Rutishauser served as Saxer's assistant for three years while undertaking research for his doctorate. He was awarded the degree with distinction in 1945 for his thesis Über Folgen und Scharen von analytischen und meromorphen Funktionen mehrerer Variabeln, sowie von analytischen Abbildungen Ⓣ. Following this he became a teacher at the Gymnasium in Glarisegg on the Swiss lakeshore of Lake Constance (Bodensee). From there he moved to the Gymnasium in Trogen in the Appenzell Ausserrhoden canton in north east of Switzerland. Although a schoolteacher, Rutishauser published papers such as Über Punktverteilungen auf der Kugelfläche Ⓣ (1945) which examines the problem of finding n points on the surface of the unit sphere so that the smallest spherical distance between any two of them is maximal; Sur les suites et familles de fonctions méromorphes de plusieurs variables Ⓣ (1947) and Sur les suites et familles de représentations analytiques du R4 Ⓣ (1947) which study meromorphic functions; and Sur le rayon d'une sphère dont la surface contient une courbe fermée Ⓣ (1948) written in collaboration with Hans Samelson.
He taught in Trogen until 1948 when Eduard Stiefel founded the Institute for Applied Mathematics in ETH. Eduard Stiefel's aim was to build an institute where the mathematical implications of computers could be studied and he employed two assistants, Heinz Rutishauser and Ambros Speiser.
Eduard Stiefel decided that the Institute for Applied Mathematics at ETH should build its own electronic computer. So he sent Heinz Rutishauser and Ambros Speiser on a fact finding visit to the United States in January 1949. Their assignment was to study the state of the art in computing in the United States and then to start a suitable project at the ETH. Rutishauser and Speiser spent most of 1949 at Harvard with Howard Aiken and at Princeton with John von Neumann. They also visited a number of other computer installations, including the ENIAC at Aberdeen, Maryland, and the MARK II at Dahlgren. Returning to ETH in December 1949, Rutishauser discovered that plans had changed since Eduard Stiefel had managed to persuade ETH to rent the use of Konrad Zuse's Z4.
Ambros Speiser explains that the :-
... specifications of the Z4, as seen in 1949, were very convincing for Stiefel, Rutishauser and Speiser. It must be borne in mind that at this time there were hardly a dozen program-controlled computers in operation, all of them in US. Less than a handful were in use for research in numerical mathematics, the others performed routine calculations. There were no doubts that Z4 could be used for serious mathematical research. ... When the Z4 machine was installed, significant work started almost immediately. Within a few years Zürich rose to be one of the foremost centres in numerical analysis. ... The creative spirit that was ever-present, the continuous expression and evaluation of new ideas, the thoroughly based academic knowledge and the sound scientific judgment were daily realities, I am almost tempted to say: This was the air that we were breathing. I can hardly believe, that Stiefel, when he decided to acquire the Z4, would have dared to hope for success of this degree! The Z4 was also extensively used in education. As early in 1951, we offered to students a course in computer programming with practical exercises on the machine. We believe we were the first on the European continent to do so. This should be taken in consideration by those who often criticize that Swiss Universities were late in recognizing the importance of informatics.Rutishauser published two papers in 1950, Über Folgen und Scharen von analytischen und meromorphen Funktionen mehrerer Variabeln, sowie von analytischen Abbildungen Ⓣ which continued his study of meromorphic functions and Eine Konvergenzverbesserung für die Newtonsche Methode Ⓣ which began his study of numerical methods. Also in 1951 the first of a series of four papers by Rutishauser, Eduard Stiefel and Ambros Speiser, Programmgesteuerte digitale Rechengeräte (elektronische Rechenmaschinen) Ⓣ appearing in 1950 and 1951. Herman Goldstine writes:-
In this series of papers the authors discuss in very considerable detail a number of the important mathematical questions that naturally arise in the design of a digital computer. These topics include possible number systems, the questions of "fixed" vs "floating" point and complementation, the arithmetic processes, the grouping of numbers to achieve higher than normal precisions, conversion between number systems, the structure of finite approximation methods, error analysis, programming and coding as well as the physical organs of a machine. In many of these considerations the authors have compared the various points of view expressed by others in the field to give a comprehensive picture of the situation as understood at the present time.He worked towards his habilitation at ETH submitting the dissertation Automatische Rechenplanfertigung Ⓣ in 1951. Herman Goldstine gives the following summary:-
