An Interview with Peter Lancaster

Higham, Nicholas J.

2005

MIMS EPrint: 2005.19

Manchester Institute for Mathematical SciencesSchool of Mathematics

The University of Manchester

Reports available from: http://eprints.maths.manchester.ac.uk/And by contacting: The MIMS Secretary

School of Mathematics

The University of Manchester

Manchester, M13 9PL, UK

ISSN 1749-9097

An Interview with Peter Lancaster1,

by Nicholas J. Higham.

Peter Lancaster, March 2005

NJH: How did you get interested inmathematics?

PL: Wow, that’s going back a long way.At school, mathematics was one of a fewsubjects that I seemed to be fairly goodat, and therefore enjoyed. After school,I made a false start. When I first wentto university I started in architecture, soI wasn’t completely wedded to mathemat-ics at that time. When the time came tomake a change I knew that I could go backto mathematics and I did. So my age was18 or 19 at that time. I hadn’t taken a verystrong interest in it before that age. ThenI got caught up in the honours programmeat Liverpool and the rest is history.

NJH: And I know that after your under-graduate degree in mathematics at the Uni-versity of Liverpool you worked in the air-craft industry in the north west of England.What sort of problems were you working on

and how did this influence your mathemat-ical career?

PL: I was in the aero-structures groupat what was then called Warton Aero-drome and was the research arm of the En-glish Electric Company, which later becameBritish Aerospace. I was involved in settingup mathematical models of aircraft struc-tures for analysis of the “flutter” problem.First as one of a group led by Ivan Yates,but then being able to take more initiativeas time went on (I was with the EnglishElectric for five years). A number of meth-ods were available. The book of Frazer,Duncan and Collar was our standard ref-erence at that time, and it had some leadson iterative methods, for example. In thisconnection, the work of A. C. Aitken wasquite influential. It was part of my job towork with people who were generating thestructural data and then to put it into themathematical model and do the comput-ing. We did both analogue and digital com-puting to produce results which, hopefully,would guarantee the safety of the aircraft.

There was one interesting occasionwhen, in a sense, we didn’t guaranteethe safety of the aircraft. Naturallythis was on a Friday. There was anincident with the aircraft being flight-tested at the time: a P1 prototype,which later became the Lightning (seewww.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 involvingthe rudder. This scared the living daylightsout of the pilot, Roly Beamont, but he man-aged to keep things under control and landthe aircraft again. Of course we were allcalled in to work flat-out over the week-end, to try to understand why it happened,and to recommend changes to the struc-ture. I remember we recommended subtle

1Interviewed at the University of Manchester, March 15, 2005. Numerical Analysis Re-

port 468, Manchester Centre for Computational Mathematics, Manchester, England, June 2005.

http://www.ma.man.ac.uk/~nareports

1

changes involving mass-balances which wewere confident would be effective. But inthe end they didn’t accept our proposals.They made an engineering change: theyjust beefed up the structure instead, andtook the expense of extra weight.

The English Electric Company aerostructuresgroup in the analogue computer room inAugust 1955. Peter Lancaster is in thedriver’s seat (top) and in the middle

(bottom). This machine was used a lot forthe flutter analysis—without the cockpit

simulator. Standing (top) is Ivan Yates, whowent on to be a director of British

Aerospace.

NJH: So you were solving eigenvalueproblems?

PL: Yes, quadratic eigenvalue problems,in particular, for vibrating systems, the

elastic system being the aircraft itself, andof course the tricky part was modelling thevelocity-dependent aerodynamics. It wasthe interaction of the aerodynamics and thestructures which was our concern and wefelt, rightly or wrongly, that we had a prettyfirm grip on the structure, but the aero-dynamics was considerably more difficult.The theory of transonic flight was in its in-fancy then. There was a strong group herein Manchester at that time including JamesLighthill and Fritz Ursell. They were thepeople who, in the 1950s, were developingthe theory of shock waves.

NJH: What size were these eigenvalueproblems?

