### Algorithms, Stability and Communication: Remembering the Great Nick Higham

*It is with a heavy heart that the mathematical community mourns the loss of Nick Higham, an esteemed colossus in numerical analysis whose departure bequeaths a monumental legacy. In my recent essay, I have compiled musings on his life and the profound impact of his work, an homage that seeks to honor his remarkable contributions to the realm of knowledge and his lasting influence. **Photo by Phil Trinh.*

In the vast domain of knowledge lies a territory where science and engineering intersect with mathematics and computing. This is a realm that may seem obscure to many, yet one we all sense exists to some degree. After all, it is well-known that physicists, chemists, engineers, and economists eventually transcribe their theories and solutions into the language of mathematics, and somehow, computers take up the task of solving these equations. We are not so naive as to believe that the chalk-filled blackboards seen in movies are the end of the story.

However, transforming a mathematical formulation into numerical results processed by machines requires an essential intermediary step: the creation of algorithms. These are systematic procedures that tackle a specific problem—be it a differential equation, a non-linear optimization, or the calculation of a matrix spectrum—and break it down into a series of operations that a computer can execute to find the solution.

Numerical analysts are the architects who design and scrutinize these algorithms. Their art and pride lie in devising procedures that deliver solutions as swiftly as lightning and with the highest precision. For over seven decades, these specialists have been the craftsmen of the algorithms that have driven the revolution in informatics and computing. Their impact spans all disciplines of science and engineering; from solving systems of linear equations to building powerful machine learning and artificial intelligence models, their legacy has been monumental. The field of numerical analysis is vast, with thousands of mathematicians dedicated to its advancement. Over the years, we have witnessed the emergence of eminent figures who have propelled the discipline to new heights and unveiled previously hidden horizons—luminaries such as Cornelius Lanczos, Isaac Schoenberg, Alston Householder, Leslie Fox, Jim Wilkinson, Gene Golub, Germund Dahlquist, Michael Powell, and Steven Orszag.

On January 20th, the community of numerical analysts bid farewell to one of the field’s giants, Professor Nicholas Higham, who passed away at the age of 62, leaving a remarkable void in the pantheon of great mathematicians.

Nicholas John Higham, a native of Salford in the county of Manchester, devoted his life and career to this area of England. He distinguished himself at the University of Manchester, where he completed his undergraduate and doctoral studies and later served as a professor. His prolific mathematical output is evidenced by more than 200 specialized articles and five books he authored, as well as the countless talks he gave at international conferences around the world. He made significant contributions to the LAPACK and NAG software libraries, developed multiple modules of numerical linear algebra for MATLAB, and his algorithms have been incorporated into other programming languages like Julia, SciPy, and Mathematica. Throughout his career, he guided over 20 doctoral students and 30 master’s students, served on various editorial boards of mathematical journals, and presided over the Society for Industrial and Applied Mathematics (SIAM).

Nick’s work has been cited more than 30,000 times according to Google Scholar, and his research career has been acknowledged with a succession of awards, from the recognition for his doctoral thesis with the Householder Award in 1987 to the distinguished Gold Medal from the Institute of Mathematics and its Applications in 2020. He was honored with memberships in renowned mathematical and engineering societies, such as the Royal Society (FRS), SIAM, and the ACM. Since 1998, he held the Richardson Chair of Applied Mathematics at the University of Manchester, and in 2018, the Royal Society awarded him the prestigious “Research Fellowship”.

I met Nick in the summer of 2007 at a reception hosted by my doctoral supervisor for that year’s finalists for the Leslie Fox Prize in Numerical Analysis. Until then, my knowledge of him was confined to his academic fame and as one of the collaborators in the Oxford research group. His presence was commanding: a man of stature and demeanor, always impeccably dressed in a suit and tie, emanating formality and warmth in equal measure, with an intellect that stood out even among a circle of brilliant mathematicians. After that encounter, I saw him about a dozen times, at conferences or during his visits to Oxford; the last time was last year, during a presentation of his most recent work on bidiagonal matrices.

Although I ran into him so many times, it was only on one occasion that I interacted with him more extensively. It was in 2009, when he graciously agreed to give a talk at the annual conference of the SIAM student chapter I had helped to found. As an organizer of that conference, I coordinated his travel from Manchester and made sure everything was ready for his presentation. His talk was on the mathematics of digital photography, a subject where his enthusiasm was palpable. It turns out, behind the aura of that virtuoso mathematician, Nick had an appreciation for the art of photography, and during that talk, he shared many of his own photographs while revealing to his audience the hidden mathematics of this art, from the different algebraic transformations between color palettes to the partial differential equations that editing programs use to correct blurred images.

During one of the coffee breaks, I spoke to him about my own doctoral project, which led to a brief exchange of emails after the event. I regret not having cultivated that contact more; often, shyness in the face of our figures of admiration paralyzes us, we hide to not make our ignorance so evident. Which is a pity, because Nick – like so many other great mathematicians – was kind and receptive to research students, happy to support others who are walking the paths they helped to forge.

Nick’s loved ones, friends, and colleagues will surely be able to recount in rich detail the more intimate facets of his life and provide a fair and complete perspective of his vast work. Although his research area does not exactly coincide with mine, my familiarity with his publications and certain aspects of his work, in my role as a numerical analyst, allows me to highlight three of his contributions that I find especially relevant.

