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A letter of C.F. Gauß to C.L. Gerling – least error squares and the Gauß-Seidel method. (Ein Brief von C.F. Gauß an C.L. Gerling – kleinste Fehlerquadrate und das Gauß-Seidel-verfahren.) (German) Zbl 1437.01012

This paper is concerned with a letter written by Gauss to Gerling which was published in its entirety in [C. Schaefer, Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling. Berlin: O. Elsner (1927; JFM 53.0023.03)]. The author of the paper under review considers three parts, namely a first part of four pages which does not need any mathematical explanation, a second part of two pages in which the Gauss-Seidel method is explained by means of an example, and a third part of also two pages concerning underdetermined linear systems. An excerpt of the second part enjoyed widespread renown due to an English translation by Forsyth, however, the full content of the letter remained largely unknown.
The author has painstakingly researched the material around the letter and gives a thorough mathematical explanation of its content. An interesting and detailed account of the sources considered is shown at the end of the paper. In the eye of the reviewer, the most relevant content of the letter is a second kind of least square method based on the method of Lagrange multipliers. However, even in the part concerning the Gauss-Seidel method new ingredients have been found, so the use of a slack variable and – not surprisingly – ingenious methods of computation like the iteration of the defects instead of the actual solution vector. As the author himself states on page 59 the concentration of so many mathematical ideas on only four pages renders the letter a very interesting document in the history of mathematics.
The mathematical expositions in this paper are complete and intelligibly presented. In fact, this paper should be read by everyone interested in the history of mathematics, the development of linear algebra, or in Gauss. The paper is written in German.

MSC:

01A55 History of mathematics in the 19th century
65-03 History of numerical analysis

Citations:

JFM 53.0023.03
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Full Text: DOI

References:

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