PL: For computing vibration modes andfrequencies the matrices would be approxi-mately 20-by-20. But for flutter itself theywere small. In retrospect, it is amazingwhat we could do with just two degreesof freedom (one bending and one torsionmode). We thought if we could get up tofour or five (for control surface problems)we were being quite ambitious. But thosedegrees of freedom were the so-called modalcoordinates, which describe the structurequite effectively and implicitly contain avast amount of information. Near the end ofmy time there I could see the beginnings ofthe movement toward finite element meth-ods where, instead of looking at modal coor-dinates, you’d look at displacements at dis-crete points on the structure as your degreesof freedom and link them with the so-calledinfluence coefficients, and this would pro-vide a representation of the structure. Sothese matrices were considerably bigger. Ithink this probably gave the finite elementmethod some impetus, but of course finiteelements developed as an effective tool quitea bit later than that.

NJH: What computing facilities wereavailable?

PL: Desk-top calculators (Monroes andone or two Brunsvigas) were the standardequipment to begin with. However, the En-

2

glish Electric Company had its aircraft di-vision in Warton (near Preston) and a dig-ital computer division in Stafford. So wewould occasionally travel down to Staffordto do our calculations on budding high-speed digital computers (the DEUCE). Ofcourse, at that time, we could use only ma-chine language. There was nothing else—nohigh level language. This probably turnedme away from writing programmes for life.Nowadays I am happy to use a high levellanguage, MATLAB in particular, but I lostmy taste for programming in those early dif-ficult years. We also had a fascinating ana-logue capability which included a life-sizecockpit: by turning a key we could varyone of the coefficients in a quadratic ma-trix function and watch the effect on theeigenvalues on a screen.

NJH: Did you have any contact withWilkinson at all at that time?

PL: I certainly knew of him and his workat the National Physical Laboratory. I firstmet him in Gatlinburg in 1961 and, later,he visited the University of Calgary on oneoccasion. This was about the time that thefirst edition of “The Algebraic EigenvalueProblem” (1965) was nearing completion; ifit wasn’t already complete. I knew it wasin production and I was very interested. Ofcourse, I got a copy at the first opportunity

NJH: I recall that Olga Taussky used totalk about working on flutter problems inthe 1940s.

PL: Yes, in the 1940s she was with theBritish Ministry of Aircraft Production andwrote Aeronautics Research Council (ARC)reports. (In the late 30s she and Jack Toddwere teaching in London University andthey married in 1939.) While I was at theEnglish Electric I saw ARC reports writ-ten by Olga. This was where I first learnedabout Gershgorin, and some of the work ofHelmut Wielandt. However, I didn’t meetOlga and Jack until that 1961 Gatlinburgmeeting. So I knew of them and their work,as with Jim Wilkinson, but didn’t meet

them until we all met in the United States.Later on, Jack and Olga made it possiblefor me to spend a year as Visiting AssociateProfessor at Caltech; this was ’65/’66—anda great experience for me.

NJH: So from industry you went to theUniversity of Singapore, where you spentfive years. Why did you choose to go toSingapore?

PL: Because it was one of the few placesthat would offer me a teaching and researchposition. I was at the English Electric be-cause this was a time of compulsory na-tional service in the UK—it was not solong after the second world war. And so Iwas working in the aircraft industry for fiveyears instead of doing two years of service inthe forces. So as the end of that five year pe-riod approached I began to look around forsomething else to do. Since my student daysI’d always aspired to a university teachingjob. I’d actually completed a Masters de-gree while I was at the English Electric, un-der the supervision of Louis Rosenhead atthe University of Liverpool, in which I sur-veyed the techniques that we used for solv-ing the flutter problem. I knew it wouldbe difficult to get into a university in theUnited Kingdom so, in the time-honouredBritish tradition, I looked further afield, toAustralia, Africa, Singapore, or wherever.And Singapore was one of the institutionsthat made me an offer, so that seemed agood place to go.

NJH: And that was where you first metRichard Guy?

PL: Yes. Richard was the Acting Headof Department in 1957 when I was ap-pointed. It was a small department but veryactive and stimulating. When I arrived Ithink there were only eight teaching staff,but we had regular seminars on a great va-riety of topics. For example, I dabbled insome problems in discrete mathematics fora while, stimulated by my colleague EricMilner in particular, and by Richard Guy.So a stimulating group was already there,

3

all willing to talk to one another about theirresearch. Also, the university was the for-mer Raffles College, so had a substantialhistory and a good library research collec-tion. Those two factors were very impor-tant while I was beginning a research ca-reer.