The first is numerical stability, which deals with how errors propagate through numerical calculations. This is key to determining the usefulness of an algorithm, because one that magnifies errors until the final solution is unrecognizable is, in essence, useless. This concept, which can be formulated with mathematical rigor, also admits an intuitive explanation: an algorithm is numerically stable if minor variations in the input data or during the computational process translate into equally minor deviations in the result.

We can find a parallel in cooking: a recipe is “stable” if, when you slightly vary the amount of salt or the cooking time, the dish remains tasty. That is to say, minor alterations in the steps or ingredients do not spoil the final result, but rather produce something very similar to what was expected. From personal experience, I would say that soups and pastas are more “stable” in culinary terms than soufflés and pastries.

The apparent imprecision of numerical algorithms and their need for stability analysis may surprise those outside the mathematical field and even cause discontent among the more purist mathematicians. However, it is a fact that the world is full of errors and, without their meticulous study, it would be impossible to achieve concrete advances in any scientific discipline.

The study of the stability of algorithms has a long and venerable history, which spans several decades and has involved mathematicians of the stature of John von Neumann and Alan Turing. Nick joined that noble tradition and emerged as one of the global leading experts. His work “Accuracy and Stability of Numerical Algorithms” has become the definitive text in the field, and everything indicates that it will continue to be so for many decades to come, thanks to his ability to infuse clarity and contemporaneity into a subject that could seem dense and arid. Among his numerous contributions to the field of numerical stability, the one that most influenced my doctoral research was his demonstration that the barycentric interpolation formula is forward stable when using a grid of points with a small Lebesgue constant—an essential finding for the Chebfun project, in which I had the pleasure to work.

However, the work of Nick that I came to know in greater depth was a series of algorithms he proposed to tackle a problem in finance. As he shared with us at a conference, at the beginning of the millennium, a fund manager presented him with a specific conundrum: how could one calculate the nearest correlation matrix from a given symmetric matrix? The correlation matrix, a critical component for financial analysis, is a data set that measures the interrelationship among different assets in a portfolio. For instance, the S&P 500’s matrix is a two-dimensional array of 500 x 500, where each row and column represent the interdependencies among the returns of the 500 largest companies in the United States. A row dedicated to a company like Microsoft would reflect its correlations with IBM, Apple, Tesla, and the other 496 companies that are also part of that stock index. The precision of this matrix is vital for predictions of risk and return in investment management.

Although theoretically the construction of this matrix should be rigorous, in practice, fund managers often adjust correlations arbitrarily, which can distort the mathematical properties required of a genuine correlation matrix. Modifying certain values alone, such as Microsoft’s correlations with certain companies, could yield technically unviable results. The task, therefore, was to find a numerical method that would maintain the integrity of the correlation matrix and at the same time reflect the desired reality as closely as possible.

Finance was not Nick’s specialty, but his mastery of numerical linear algebra made him the ideal candidate for this challenge. In 2002, he published a pioneering paper, “Computing the Nearest Correlation Matrix – A Problem from Finance,” which not only meticulously defined the problem but also offered an algorithm of alternating projections capable of converging to the ideal solution. This paper became one of the most referenced in his career, with over a thousand citations on Google Scholar. Nick continued to refine his research in this field over the following two decades, with publications that significantly expanded the scope of his work. But as often happens in the world of mathematics, what started as an application in finance spread to many other areas. His algorithms for finding the nearest correlation matrix are now used in fields as diverse as climate change, genetic engineering, structural engineering, and hydrology.

The last aspect of Nick’s work that I wish to highlight is the aesthetic one. His writings radiate a clarity, sharpness, and beauty that are seldom found in academic writing and are even rarer in the field of mathematics. Each sentence and equation are sculpted with diligence, as if Nick undertook a craftsperson’s quest to distill what is fundamentally significant, freeing the reader from the fog of confusion or superfluous embellishments, allowing them to grasp his ideas with ease. Such was his meticulous attention to detail that he was the author of the “SIAM Style Manual”, a guide for editors of the thousands of academic papers published each year by this society of mathematicians.

That same care he applied to his talks, which were always fascinating and enlightening, no matter how complex the material. Or even how mundane: I once attended a lecture he gave to postgraduate students on the topic of “The Transition from University to Work”. What would typically be a routine vocational presentation became, in his hands, a detailed, entertaining, and tremendously practical exposition on how to advance in a research career.

Nick Higham’s work is extensive and profound, and his legacy impacts multiple disciplines. These days I have revisited his books and articles, and I marvel at the magnitude of what a brilliant mind like his could achieve over four decades of a career. His personal page at the University of Manchester is a treasure trove of his labor, with broad access to his works, presentations, and computer codes, all pristinely organized. As I navigated it, I realized that Nick had been writing a superb blog for a decade, touching on fascinating mathematical topics. I lost myself for hours in an endless stream of his entries, reading them from back to front. When I finally arrived at the very first one he wrote, on January 1, 2013, I couldn’t help but smile: it was a short review inviting readers to a book written by my doctoral supervisor, for which I proudly served as a proofreader.

Nick Higham was a true champion of algorithms and numerical analysis. Though he left us prematurely, his work remains as a celebration of his life and an inspiration for future generations of mathematicians seeking to expand the horizons of knowledge.