**Patrick Keast**was known to most people as

**Pat Keast**. His parents were John Keast (born in Broxburn in 1913), a bus driver, and Rose Gorman (born in Broxburn in 1918). John and Rose were Roman Catholics who married in 1940 in Broxburn, a town in West Lothian, Scotland, located about 20 km west of Edinburgh. Their wedding was registered at the near-by Uphall, West Lothian. The family, including Pat and his two brothers John (born 1940) and Ronald (born 1947), continued to live in Broxburn. Pat was educated at St Mary's Academy in Bathgate and he graduated from this school in 1960. At the Academy, Keast met Kathleen Duffin; they married in 1967.

In October 1960, Keast began his studies at the University of Edinburgh taking courses in mathematics, physics and chemistry. He specialised in mathematics from his second year of study and was taught by, among others, William Edge, David Monk, Jim Fulton, Ivor Etherington, Michael Osborne, and John Cossar. It was Jim Fulton's honours course on numerical analysis which was particularly influential in inspiring Keast to become a numerical analyst. Mike Osborne, who came to Edinburgh in July 1963 as Assistant Director of the University Computer Unit, contributed some lectures on the numerical solution of differential equations to Fulton's honours course. At Edinburgh University, Keast was in a class which contained a number of students who went on to academic careers including Colin Campbell, whose career was at the University of St Andrews, and Alistair Watson, whose career was at Dundee University. In fact, out of around 40 students, ten were awarded First Class Honours degrees in 1964. Keast was one of the best of these First Class students.

After graduating with his first degree, in 1964 Keast was appointed as an Assistant Lecturer in Mathematics at the University of St Andrews. He undertook research in numerical analysis advised by Ron Mitchell who was in the Department of Applied Mathematics and headed the numerical analysis group at St Andrews. Also in the group was Jack Lambert who had been appointed as a lecturer in 1959 and had written a thesis advised by Ron Mitchell. Graeme Fairweather, who graduated with a First Class Honours degree from St Andrews in 1963, was another of Mitchell's students at this time and he became a close friend of Keast. They later collaborated on a number of research projects. Another student working towards a Ph.D. in numerical analysis was Brian Shaw, who also graduated with a First Class Honours degree from St Andrews in 1963, and was being advised by Jack Lambert. It was in 1964-65, Keast's first year at St Andrews, that I [EFR] first got to know him. In that year I was in my final year as an undergraduate.

In 1966 Keast, in collaboration with his thesis advisor Ron Mitchell, published the paper *On the instability of the Crank Nicolson formula under derivative boundary conditions*. The authors wrote the following abstract:-

Keast also worked with Edward Copson who was the Regius Professor of Mathematics at St Andrews. They published a joint paper in 1966 entitledFinite-difference solutions are considered for the heat conduction equation in one space dimension subject to general boundary conditions involving linear combinations of the function and its space derivative. It is shown that under such conditions, instability can often arise even although "stable" formulae of the Crank-Nicolson type are used. In particular, the persistent error discussed by Parker and Crank(1964)is shown to be a weak case of this more serious instability.

*On a boundary-value problem for the equation of heat*. In the following year, 1967, Keast published his second paper with Ron Mitchell,

*Finite difference solution of the third boundary problem in elliptic and parabolic equations*. The authors wrote the following abstract:-

In 1967 Keast was awarded a Ph.D. for his thesis on numerical analysis and he was promoted to lecturer. We noted above that Keast married Kathleen Duffin in Bathgate, West Lothian, in 1967. They had two sons Liam (born in St Andrews in 1969), who became a software engineer, and James (born 1975) who also became a software engineer.Finite difference methods(including the Peaceman-Rachford method)are considered for the solution of the third boundary value problem for parabolic and elliptic equations. Conditions on the coefficients involved in the boundary conditions are obtained from the stability requirements of the difference methods and shown to coincide with those necessary for asymptotic stability of the differential system.

Colin Campbell, who had been in the same class as Keast as an undergraduate at Edinburgh, had spent 1964-65 studying for a Master's degree at McGill in Canada before being appointed to St Andrews in 1965. He and Keast wrote the joint paper *The stability of difference approximations to a selfadjoint parabolic equation, under derivative boundary conditions* which was published in 1968. The authors wrote the following abstract:-

Keast was a keen sportsman and an enthusiastic cyclist taking part in competitive events. I [EFR] was appointed to the Department of Mathematics at St Andrews in 1968 and became one of Keast's colleagues. I remember him saying to me that he was now able to afford a much more expensive racing bicycle than when he was younger but his times in races had not improved - although in his 20s, he complained of getting old! Also at St Andrews Keast played squash and taught Colin Campbell to play as Colin explains [1]:-A self-adjoint parabolic equation in one space variable is considered, under boundary conditions which involve the function and its space derivative. A type of numerical instability can arise, which is traceable to the boundary conditions, and which is caused by the existence of unbounded solutions of the original differential equation.