While talking about this period, Ishould also say that Alexander Ostrowskihad a strong influence on my career. I wasworking on iterative methods for eigenvalueproblems and came across the work of Os-trowski. And I found a way to general-ize some of his analysis to eigenvalue prob-lems for matrix polynomials. I wrote to himabout it. He was very kind and wrote backat some length and I like to think that I de-veloped a little analytical style from what Ilearnt from Ostrowski. He was also one ofthe prime movers in the Gatlinburg meet-ings, so in 1961, when I was actually onleave in the UK, he arranged for an invi-tation for me to the ’61 Gatlinburg meet-ing, where I met not only Ostrowski, butmany of the other famous names in the field:Olga Taussky and Jack Todd, Jim Wilkin-son, Leslie Fox, Collatz, Bauer, Henrici, andso on.

NJH: So that was when the Gatlinburgmeeting was in Gatlinburg. Would it be thefirst?

PL: Yes, I think the meeting of 1961 wasthe first. And, of course, Householder wasthe principal organiser, so I got to know himtoo.

NJH: In 1962 you moved to Calgary.Was this the time that the mathematics de-partment was actually being formed, or wasit already in existence?

PL: Yes, it was just being built up.At that time there wasn’t a full degreeprogramme in mathematics (although mymemory might fail me here). There cer-tainly wasn’t a full degree programme in en-gineering. We taught just the first two yearsof engineering mathematics and then thosestudents would go to Edmonton to complete

their degree. The science programme mighthave been a little more advanced, but itwas certainly the plan to develop a full pro-gramme over a period of just a few years.The Head of Department at that time wasJohn Peck, who is alive and well in BritishColumbia. John began life as a functionalanalyst, but became a computer scientist.He was one of the prime movers in the Algollanguage, and then became the first head ofcomputer science at UBC. I think he hadthe right ideas and ambitions for the Cal-gary department. Again, it was a good en-vironment for me. (The University of Cal-gary became an autonomous institution in1965.)

NJH: How did your 1966 book, Lambda

Matrices and Vibrating Systems, comeabout?

PL: That began life as my Ph.D. disser-tation. In those days—and it may still bepossible in British universities, or universi-ties in that tradition—one could register fora degree while teaching. My first appoint-ment in Singapore was as an assistant lec-turer, but after a couple of years it was clearthat my research was going quite well and Iwas managing to publish some papers. ButI didn’t have a doctorate and I was wonder-ing about registering for such a degree. Af-ter talking to Richard Guy and AlexanderOppenheim, who was the Vice Chancellorat the time (another mathematician), I de-cided to register for a Ph.D. in Singapore.I completed that dissertation before leav-ing Singapore in ’62, and the two externalexaminers were analyst Ian Sneddon fromGlasgow, and Leslie Fox, a numerical an-alyst from Oxford. Ian Sneddon liked thesubject matter and suggested that I writeit up as a monograph in the Pergamon se-ries that he was editing. So that’s how ithappened.

NJH: Only three years later you pub-lished the Theory of Matrices book. Howdid that come about, and what influencehas the book had?

4

PL: In Calgary in the 1960s, partly be-cause we were a new department and wewere setting up new programmes, I was ableto give a senior linear algebra course and,given my background, it tended to empha-size matrix analysis. I gave that course per-haps three or four times through the 1960s,and it seemed to me that there was a marketfor a textbook like that. I was strongly in-fluenced by Gantmacher, but Gantmacher’svolumes were hardly a textbook, and so oneof my objectives was to write a textbook inthe manner of the Gantmacher volumes. Itseemed to meet a need and sold quite well,and perhaps because of the association withGantmacher the Russians liked it and it wastranslated into Russian and became betterknown in the Soviet Union than it was inthe West.

NJH: Was this a legal translation?PL: No it was illegal. It was done shortly

before the Soviet Union signed internationalcopyright agreements. So although theypublished sixty thousand copies in two dif-ferent printings it did nothing for me finan-cially.

