​To some, “Schrödinger" and “Faddeeva" are mere surnames. To Stellenbosch University (SU) distinguished professor André Weideman, they represent mathematical equations and formulas that he has solved and computed during his 40-year career in applied mathematics. These have led to some of his most cited research papers on numerical analysis and scientific computing, and contributed to his receiving his first A1 rating from the National Research Foundation in 2022 as a world-class researcher.

He joined the SU Department of Mathematical Sciences in 1999, after spending many fruitful and stimulating years in the 1990s as an associate professor at Oregon State University in the USA. At the time, Prof Weideman, his wife, Marì and three children returned to South Africa to be closer to family.

Recognition

In recent years, quite a few other honours have come his way.

In 2019 he received a medal for research distinction from the South Africa Mathematical Society, and in 2020 the SU Chancellor's Award for Research. A year later, he was promoted to distinguished professor.

Prof Weideman is still the only person from Africa yet to be elected as a Fellow of the prestigious international Society for Industrial and Applied Mathematics (SIAM). When selected as such in 2017, he was commended for his ability to produce “powerful and elegant numerical algorithms derived from complex analysis."

Having subsequently served on the SIAM Fellows Selection Committee he has intimate knowledge about what it takes to be accepted as a SIAM fellow.

“It is a very humbling experience," he notes.

When opportunity knocks

Applied mathematics, Prof Weideman explains, is about solving mathematical problems that arise in real world applications in, for instance, engineering, finance, and medicine.  Such solutions should be fast and accurate when implemented on a computer, and allow for minimal machine generated roundoff errors.

Many of his applied maths colleagues work side by side with people in industry to solve engineering or production related issues. Prof Weideman works on more general classes of problems rather than specific ones, thereby hoping that he can help many others to do their jobs better.  

His work has proved to be valuable to scientists and engineers working in fluid dynamics, spectroscopy, astrophysics and laser optics.

He is a regular reader of journal articles in his field – and it's there that he often comes up with problems to solve. 

“I don't choose my research fields; they choose me," he notes. 

In that sense he is a “when opportunity knocks" type of person.  

“When I see a mathematical equation or formula, I'll start thinking how to compute it efficiently. Only after working on it I might realise how and where the solution can be applied.

“Of course, for every one paper published, there are a thousand problems that I couldn't solve," he notes dryly about the nature of things. “You may think you have a brilliant idea, but if you cannot demonstrate that it's better than the existing ones it ends up in the wastepaper basket."

Some might find his work daunting, and even his way of working, but it's no such thing to Prof Weideman.

“You can work by playing. I am lucky that my work is my hobby," says the son of a mathematics teacher from the Free State.

Prof Weideman admits that he has often done some of his best work “over Christmas and New Year", when he has fewer lecturing and administrative duties. 

Most of his papers appeared in top journals in applied mathematics, numerical analysis or scientific computing.   

He has also worked on algorithm and software development. Many of his codes are available online in the programming languages MATLAB and Python.

A 1994 paper on the computation of the complex error function (also known as the Fadeeva or plasma dispersion function) has become the standard against which all new methods for computing this function are compared.

A paper from 2000 on spectral differentiation matrices, available online along with a software package DMSUITE, has attracted just under 1000 citations on Google Scholar. Reviewers on the MathWorks site gave it a 5-star rating and added comments such as "intuitive, well documented and extremely useful!"

He serves as an associate editor for two international journals, Electronic Journal of Numerical Analysis and Numerical Algorithms, and is an editorial board member for South Africa's Quaestiones Mathematicae. He referees papers for some of the most prestigious journals in his field, and also reviews book proposals for SIAM and Springer.

For the last two decades, he has served in the main organizing committee of SANUM, the South African Numerical and Applied Mathematics Symposium.

“This has grown to be a truly international meeting, with many leading international speakers."

Among his recent research highlights count co-organising a four-month research program on Complex Analysis at the prestigious Isaac Newton Institute in Cambridge in the UK. He was also an invited speaker at the 2019 ICIAM meeting in Spain, which attracted more than 4000 participants. It is the biggest meeting of its kind in industrial and applied mathematics.  

Early years

Prof Weideman started considering a career in mathematics after participating in the national mathematics Olympiad during matric at Hentie Cilliers High in Virginia in the Free State in 1975.

“Somewhere in my office I still have the book prize I received. It made me realise that I could try maths - without me yet really knowing what applied mathematics was."

During his BSc studies at the University of the Free State in the late 1970s, the laborious system of punch cards was still the norm in computer programming. He completed his master's degree in 1980, using a mainframe computer.  

He bought his first computer, an Apple IIe with a 64kb memory system and a floppy drive, around 1983, before starting to work on his PhD. His supervisor, Prof Ben Herbst, would later become a colleague at SU.

The Apple still has pride of place in his office, along with other computer paraphernalia that spans his research career.

His very first attempt at publishing a paper was done on that Apple. It was rejected by the journal.  Unjustly so, he still claims to this day.

“Total immersion"

“To be a true researcher, is all about total immersion," says Prof Weideman.

He has carried this idea with him since mid-1986, when as a newly capped PhD graduate, he spent a formative year at the Massachusetts Institute of Technology (MIT) in the USA.

“I was blown away by the fact that a university library could be open 24 hours a day!"

At MIT he met one of his lifelong mentors, Prof Nick Trefethen.

“We started a collaboration that endures to this day," he adds thankfully.

During lockdown in 2020, for instance, the two, together with Prof Yuji Nakatsukasa at Oxford University, spent hours on Zoom collaborating on a new paper.

“Despite the challenges of full-time online teaching, we managed to write a paper on exponential node clustering in various rational approximation methods.

“The eventual paper, published in Numerische Mathematik, formed a prominent part of Prof Trefethen's prizewinning 2020 John von Neumann lecture, one of SIAM's highest accolades. My contribution to it was acknowledged."

In 2014 the duo published a paper in SIAM Review (SIREV), the flagship journal of the Society of Industrial and Applied Mathematics, on the exponential convergence of the trapezoidal rule.

It's a paper with a long history.

“It was triggered by another I wrote on the numerical integration of periodic functions, published in American Mathematical Monthly in 2002.

“During my sabbatical to Oxford in 2004/5, Trefethen suggested we turn this into a bigger paper. We did this, off and on, over the next eight years. It culminated in the 74-page behemoth published in 2014. Even though it can be categorized as a review paper, it contains a lot of new material and previously unnoticed connections between different areas of computational mathematics."

He's quite proud of the legwork that went into getting the necessary history cited in the paper just right.

“Not sparing the cost and time, we even visited the library at Cambridge University to track down an obscure reference."

On the point of obscure: this avid supporter of the Cheetahs rugby team once received a phone call from a Maties rugby coach. He was searching for the perfect kicking position so that the goal kicker could maximise the angle to the posts and therefore the chance of success. 

Wryly Prof Weideman just answered: “Well, you've at least reached the right academic department for such a question. The good news is that it's already been solved. The bad news is that it will simply take too much thinking on the field to put it into practice. Just tell your flyhalves to trust their instincts."