Keast's first single author paper wasI learned to play squash from Pat. At the end of one of Pat's20mile cycle runs I would join him on the squash court and he could still run energetically. I was determined to beat him within20sessions and I just about managed it. As well as being a colleague at St Andrews we even shared an office close to the main building(47North Street)which is long since demolished.

*The third boundary value problem for elliptic equations*(1968). In that paper, Keast gives an excellent summary of its contents:-

In 1968-69 Keast made his first visit to Canada when he was a Visiting Assistant Professor at the University of Toronto. He returned to St Andrews for the year 1969-70 but, in 1970, he moved to Canada when he was appointed as an Assistant Professor of Mathematics in the Physical Sciences Division of Scarborough College, part of the University of Toronto. He continued his enthusiasm for cycling and became an active member of the Scarborough Cycling Club. My colleague Colin Campbell and I spent September 1976 in Toronto undertaking research with Donald Coxeter. While in Toronto, we met up with Pat Keast who invited us to an excellent meal at the University of Toronto staff club. I remember it was a very enjoyable occasion when I had chicken livers for a main course.In[Finite difference solution of the third boundary problem in elliptic and parabolic equations], the author discussed the numerical solution of the heat conduction equation in an open rectangular region, under boundary conditions involving a linear combination of the function and its normal derivative. It was shown that the instability which was observed in the difference methods examined, could be traced to the existence of solutions of the differential equation which grew, asymptotically, with time. This numerical instability was important in the solution of the heat equation only for large values of the time, and so did not affect calculations which were carried out over a few time steps. But, in the numerical solution of elliptic equations by iteration, such asymptotic instability will prevent convergence of the iterative process to the original system of equations. It is the purpose of this note to demonstrate this fact, and also to discuss the solution of Laplace's and Poisson's equations, when the Laplacian operator is singular, in a sense to be defined. It will be shown that, for certain boundary conditions, the numerical solution of Laplace's equation is best obtained by direct methods, rather than by iterative methods. In addition, a condition for the existence of a solution of the singular problem is obtained.

Keast moved to Halifax, Canada, in 1983 when he was appointed as a full professor in the division of Computing Science of Dalhousie University, at that time a newly established division within the Mathematics Department. He taught both mathematics courses and computer science courses at Dalhousie University. His two sons Liam and James attended Halifax West High School, Liam graduating in 1987 and James in 1992.

From 11 to 16 August 1986 a NATO Advanced Research Workshop entitled 'Numerical Integration: Recent Developments, Software and Applications', was held at Dalhousie University, Halifax. Keast was a main organiser of the Workshop which was attended by thirty-six scientists from eleven NATO countries. Thirteen invited lectures and twenty-two contributed lectures were presented and a volume, jointly edited by Patrick Keast and Graeme Fairweather, was produced containing twenty-five articles based on the lectures given at the workshop.

When the division of Computing Science split from the Mathematics Department to become a department of its own, Keast became a member of the Mathematics Department. On 1 July 2003, Keast became Chairman of the Mathematics Department taking over from Richard Nowakowski. After one year in the post, he wrote in his report [3]:-

In the report at the end of his second year as Chairman he wrote in May 2005 [4]:-This has been an exciting year for the Department of Mathematics and Statistics, as the contents of this year's Chase Report show. The year seems to have raced by, at a breathtaking pace and, although it feels as if I have just begun my term as Chair, I am now one-third of the way through. ... During this spring and summer we will be hosting no less than4national conferences, with several hundred participants from many countries.

In May 2006 his term as Chairmen was about to come to an end when he wrote his report [5]:-This is my second year as Chair of the Department of Mathematics and Statistics, a busy year with many changes taking place, and many new faces about to arrive. ...