Front covers of 1966 (Pergamon) and 2002(Dover) editions of Lambda-Matrices and

Vibrating Systems.

NJH: When did your collaboration withIsrael Gohberg begin?

PL: It began in February of 1975 andit came about because I knew of the workof the school of M. G. Krein and Gohberg

through the 1960s. In fact, Krein had be-come aware of my Lambda Matrices bookand, through an intermediary, asked for acopy of it, which I sent to him and, in re-turn, he sent me a copy of one of his mono-graphs. So I knew of Gohberg and thenI heard through the grapevine that Chan-dler Davis in Toronto had arranged for Go-hberg to visit Canada shortly after emigrat-ing from the Soviet Union. So he emigratedfrom Kishinev, Moldavia, to Israel in thesummer of ’74 and then in early ’75 he wasalready visiting Canada. I invited him tocome and spend some time in Calgary dur-ing that visit. That’s where our very richfriendship and collaboration began.

NJH: You went on to write three mono-graphs in the 1980s with Gohberg and Rod-man. How did that develop from that initialmeeting?

PL: Our actual collaboration began a lit-tle later. I had a sabbatical in ’75/’76 in theUniversity of Dundee and Gohberg cameover to Dundee. At that time I was pub-lishing some papers on what we now callJordan pairs and triples—ways of encapsu-lating the spectral data for matrix polyno-mials. He liked that and could see inter-esting ways of taking advantage of it, usingand developing it, and that’s where the col-laboration began. At the same time LeibaRodman was his Ph.D. student in Tel-Aviv,working in a similar direction, and so thecollaboration between the three of us grewout of that. Then I had Leiba visit Cal-gary as a postdoctoral fellow from ’78 to’80 when a lot of the ground work for thetwo later volumes was prepared.

NJH: Did you develop the books by firstof all publishing the material in papers, oris there material that went directly into thebooks?

PL: Quite a lot went into papers first,although there was always a substantialamount of material that went directly intothe books. There was a series of papers inLinear Algebra and its Applications, for ex-

5

ample. At the time, I think it’s fair to saythat we, but certainly I, felt the factoriza-tion theory was the most significant math-ematical aspect of what we were doing. Wehad a line of attack on that which seemedto be very fruitful and the core of that workwas published in a paper in the Annals ofMathematics in 1980. That led to so manyother things. You might say my research ca-reer began with quadratic polynomials andthen the step from there to general poly-nomials of arbitrary degree is very natural,then to singular polynomials, and then torational and analytic matrix functions andso on. So we fairly quickly moved into abroader spectrum of mathematics, includ-ing not only the canonical Jordan structuresbut also questions of perturbation theory,which play a central role in many applica-tions. In that way we made connectionswith classical work of Rellich and Kato, forexample.

NJH: You have written quite a lot of pa-pers with coauthors. Do you have a partic-ular way of working?

PL: By accident I think. I don’t have adesign that I repeat. I find that a ratherdifficult question to answer. In more re-cent years the collaborations have comeabout because younger people have visitedCalgary, perhaps at my invitation, eitheras postdoctoral fellows or Ph.D. students.And so quite a lot of collaboration devel-oped in this way with a younger genera-tion. And, naturally, there is a certain pat-tern there which accounts for some of thecollaborations. Others that don’t quite fitthis mould would be more senior peoplewho have visited Calgary for a longer pe-riod. These include Pal Rozsa, Ludwig El-sner and, last but not least, Alek Markus,who is a mine of information on operatorand matrix polynomials; I have really en-joyed this collaboration and learned a lotfrom it. But for me the secret is alwaysface-to-face meetings and spending time to-gether. I think perhaps I am a slow worker;

it takes time for ideas to develop and evolve.So longish periods for meetings and interac-tion are needed to develop collaborations.

NJH: You spent sabbaticals at variousinstitutions. Do any of those stand out andwhy?

