# Germund Dahlquist

### Quick Info

Uppsala, Sweden

Stockholm, Sweden

**Germund Dahlquist**was a Swedish mathematician who worked in the theory of numerical analysis as applied to differential equations.

### Biography

**Germund Dahlquist**'s father was a minister in the Church of Sweden, the established Lutheran church in Sweden. Germund's mother was a poet, but also wrote a number of well-known hymns. Dahlquist entered Stockholm University in 1942 to begin his study of mathematics. The person who had the biggest influence on him was Harald Bohr who was teaching there having been forced to leave Denmark due to the German invasion during World War II. Björck, Gear and Söderlind write [1]:-

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 centred 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 first degree from Stockholm University in 1949 having written a thesis

*On the Analytic Continuation of Eulerian Products*which he published in 1952. Deciding against continuing to undertake research for his doctorate, he joined the Swedish Board of Computer Machinery as an applied mathematician and programmer. In 1951 a decision was taken that Sweden should build its own digital computer. Named BESK (Binary Electronic Sequential Calculator), the computer was modelled on the ideas put forward by John von Neumann and Herman Goldstine in their ideas for the EDVAC (Electronic Differential Variable Computer) which they produced in the late 1940s. BESK came into operation in December 1953 and Dahlquist used the machine to solve differential equations. This led him to a deep study of numerical analysis. He also joined a group using BESK to solve problems relating to weather forecasting. Their methods led to a speeding up of the process of reducing the data and in September 1954 they produced the first 24 hour forecast produced on the day in which the input data was collected.

During this time Dahlquist wrote a number of papers such as

*The Monte Carlo-method*(1954),

*Convergence and stability for a hyperbolic difference equation with analytic initial-values*(1954), and

*Convergence and stability in the numerical integration of ordinary differential equations*(1956). He submitted his doctoral thesis

*Stability and error bounds in the numerical integration of ordinary differential equations*to Stockholm University in 1958, defending it in a viva in December. In the thesis he [1]:-

.... 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.His thesis advisor was Fritz Carlson but he also received strong support from Lars Hörmander. The thesis was published in the following year.

From 1956 to 1959 Dahlquist had been head of Mathematical Analysis and Programming Development at the Swedish Board of Computer Machinery. Following the award of his doctorate in 1959 he was appointed to the Royal Institute of Technology in Stockholm. He spent the rest of his career at this institution where the Department of Numerical Analysis, an offshoot the Department of Applied Mathematics, was founded in 1962. In the following year Dahlquist became Sweden's first professorship in Numerical Analysis when he became the professorial head of the Department which at this stage had six members of the academic staff. In the same year of 1963 he published

*Stability questions for some numerical methods for ordinary differential equations*, an expository paper on his fundamental results concerning stability of difference approximations for ordinary differential equations. In the same year he published

*A special stability problem for linear multistep methods*which introduced $A$-stability and became one of the most cited papers in numerical analysis. It was published in BIT, a journal which Dahlquist had played a major role in founding in 1961. He served as an editor of BIT for over 30 years.

In 1969 a collaboration with Ake Björck led to the publication of

*Numeriska metoder*(Numerical methods) published in Swedish:-

This is a substantial, detailed and rigorous textbook of numerical analysis, in which an excellent balance is struck between the theory, on the one hand, and the needs of practitioners (i.e., the selection of the best methods - for both large-scale and small-scale computing) on the other. The prerequisites are slight (calculus and linear algebra and preferably some acquaintance with computer programming) so that some of the finer theoretical points (those at which numerical analysis becomes applied functional analysis, for example) are outside the scope of the book. However, the class of readers for whom the book is intended are admirably served.The importance of the book is easily seen from the fact that a German translation appeared in 1972 under the title

*Numerische Methoden*, an English translation was published two years later under the title

*Numerical methods*, and a Polish translation was published as

*Metody numeryczne*in 1983. Several later editions and reprints of the various translations of the classic text continued to appear as well as a Chinese translation in 1990.

