## Obituary: Germund Dahlquist

**May 1, 2005**

Germund Dahlquist at Stanford University, 1985.

Germund Dahlquist, a mathematician and numerical analyst at the Royal Institute of Technology, Stockholm, Sweden, died on February 8, 2005, a few weeks after his 80th birthday. He is survived by his wife and four children, and their families.

From 1951 onward, Dahlquist made ground-breaking contributions to the theory of numerical methods for initial value problems in ordinary differential equations. His work has had a profound impact and has shaped computational practice for time-dependent problems, especially in stiff differential equations.

Germund Dahlquist was born on January 16, 1925, in Uppsala, Sweden. His father was a vicar in the Swedish church, and his mother wrote poetry, including several well-known hymns. In 1942, at the age of 17, Dahlquist started his studies of mathematics at Stockholm University. One of his teachers there was Harald Bohr (younger brother of the famous quantum physicist Niels Bohr), in exile from occupied Denmark during the World War II years. Harald Bohr was remarkable not only as a mathematician, but also as a very humane and generous person. These are qualities that Dahlquist shared to a high degree. Bohr took the time to discuss mathematics with his young student and inspired Dahlquist’s early interests, which centered on analytic number theory, complex analysis, and analytical mechanics. Dahlquist would later refer to the profound influence on his view of mathematics of that early time with Bohr.

Dahlquist received a licentiat degree in 1949, with a thesis titled “On the Analytic Continuation of Eulerian Products.” He would never give up this early research interest, but in 1949, rather than continuing his studies toward a doctoral degree, he joined the newly formed Swedish Board of Computer Machinery. Employed initially as an applied mathematician and programmer, he served later, from 1956 to 1959, as head of Mathematical Analysis and Programming Development.

In 1951, a time when no commercial computers were available, a Swedish effort to build an electronic computer was begun. BESK (the Binary Electronic Sequential Calculator) was similar in design to (but not a copy of) the first generation of von Neumann’s Princeton computers. BESK became operative in December 1953 and was for a short time the world’s fastest computer. In his work with BESK, Dahlquist became involved in numerical analysis, and he soon began to explore various difference methods for solving differential equations. He also joined a group working to develop methods for numerical weather forecasts, led by the Swedish-American meteorologist Carl Gustaf Rossby at the International Meteorological Institute of Stockholm University. In September 1954, several years ahead of similar efforts in the USA, the world’s first 24-hour forecast based on weather observations made the same day was carried out on BESK.

In the early 50s, numerical computing was largely an uncharted field of research. Although numerical methods certainly existed, it was no longer possible to have direct human insight into the computational process. New anomalies were also frequently encountered, simply because the computations had reached an unprecedented level of complexity. A better theoretical understanding was needed, in particular of such notions as stability and conditioning.

Dahlquist’s work at the Swedish Board of Computer Machinery led to his first publications in numerical analysis. In 1951, having explored the weak (in)stability of the explicit midpoint method, he attended a GAMM meeting in Freiburg to present his result. As he recounts in “33 Years of Numerical Stability,” the day before his talk he was contacted at his hotel by Heinz Ru-tishauser, who had just seen Dahlquist’s abstract. It turned out that they had independently obtained the same (still unpublished) result! In 1956, Dahlquist presented his complete study of O-stability of linear multistep methods, creating the convergence theory for such methods. This work was akin to that of Peter Lax, who in 1955 had established the Lax principle, which would later become the standard approach to the convergence of discretization methods.

In December 1958, Dahlquist defended his PhD thesis, “Stability and Error Bounds in the Numerical Solution of Ordinary Differential Equations,” in which he took several original approaches. He introduced the logarithmic norm (also introduced independently by Lozinskii in 1958), which he used to derive differential inequalities that discriminated between forward and reverse time of integration. Dahlquist was now in a position to derive more realistic error bounds for problems that might not even be well-posed in reverse time. In such cases, the classic Lipschitz convergence analysis would have failed. Dahlquist was to use this idea throughout his research in stiff differential equations.

Dahlquist’s research in the years between 1956 and 1958 instantly made him an international authority on the numerical solutions of ODEs. The basic theory he developed soon became available to a wider audience through Peter Henrici’s 1962 monograph, a text that set the stage for modern ODE theory.

Around this time, Dahlquist played an important role in establishing the Nordic journal BIT, which completed its first volume in 1961. Dahlquist served on the editorial board for 30 years, and also chose BIT for his most important papers. It was there that he published his classic 1963 paper on A-stability, one of the most frequently cited papers in numerical analysis. Again, he introduced a new requirement: that the method be numerically stable for all mathematically stable problems. He then showed that to be A-stable, a multistep method could not have order greater than two and that, of all second-order methods, the trapezoidal rule produced the smallest error. A deep argument in complex analysis was necessary for the proof.

Together with the results on the maximum order of an O-stable method, the fundamental limits on difference methods had now been established; they became known as the “first and second Dahlquist barriers.” In particular, the second Dahlquist barrier created a cottage industry of numerical analysts seeking alternative methods for achieving A-stability. It also generated a whole alphabet of less restrictive stability notions, as researchers attempted to bypass the second Dahlquist barrier.