PL: The one we’ve already mentioned isthe first to come to mind; that was the Uni-versity of Dundee in ’75/’76. I think partlybecause that was the period when the col-laboration with Israel Gohberg evolved, butalso at that time I was very interested in ap-proximation theory and so I was fascinatedto be there in a group of people, almost awhole department as I remember, who wereinterested in either differential equations ornumerical analysis. This was a very stim-ulating environment to be in. I talked atsome length with Ron Mitchell about fi-nite element methods, and the notion ofupwinding schemes for appropriate differ-ential equations was just being formulatedat that time. I took quite an interest in fi-nite element methods but also in the morefundamental aspects of approximation the-ory, the questions of approximation in twoor three dimensions, and construction of in-terpolation schemes on square or triangularelements.

NJH: Do you have any comments on howthe relationships between core linear alge-bra and numerical linear algebra have de-veloped or changed over the years?

PL: I certainly find that the middleground between those is very interesting.My sense is that in recent years, in the lastdecade or two, core numerical analysis hasrather moved away from core linear alge-bra. Perhaps there’s been a little divergenceof interest there. For example, the workthat we (and many other people) have doneon indefinite scalar products and the use ofthe geometry of subspaces of various kindsremains to be exploited numerically. Myfeeling is that the theory has gone quite along way in the last 15 or 25 years, but thenumerical absorption of this, or the advan-

6

tage that has been taken of this, could havebeen stronger. Perhaps it simply meansthat there is research to be done there inthe near future to fill this gap.

NJH: Have you ever felt a result, paperor book of yours has not received the atten-tion it fully deserved and if so which, andwhy?

PL: That’s a difficult one. With a goodfriend and colleague in Calgary, Kes Salka-uskas, we wrote an introduction to trans-form theory about eight or nine years agowhich went up like a lead balloon. I havea gut feeling that it’s worth more thanthat, and that it should have a betterplace somewhere in undergraduate teach-ing. This book was aimed at trying topresent transform theory to an audience ofpeople with minimal prerequisites in anal-ysis, so it seemed to me that the potentialaudience was really very large. Probablyan explanation for its lack of success is thatit was based on a course that we ourselvesdesigned, primarily for geophysicists in theCalgary environment. Therefore it’s not atext that fits neatly into a standard cur-riculum and, consequently, hasn’t had a lotof adoptions. But I hope and think thatit’s probably worth more—and maybe itstime will come. (See Transform Methods in

Applied Mathematics, with Kes Salkauskas,Wiley-Interscience, 1996.)

NJH: Another book that will be lessknown to your linear algebra colleagueswould be your book on splines. Was thatgraduate level or undergraduate level, andwhat was the aim of that book?

PL: Well again, that developed from ex-tension courses for geophysicists working inindustry in Calgary. Of course, it is notjust splines but it was one of the earlier in-troductions to splines and more general sur-face fitting. That did fairly well. I’m notsure that it’s still in print, but it had quitea wide circulation in its time. Soon after wewrote the book the field moved very rapidlywith the introduction of wavelet analysis.

(See Curve and Surface Fitting, with KesSalkauskas, Academic Press, 1986).

NJH: Do you have a favourite paper orfavourite book?

PL: I’m most proud of the paper in theAnnals of Mathematics (1980) that we re-ferred to before. That and the paper withLeiba Rodman on Riccati equations (also1980). I feel that our theorem there on thecharacterization of solutions via invariantsubspaces was elegant and timely. I am alsoproud of the text The Theory of Matrices

(2nd edition with Miron Tismenetsky), andof its longevity: thirty-six years in print.

NJH: You’re still as active as ever, elevenyears after retirement. To what do you at-tribute your enthusiasm for research, andyour ability to develop new research topicsand find new collaborators?

PL: Well it’s really quite simple. I en-joy it and I see no reason to stop. I’m veryfortunate in that I continue to get the sup-port of the University of Calgary and theCanadian research council. Of course, thatoils the wheels and provides the necessaryinfrastructure: a place to sit, a place to getyour computer services and all the other ser-vices we need, the potential to go on super-vising graduate students, and the freedomto make and accept invitations to collabo-rate. I’ve had maybe three Ph.D. studentssince I retired and three or four postdoc-toral fellows coming and going. And I sim-ply enjoy it. Not only the sense of satis-faction and accomplishment one gets fromsucceeding—achieving some small successesin research—but also the continued expo-sure to the younger generation is very stim-ulating and helps to keep us older folks aliveand active.