The book

*Numerical Methods in Scientific Computing, Volume 1*by Germund Dahlquist and Ake Björck was published in September 2008. Björck writes in the Preface:-

In 1984 the authors were invited by Prentice-Hall to prepare a new edition of [Numerical Methods]. After some attempts it soon became apparent that, because of the rapid development of the field, one volume would no longer suffice to cover the topics treated in the 1974 book. Thus a large part of the new book would have to be written more or less from scratch. This meant more work than we initially envisaged. Other commitments inevitably interfered, sometimes for years, and the project was delayed. The present volume is the result of several revisions worked out during the past 10 years. Tragically, my mentor, friend, and coauthor Germund Dahlquist died on February 8, 2005, before this first volume was finished. Fortunately the gaps left in his parts of the manuscript were relatively few. Encouraged by his family, I decided to carry on and I have tried to the best of my ability to fill in the missing parts. I hope that I have managed to convey some of his originality and enthusiasm for his subject. It was a great privilege for me to work with him over many years. It is sad that he could never enjoy the fruits of his labour on this book.Nick Higham, in a review of the book, writes:-

This work is a monumental undertaking and represents the most comprehensive textbook survey of numerical analysis to date. It will be an important reference in the field for many years to come.In 1990 Dahlquist retired from the Royal Institute of Technology in Stockholm, but remained very active in research. As an example let us note the publication of

*On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules*in BIT in three parts (1997, 1997, 1999). Dahlquist writes:-

Three methods, old but not so well known, transform an infinite series into a complex integral over an infinite interval. Gauss quadrature rules are designed for each of them. Various questions concerning their construction and application are studied, theoretically or experimentally. They are so efficient that they should be considered for the development of software for special functions. Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation of power series outside the circle of convergence, and to ill-conditioned power series. [The second part] is mainly concerned with the derivation, analysis and applications of a summation formula, due to Lindelöf, for alternating series and complex power series, including ill-conditioned power series.The authors of [1] tell of two other aspects of Dahlquist's life outside of mathematical research, namely his work for Amnesty International and his love of music. Let us quote from their paper the delightful stories they recount regarding these two interests:-

As an active member of Amnesty International during the 1970s, Dahlquist worked to help scientists who were politically persecuted, in some cases travelling 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!Next the story regarding his love of music:-

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 favourites, 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!"Dahlquist received many honours for his outstanding contributions. He was elected into the Royal Swedish Academy of Engineering Sciences in 1965. He was a plenary speaker at the International Congress of Mathematicians at Berkeley in 1986. The Society for Industrial and Applied Mathematics (SIAM) named him their John von Neumann lecturer in 1988. In 1995, on the occasion of his 70th birthday, SIAM established the 'Germund Dahlquist Prize' to be awarded biennially:-

Awarded to a young scientist (normally under 45) for original contributions to fields associated with Germund Dahlquist, especially the numerical solution of differential equations and numerical methods for scientific computing.He was awarded the Peter Henrici Prize in 1999. The following announcement of the award appeared:-

Yesterday, the Eidgenössische Technische Hochschule Zürich and Society for Industrial and Applied Mathematics presented 'The 1999 Peter Henrici Prize' to Germund Dahlquist for his outstanding research and leadership in numerical analysis. He has created the fundamental concepts of stability, A-stability and the nonlinear G-stability for the numerical solution of ordinary differential equations. He succeeded, in an extraordinary way, to relate stability concepts to accuracy and proved the deep results which are nowadays called the first and second Dahlquist barrier. His interests, like Henrici's, are very broad, and he contributed significantly to many parts of numerical analysis. As a human being and scientist, he gives freely of his talent and knowledge to others and remains a model for many generations of scientists to come.He received honorary doctorates from Hamburg University (1981), Helsinki University (1994), and Linköping University (1996).

### References (show)

- A Björck, C W Gear and G Söderlind, Obituary: Germund Dahlquist.
- G Dahlquist, 33 years of numerical instability: Part 1,
*BIT**Numerical Mathematics***35**(1) (1985), 188-204. - G Söderlind, Preface,
*BIT**Numerical Mathematics***46**(2006), 453-454.

### Additional Resources (show)

Other websites about Germund Dahlquist:

### Honours (show)

Honours awarded to Germund Dahlquist

Written by J J O'Connor and E F Robertson

Last Update December 2008

Last Update December 2008