glasgow, and Leslie Fox, a numerical analyst from Oxford. Ian Sneddon liked the subject matter and suggested that I write it up as a monograph in the Pergamon series that he was editing. So that's how it happened.

*NJH: Only three years later you published the *Theory of Matrices* book. How did that come about, and what influence has the book had?*PL: In Calgary in the 1960s, partly because we were a new department and we were setting up new programmes, I was able to give a senior linear algebra course and, given my background, it tended to emphasize matrix analysis. I gave that course perhaps three or four times through the 1960s, and it seemed to me that there was a market for a textbook like that. I was strongly influenced by Gantmacher, but Gantmacher's volumes were hardly a textbook, and so one of my objectives was to write a textbook in the manner of the Gantmacher volumes. It seemed to meet a need and sold quite well, and perhaps because of the association with Gantmacher the Russians liked it and it was translated into Russian and became better known in the Soviet Union than it was in the West.

* * *

*NJH: Do you have any comments on how the relationships between core linear algebra and numerical linear algebra have developed or changed over the years?*PL: I certainly find that the middle ground between those is very interesting.

My sense is that in recent years, in the last decade or two, core numerical analysis has rather moved away from core linear algebra. Perhaps there's been a little divergence of interest there. For example, the work that we (and many other people) have done on indefinite scalar products and the use of the geometry of subspaces of various kinds remains to be exploited numerically. My feeling is that the theory has gone quite a long way in the last 15 or 25 years, but the numerical absorption of this, or the advantage that has been taken of this, could have been stronger. Perhaps it simply means that there is research to be done there in the near future to fill this gap.

* * *

*NJH: Have you ever felt a result, paper or book of yours has not received the attention it fully deserved and if so which, and why?*

PL: That's a difficult one. With a good friend and colleague in Calgary, Kes Salkauskas, we wrote an introduction to transform theory about eight or nine years ago which went up like a lead balloon. I have a gut feeling that it's worth more than that, and that it should have a better place somewhere in undergraduate teaching. This book was aimed at trying to present transform theory to an audience of people with minimal prerequisites in analysis, so it seemed to me that the potential audience was really very large. Probably an explanation for its lack of success is that it was based on a course that we ourselves designed, primarily for geophysicists in the Calgary environment. Therefore it's not a text that fits neatly into a standard curriculum and, consequently, hasn't had a lot of adoptions. But I hope and think that it's probably worth more--and maybe its time will come. (See *Transform Methods in Applied Mathematics*, with Kes Salkauskas, Wiley-Interscience, 1996.)

* * *

*NJH: You're still as active as ever, eleven years after retirement. To what do you attribute your enthusiasm for research, and your ability to develop new research topics and find new collaborators?*

PL: Well, it's really quite simple.

I enjoy it and I see no reason to stop. I'm very fortunate in that I continue to get the support of the University of Calgary and the Canadian research council. Of course, that oils the wheels and provides the necessary infrastucture: a place to sit, a place to get your computer services and all the other services we need, the potential to go on supervising graduate students, and the the freedom to make and accept invitations to collaborate. . . .