NJH: You’ve been involved in the PacificInstitute of Mathematical Sciences since itsconception, I believe. How important hasPIMS been to you and to Canadian Math-ematics research?

PL: For my research personally, it cer-tainly had a beneficial effect through the

7

development of the Banff Research Stationand the workshops that I have been able toattend there over the last couple of years. Itseems clear that my career would have goneon without it, but looking more broadlyat the national scene PIMS has had a ter-rific impact on Canadian mathematics. For-merly, there was some sense of isolation inthe west of Canada. The simple physicalproblem of separation by some thousandsof kilometres from the power houses of Ot-tawa and Toronto, in terms of mathemat-ics or politics, posed certain problems. Thecreation of PIMS has formed a balancingstructure and a source of stimulation thatwe can relate to without having to crossa continent. So it’s had great impact on,particularly, the two western provinces, Al-berta and British Columbia, and, of course,to a lesser degree on the rest of Canadatoo. But the connections with the PacificRim are also very interesting and obviouslystimulated by PIMS. For example, there arestronger links with Japan, Mexico, Chile,and even Australia (which is also a PacificRim country). This applies not only in re-search, but in all the aspects of PIMS activ-ities; in education, industrial connections,and so on.

NJH: I know you love the outdoors andknow the Canadian Rockies very well, andmany visitors have enjoyed your hospitalityin your log cabin near Banff in the lakes andmountains. How important have these out-door activities been to you and how havethey meshed with your research?

PL: Well, I see these activities, and myfortunate ability to continue these activi-ties, as very important. It’s my form of re-laxation and recuperation, and it plays animportant role in sustaining me. I certainlyget very fidgety if I’m not able to go outand indulge these activities for a week ortwo. So, for me personally, it has been andcontinues to be very important. And fortu-nately my health allows me to continue toindulge this. By the way, this phenomenon

is not unusual in the broader mathemati-cal community. There are quite a few col-leagues out there who share this enthusiasmwith me, and this suggests that the twointerests (of mathematics and mountains)support one another.

NJH: Have any theorems been con-structed during hikes, maybe with col-leagues?

PL: Particularly with Israel Gohbergand Leiba Rodman, who have spent pro-longed visits in Calgary. We have done sev-eral of the traditional hikes together. I re-call discussing this, that, or the other issueof our current preoccupation, whether thenotion will go this way or that way, will thiswork or that work? That kind of discussioncertainly took place among the three of usand, to a lesser degree, I’m sure it has hap-pened with other colleagues and collabora-tors.

NJH: Your Lambda Matrices book wasreprinted by Dover in 2002. In the last fewyears there has been a resurgence of inter-est in quadratic and more generally poly-nomial eigenvalue problems. Are you sur-prised that the book is of renewed interestafter 36 years?

PL: Yes—surprised, and very pleased.It may also suggest that it was underval-ued earlier, through the ’70s. It went outof print with Pergamon press very early.I think they only printed a few hundredcopies, but it continued to get citationsthrough the rest of the twentieth century,mostly from the engineering community. Ithink there is some interesting expositionthere which is only recently being recog-nised by the numerical analysis community.

NJH: You’ve written ten books accord-ing to my reckoning, including one secondedition. Do you have any tips or advice forauthors about book writing?

PL: Keep careful notes when you’re lec-turing. It may or may not lead to bookwriting but it certainly helps if that’s thedirection you want to go and if the book

8

you want to write is something you’ve de-livered in lectures. I’m not sure what elseto say. I don’t want to suggest that it’s easywriting books. One has to be prepared totake the time and effort to write and thenre-read and then re-write if necessary. Thisprocess is not to be underestimated. Per-haps another positive piece of advice wouldbe to find good collaborators. Two headsare almost always better than one, partic-ularly in the context of book writing. Thebroader perspective that you get from hav-ing two authors rather than one is generallyvery valuable.

NJH: What about the rate of writing,because it seems to me if you’re not quiteon the same wavelength as a co-author interms of speed and the ability to focus 100%on the book for a while, it could be ratherdifficult.