For some time various people have been interested in the problem of mechanizing, insofar as possible, the process of coding. In this paper the author analyses such a system and considers its merits. He establishes a machine procedure for handling the various portions of an arithmetic formula such as parentheses, etc., and discusses how these may be combined to produce codes. He then treats inductive processes and the complications that they entail.In 1955 Rutishauser was appointed extraordinary professor at ETH and, in 1962, he was named Professor of Applied Mathematics. Bauer in  explains Rutishauser's involvement in the development of the language ALGOL:-
[It] was natural that Rutishauser, Samelson ... and I did intensify more and more our cooperation in the area of what became called compiler-building. This had the consequence that we were forced to agree on language constructs, and again the lead of four years in practical programming Rutishauser had, thanks to the Z4, was his great asset. Rutishauser's reputation also had the effect that his appeal at the GaMM-meeting [GaMM is the Gesellschaft für Angewandte Mathematik und Mechanik] in 1955 to create a common programming language found the support of the GaMM president; within the 'Fachausschuss Programmieren' a working group with Zürich, Munich and later Mainz members was established that pushed the language creation. Rutishauser was not less active than Samelson and myself: the first larger working meeting of the group, that meanwhile called itself ZMD-Gruppe, took place in autumn 1957 in Lugano, organized by Rutishauser. At that meeting it was decided, in the name and with the approval of the GaMM, to approach the American ACM and the British Computer Society with the proposal of a joint conference for the creation of an internationally-based programming language for scientific computation. Unfortunately, the BCS did not react; the ACM, however, under its president John Carr III was cheerful and it came to a one-week ACM-GaMM conference in May 1958 at the ETH Zürich, thanks to the support Eduard Stiefel had given. While the English working title was IAL (International Algebraic Language), the name Algorithmic Language ALGOL was already chosen in the publication, edited by A J Perlis and K Samelson, in the newly founded Springer journal 'Numerische Mathematik', Vol 1 (1959). The further course of development of ALGOL saw Heinz Rutishauser also taking part in the conference held January 11-16, 1960 with thirteen experts from the USA, Germany, Switzerland, Netherlands, England, Denmark, and France in Paris. It produced ALGOL 60, a milestone. Rutishauser's share was more than a thirteenth.Of all the algorithms which Rutishauser invented, the most famous if the qd-algorithm. He presented this algorithm in the 74 page booklet Der Quotienten-Differenzen-Algorithmus Ⓣ. G E Forsyth writes:-
This is a comprehensive exposition of the author's QD algorithm. ... The QD algorithm represents a number of computational schemes for doing a surprising number of jobs: e.g., getting all eigenvalues of a matrix from its Schwarz constants, getting the zeros of a polynomial from its coefficients, finding the poles of a function from its power series, obtaining partial fraction representations of functions, and so forth. The present booklet devotes some attention to the problems of doing QD on automatic computers, and belongs in every numerical analysis library.He remained at ETH Zürich for the rest of his life, becoming one of three professors in the Group for Computer Science set up at ETH in 1968. In 1974 this Group became the Computer Science Institute finally becoming the Division of Computer Science in 1981. Rutishauser, however, did not live to see these later developments. Bauer writes in  about Rutishauser's health problems:-
However, his health was not the best. This I became aware of already in 1955 when Rutishauser stayed in our house. In the night of Wednesday, February 16, 1955, he developed a heart problem. Fortunately, the next day it was stabilized, but he was forced to take a longer recreation stay in Klosters. He also recovered rather quickly from a heart infarction in the autumn of 1964, and he was granted a few more years. But when towards the end of the 60's a professorship was open for the Munich Leibniz Computing Center and Rutishauser was chosen by the Ludwig-Maximilians-Universität and the Bayerische Akademie der Wissenschaften, he did not accept. He let me know that his health would not allow him to accept this honourable offer. In 1969 he became seriously ill. He died on November 10, 1970, sitting at his desk, with acute heart failure.
Article by: J J O'Connor and E F Robertson
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