In his May 2007 report, the new chairman, Karl Dilcher, wrote [6]:-It is said that time passes quickly when you are having fun. Whether that means that time drags when you aren't, I do not know, but I will say that the last three years have gone by incredibly quickly. My term as chair ends on June30, when Dr Karl Dilcher takes over. ... Our Honours programmes continue to grow. Especially gratifying is the increase in enrolment in the Statistics Honours programme, which is at the highest it has ever been. The Graduate programmes are also growing, and we are attracting a number of excellent students in both Mathematics and Statistics. Our Outreach Programmes report describes a very good year of activities by the Outreach team, involving high school students and teachers. All in all, I think that the Department has had a wonderfully successful year, and I have had an enjoyable term as chair. I shall miss being in the centre of all the activity. I have had a good three years, made so by the hard work of the members of the Department, efforts that it is easy to take for granted because nobody complains! I would like to thank everyone for their help, advice and understanding, and for the willingness to take on tasks uncomplainingly, and for carrying them out efficiently. I think few chairs are so lucky as I have been.

Keast had many other roles in the Dalhousie Mathematics Department including Mathematics Graduate Coordinator and an organiser of the Annual Bluenose Numerical Analysis Day until 2009. The 2008 meeting took place at Dalhousie on 13 June with Keast as the local organiser. He was one of the developers of a new website for the Department and then, starting in June 2006, he handled the maintenance of the site including additions, troubleshooting, and alterations.Pat Keast's3-year tenure as chair marked a period of rapid change, with numerous retirements and some resignations, and with just as many excellent new appointments. Such times of change often bring with them the danger of turbulences and uncertainties, but under Pat's capable leadership this beginning transition in the department's demographics went very smoothly, and this is still the friendly place it has always been.

As a lecturer, Keast was much appreciated by his students. The following comments were all made by students taking his courses between 2002 and 2006: "Extremely knowledgeable on the subject, has high expectations, very approachable"; "A great professor, cares about students, really nice guy"; "One of my best profs so far. The subject is difficult but he teaches it very well. Definitely knows his stuff and has a lot of enthusiasm for the subject. Very helpful outside of class and shows a lot of concern for his students"; "DEFINITELY knows his stuff. Helpful and easy to talk to. Assignments are fair but his tests can be BRUTAL at times. One of my all-time favourite profs, he'll work you hard but you'll be all the better for it"; and "Pretty funny guy. Very helpful and willing to help outside of class. Not many profs actually want to help outside their hours, he did. Lectures are easy to follow".

In 2007 Keast retired although he continued to undertake work for the Department. He was the director of the Atlantic Association for Research in the Mathematical Sciences Summer School at Dalhousie in 2007. As mentioned above, he organised the Bluenose Numerical Analysis Day at Dalhousie in 2008.

We must mention Keast's work for the Church, particularly in the communities of St Pius X and St Benedict's and through his support for the poor in working for the St Vincent de Paul Society. Although his health had not been good from the time of his retirement, in fact it had been better for a year before his sudden death in 2016. A funeral mass was celebrated at St Benedict's Parish in Halifax on Tuesday 2 August, the day that would have been his 74th birthday.

Let us record two tributes paid to Keast after his death. Carol Stewart writes:-

Maria Lantin writes:-I was a mature student at Dalhousie, and knew Professor Keast. When he discovered I was soon graduating he was overjoyed and pleased for me. He had a passion for teaching and opened up the minds of those who were eager to learn. It came naturally to him. He was a very popular professor amongst students and colleagues and received much respect from everyone.

Finally, let us note the following report in theDr Keast. this is what I called him. When I graduated he told me I could call him Pat. I could not do it. He'll always be Dr Keast to me. This belies the tenderness I feel for this man who believed in me and was a feminist ahead of his time. He was strong. He was dedicated. He was brilliant and funny. He was absolutely relentless in his demand for commitment and curiosity. I have many good memories. Life shaping experiences. Thank you Dr Keast. Thank you Pat. You were a light, a warrior of the most gentle kind.

*Sunday Herald*, a Scottish newspaper, in 2004 [11]:-

Not only does this show his sense of humour, but it also shows that he continued to keep in contact with Scottish affairs throughout his time spent in Canada.Pat Keast of Broxburn, now living in Halifax, Nova Scotia, was reading a local newspaper report about a driver accused by the police of illegally having alcohol in his car. He was found guilty, but Pat reckons he knew why the driver felt he had to have a go and have his day in court. The driver was called David Eisnor, and the constable was Walter Goliath. Sometimes a sling is not enough.

**Article by:** *J J O'Connor* and *E F Robertson*