PL: Yes, certainly it would be hard tofind two people who could produce serioustext at about the same rate. Developmenttends to be spasmodic. In my collabora-tions the writing has generally been divided.Each of the authors has a favourite part ofthe subject matter and will write the firstdrafts of this, that, or the other chapter.Then the other author(s) get to see this,comment on it, and suggest further devel-opments from it. These are the benefits ofthe collaborative process. At the very least,the material gets a careful reading by oneother person. In a good collaboration yourideas will not only get a good reading, theywill also be pushed a little further and de-veloped.

NJH: You’re currently putting the fin-ishing touches to the second edition of your1982 book Matrices and Indefinite Scalar

Products with Gohberg and Rodman. Havepersonal computers and LATEX made bookwriting easier?

PL: My first reaction is yes, because weare deeply involved with LATEX, particularlyin the edition we are working on right now.This is the way in which we operate. Be-

ing rather naive about computer science ingeneral, it seems miraculous to me that a300 page draft can be whisked back andforth around the world to one keeper or an-other. This is part of our current collabo-ration with the three authors; the masterversion can move from the care of one indi-vidual to another and (at least while I amin Manchester) the three of us are on differ-ent continents. And the geography presentsno difficulties. So my first reaction is yes,it’s enormously helpful. My second reac-tion is perhaps a little nostalgic. It wasalso very nice to write material yourself,and in the earlier days of collaboration youwould literally have a manuscript, your co-authors would see this and comment, andthe manuscript would evolve as it passedbetween the co-authors. Then you wouldhand it over to the typist, hopefully a skil-ful one (I’ve been very fortunate in this re-spect), and you could forget about it un-til the typescript came back. There wassomething to be said for that era too. Itwasn’t quite so absorbing as handling yourown TEX files turns out to be.

NJH: What are your future researchplans?

PL: As you know, I’m really caught upin these inverse quadratic eigenvalue prob-lems and I feel now that I’m just getting ontop of them. I feel that I’m getting a goodperspective and, for the immediate future,I am interested in polishing this materialand bringing it together into a unified bodyof knowledge. I hope for the health andstrength to go on with that and also withthe work on pseudospectra (with PanosPsarrakos and Lyonell Boulton), which isquite fascinating. But then the work on abiological problem, the Markov process de-scribing phylogenetic trees (together withErich Bohl), is also very much in the fore-front of my mind. I am eager to see thatin print and I would very much like to seeit developed further. But whether time andcircumstance will allow me to develop three

9

or four projects simultaneously remains tobe seen.

NJH: Might we see another book?PL: You might, yes. There again, the

inverse eigenvalue problems might be a can-didate. We feel that’s relatively new mate-rial and, as I was saying before, perhaps itshould all be brought together into a sys-tematic form that could conceivably be amonograph. If so, this might be togetherwith my recent collaborator, Uwe Prells.

NJH: Is there anything else that youthink I should have asked?

PL: You might have asked about earlyinfluences, other than mathematical. Therewere two or three career turning points thatI can identify which don’t necessarily haveanything to do with mathematics. One ofthese has to do with the curate of the An-glican church that I attended when I was ateenager. There was some question aboutwhether I was going to be able to continuewith a university education. My parentsweren’t all that enthusiastic; I didn’t comefrom a family with either money or an aca-demic background. But the efforts of WillPugh, the Anglican minister in question,were quite critical in getting me into themathematics programme at Liverpool, andallowing me to pursue that career.

Also, I can’t over-emphasize the impor-tance of partnership—in the marital sense.Having the back-up of a family, and an un-derstanding wife, in particular, is for me asine qua non.

Related References

• Israel Gohberg. Peter Lancaster, myfriend and co-author. volume 130 ofOperator Theory: Advances and Ap-

plications, pages 23–27. Birkhauser,Basel, Switzerland, 2001.

• Peter Lancaster. My life and math-ematics. volume 130 of Opera-

tor Theory: Advances and Applica-

tions, pages 1–7. Birkhauser, Basel,Switzerland, 2001.

• Special issue of Linear Algebra and its

Applications in honor of Peter Lan-caster, Edited by H. Bart, I. Koltra-cht, A. Markus and L. Rodman, Vol-ume 385, Pages 1-476 (July 2004), in-cluding a collection of Personal Notesby Peter’s family and friends.

10