Werner Romberg Vereinfachte numerische Integration

Abstract

Werner Romberg (1909–2003) was born in Berlin and got his university education at Heidelberg (1928–1930), and Munich (1930-1933). The supervisor of his graduate studies was Arnold Sommerfeld. Being critical to the Nazi regime in Germany he had to leave the country just after his PhD defense in 1933. He spent some years in the Soviet Union and Czechoslovakia before he came to Norway in 1938, much thanks to Professor Hylleraas at the University of Oslo. He worked as an assistant for Hylleraas until 1949, only interrupted by a stay in Uppsala, Sweden during the second world war. In 1949 Romberg got a position as "dosent" in physics at the Norwegian Institute of Technology (NTH) in Trondheim and established an education program in mathematical physics. It was in these years he wrote the famous contribution in DKNVS Forhandlinger. In 1960, a chair in applied mathematics was established at NTH, and Romberg’s experience with computers toghether with his broad scientific background made him a good candidate for the position, which he was offered and accepted. In his new position, Romberg gave courses in many areas of applied mathematics, and he established the field of numerical analysis at NTH. In 1968 a professorship dedicated to him was established in Heidelberg, and Romberg left Norway for his new post which he held until he retired in 1978.

In his celebrated 1955 paper, Romberg considered numerical approximations of the definite integral. His idea was to accelerate the convergence of the composite trapezoidal rule by means of extrapolation. Although similar methods were already known at the time, going back to Colin Maclaurin, 1742, Romberg developed a more systematic approach. His method consisted in computing composite trapezoidal approximations by successively halving the step size, and he derived a simple formula for eliminating the terms in the error expansion, combining the computed approximations. An added benefit from the halving strategy was that the total cost of the method became essentially the same as in the more accurate trapezoidal calculation. The algorithm of Romberg became widely known after Jean-Pierre Laurent presented a rigorous analysis of the method in 1963, identifying it as a special case of the more general extrapolation technique of Richardson.