Dahlquist’s A-stability paper also contained the embryo of theories to come. He had realized that it ought to be possible to extend his multistep stability results to nonlinear dissipative problems. Exploration of these ideas took many years, and in 1978 he succeeded in showing that the nonlinear G-stability property is in fact equivalent to A-stability. These ideas on nonlinear stability inspired researchers in Runge–Kutta methods, especially John Butcher, who quickly established a similar theory of nonlinear “B-stability” for Runge–Kutta methods. A rich theory for stiff differential equations was now in place. This had a significant impact on computational practice, as did the state-of-the-art software based on modern computational techniques then becoming widely available.

Beginning in 1959 and continuing up to and after his retirement in 1990, Dahlquist worked at the Royal Institute of Technology (KTH) in Stockholm. There, he started the Department of Numerical Analysis and Computer Science (NADA) and built a broad scientific research and educational program. In 1963 he was appointed full professor of “Computer Sciences, in particular Numerical Analysis,” the first position of its kind in Sweden. He developed the first curriculum in numerical analysis in Sweden. His and Åke Björck’s book Numeriska Metoder (in Swedish) appeared in 1969. With the encouragement of George Forsythe, a revised and extended version titled Numerical Methods was published in 1974 by Prentice–Hall; many universities in the USA used it for many years as a graduate textbook. It was later translated into several other languages, including German, Polish, and Chinese. Over the years, Dahlquist had 26 PhD students whose research covered a wide range of topics, and who in turn have supervised some 70 graduate students.

Dahlquist’s heart was very much in the practical application of mathematics. In 1965 he was elected to the Royal Swedish Academy of Engineering Sciences. During the 60s and 70s, he played an important role in procuring computers for Swedish universities. From 1972 to 1977, he chaired the research board of the Swedish Institute of Applied Mathematics, an organization created to bridge the gap between academic research and new applications in industry.

He also encouraged a couple of his students to start the software company Comsol. They constructed a PDE toolbox for MATLAB based on codes that Dahlquist had developed for a graduate course. Dahlquist’s codes had a brilliance and potential that were immediately recognized. Today, some 15 years later, Comsol’s main product, Femlab, is a leading multiphysics software system. The Comsol group now has subsidiaries in eight countries all over the world, with a total of some one hundred employees.

During the 1960s and 1970s, the NADA Department at KTH welcomed a stream of prominent international guests and held many workshops. Dahlquist also made many international visits, to Europe, the USA, Australia, New Zealand, and China. Most notably, he visited Stan-ford University in 1968 and 1977–78, and held a five-year part-time position from 1982 to 1986. In 1986, he gave a plenary presentation at the International Congress of Mathematicians. In 1988, SIAM named him the John von Neumann lecturer, the most prestigious of its prizes. In recognition of his pioneering work in numerical analysis and, in particular, his work on the numerical solution of differential equations, SIAM established the international Germund Dahlquist Prize in his honor, on the occasion of his 70th birthday (1995). In 1999 he received the Eidgenössische Technische Hochschule and SIAM Peter Henrici Prize for outstanding research and leadership in numerical analysis. He received three honorary doctorates, at Hamburg (1981), Helsinki (1994), and Linköping (1996).

As an active member of Amnesty International during the 1970s, Dahlquist worked to help scientists who were politically persecuted, in some cases traveling to offer his encouragement and recognition in person. He used to tell the story of his intervention on behalf of a Russian mathematician who, in despair, had made a thoughtless public statement to the effect that the Soviet Union was “a land of alcoholics.” Guriy I. Marchuk, who had visited Stockholm University in the 1960s, was then president of the USSR Academy of Sciences and vice-chair of the USSR Council of Ministers. Dahlquist wrote to Marchuk pleading the dissident’s case. After a long time with no response, two staff members of the Soviet Embassy called at Germund’s office one day, bringing greetings from Marchuk and a package, that turned out to contain . . . two bottles of vodka!

Germund had a keen interest in music, mainly classical but also jazz music. He would often happily sit down at the piano and entertain his colleagues with a few old standards, starting with “On the Sunny Side of the Street” and ending with “As Time Goes By.” But his knowledge went much deeper. On one visit to the USA, with a few colleagues in a fine restaurant, Germund heard a female bar pianist whose music was obviously the highlight of the evening for him. When it was time to leave, Germund told the pianist how much he had enjoyed her stylish playing, adding that it had reminded him of one of his favorites, the great jazz pianist Art Tatum. The pianist was duly flattered, but it was Germund who was surprised when she answered: “Art Tatum was my father!”

By his friends and colleagues all over the world, Germund Dahlquist is much missed but will be remembered as a unique personality, admired not only for his scientific achievements but also for his good humor and warm and generous personality. Always open to new ideas, he set a fine example for young scientists and PhD students, as attentive to them as he was to his peers. Both as a human being and as a scientist, he will be a model for many generations of scientists to come.—Åke Björck, Linköping University, C. William Gear, Princeton University, and Gustaf Söderlind, Lund University.

**Selected Publications**

Germund Dahlquist, *On the analytic continuation of Eulerian products*, Ark. Mat., 1:36 (1951), 533–554.

Germund Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, PhD thesis, Stockholm, 1959.

Germund Dahlquist, *A special stability problem for linear multistep methods*, BIT, 3:1 (1963), 27–43.

Germund Dahlquist and Åke Björck, *Numerical Methods, Series in Automatic Computing*, Prentice–Hall, Englewood Cliffs, New Jersey, 1974.

Germund Dahlquist, *G-stability is equivalent to A-stability*, BIT, 18 (1978), 384–401.

Germund Dahlquist, *33 years of numerical instability: Part 1*, BIT, 35:1 (1985), 188–204.