# Graduate School:

1963 – 1967

In 1963 I applied for admission to the graduate program at Stanford University, following the advise and recommendation of Gonzalez Domínguez.

He had been a schoolmate of George Forsythe at Brown University, where both had been students of Jacob D. Tamarkin. I was accepted with a full assistantship, obtained a Fullbright Travel Fellowship, and moved to Palo Alto arriving on September 6, 1963. I was assigned to the Computer Sciences Department, although I was in the Mathematics program since there was yet no academic program in Computer Sciences.

The first day at Stanford I met George Forsythe, Peter Henrici who was visiting, J. Ben Rosen and Gene Golub, and my life changed forever. I started taking courses in modern numerical analysis and using advanced computer equipment, namely: a Burroughs 5000 and IBM 7094 and also took advantage of the congenial atmosphere at Stanford. Gene Golub was starting to hone what it would become his style of being more than helpful to students in general and to foreign students in particular. A number of my fellow students at the time have done quite well, among others: J. Varah, J. Daniel, C. Moler, R. Bartels and P. Businger.

I had the opportunity of working with J.B. Rosen, whom seeing my previous experience on least squares, assigned me to the development of one of the first numerical methods to compute the pseudoinverse of a matrix, which we published as an internal Stanford report:

This algorithm also obtained the minimal and basic solutions of rank-deficient problems. It would be interesting to know if it could be helpful in the field of compressed sensing, since the basic solution is a sparse solution to an under-determined linear system that can be used to initialize some of the algorithms to calculate sparser solutions.

Also, around that time, Gene Golub came from a conference where he had heard Milton Lees prove a theorem about the convergence of a one step difference correction method for a scalar ordinary differential equation subject to boundary conditions. He was enthralled with this result and asked me to try to reproduce it. At the time I did not know what a boundary value problem was and, in any case, I was busy with other things, so I filed it for later attention. A few months later I came upon this at a good time and asked Gene for some starting point to learn about the numerical solution of ordinary boundary value problems and difference corrections, which by the way, were not the astronomers difference corrections I had worked with earlier. Surprisingly, later developments came to connect all that under the umbrella of Stetter’s general defect correction principle, which evolved from the work I was about to start and from work that Zadunaisky was doing in parallel in Buenos Aires.

Gene luckily pointed me to Leslie Fox‘s books, which were fundamental in providing me with a quick understanding of the work on the subject. Coupling this material with the training on Henrici‘s asymptotic theory, led me to a proof, different from Milton Lees, which started a long stint in the area and led eventually to my doctoral thesis, many papers and some fundamental pieces of software through the next 15 years and couple of countries.

Curiously enough, Henrici had also a “proof” of this theorem as an exercise in his book on the Numerical Solution of Ordinary Differential Equations. Unfortunately, although I had taught from Henrici’s book in Argentina, I had concentrated on initial value problems, and never made it to the boundary value problem chapter until this point in time.

The first publications on this subject were written at Stanford:

In 1965 I received my Master in Mathematics from Stanford and decided to move on. I was lucky again to connect with Ben Rosen at the IFIPS meeting in New York, to which George Forsythe had generously sent me. Ben had moved to Wisconsin and he offered me a job at the Mathematics Research Center that I accepted. We moved to Madison in the fall of 1965 and with this full time job we felt very rich.

Madison was very scientifically active at this time. The place was humming, with its large Mathematics Department, its Computer Science Department with Numerical Analysis and Optimization and with its thriving Mathematics Research Center. Besides, it was too cold to do anything else but work. A large number of short and long term visitors went by and it was in general a very exciting and inspiring time.

I continued work on deferred corrections and started writing my thesis for Buenos Aires, under the nominal direction of Zadunaisky. The job at the MRC left me with a lot of freedom and allowed me essentially to do my own research, with the added advantage of having their strict quality control in the production of reports applied to my work. I also had access to a Control Data 3600 computer, which was quite helpful.

The scheme was that I would write chapters of the thesis in Spanish and send them to Zadunaisky for his approval. I also received some of the work he was doing in the area of error estimation for initial value problems and started to notice some striking similarities in our approaches. However, I failed to appreciate what he was doing, which had departed in some fundamental aspects from my point of view.

I paid dearly for this slip, since years later, when Zadunaisky’s theory had matured, Hans Stetter saw its importance and took the opportunity with his school to improve and in many respects supersede, deferred corrections, by its very near cousin defect corrections, Zadunaisky’s invention. They adopted most of my terminology and methodological approach and in a barrage of papers buried most of my earlier contributions.

During this period I also had the fortune to meet and interact with a number of prominent local and visiting scientists. To name a few: Eduardo Zarantonello, Jean Pierre Aubin, I. Schoenberg, Larry Schumaker, L. Rall, D. Greenspan and Jim Daniel.

But let us go back to the correct time frame. Thanks to the Madison atmosphere a number of publications started to appear:

By the summer of 1966 I had essentially collected enough material for my thesis, when some sad events in Argentina (“the night of the long sticks,” when the military government of Ongania assaulted the School of Sciences), produced the disbanding of most of my colleagues from the University of Buenos Aires. Louis Rall, then the assistant-director of MRC, had read my thesis and recommended it highly. Thus, when I approached him asking about the possibility of getting my PhD in Wisconsin, he was quite enthusiastic and extremely helpful. It took a number of exceptional rulings, since I was not a student, but I managed to re-write my thesis in English and fulfill all the pre-requisites for the PhD degree, which I obtained in January 1967. Incidentally, this was the first PhD in Computer Sciences awarded by the University of Wisconsin.

Seymour Parter was my formal advisor, but to this day he acknowledges no responsibility or takes any credit for the content of my thesis. I must say that during my research, Donald Greenspan had some significant input, specially in relation to the test problems used. In terms of influence from the literature, Leslie Fox’s work on difference corrections was seminal. A paper by H. Stetter on a functional analysis approach to asymptotic expansions came at the right time to give me the basis for a general theory for deferred corrections and Richardson‘s extrapolation.

Also it was challenging to compete with the team of Varga et al that were developing the first finite element techniques for boundary value problems and show them some of the remarkable high accuracy and high efficiency results that finite differences with deferred corrections could produce. I was surprised and disappointed by their reluctance to acknowledge this obviously relevant work. I learned later a maxim by Martin Schultz, one of the members of that team, that explained what was happening: “That is not how the game is played. Your results are too good; if we mention them that will put us out of business.” OK!

Then it came the time for my:

This also resulted in some additional publications:

This long paper contains some hidden gems. In particular, it has a complete treatment of the convergence of nested iterations, as they occur for instance when the linear problems in Newton’s method are solved incompletely.

This paper has been rediscovered recently by Bertil Gustafson (University of Uppsala), as the lonely antecedent of the application of deferred corrections to initial value problems, that he is extending to hyperbolic systems.

An interesting side story was that Alexander Ostrowski came to spend some time at the MRC, and I was assigned to be his assistant again. As such, I was witness to a great mind doing fundamental work on the convergence of descent methods for minimization and also collaborated with him in some research and software development for the solution of polynomial equations, which resulted in a report:

As soon as I got my degree I was offered a three years assistant professor position that I accepted. Unfortunately, I was in the US as an exchange visitor and it was very hard to change to a permanent resident status. I tried, but time was flying and I could not get any reassurances from the INS. At this time we received a very generous offer of jobs in Venezuela, where some of our Argentine friends have landed. Manuel Bemporad and Carlos Domingo were instrumental in getting us there, even after a fairly large earthquake produced heavy damage in Caracas, July 1967.