Notes

Journal:
Math. Comp. **5** (1951), 255-258

DOI:
https://doi.org/10.1090/S0025-5718-51-99414-8

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**[2]**See R. V. Southwell,*Relaxation Methods in Engineering Science*. Oxford, 1940, and L. Fox, ``A short account of relaxation methods,''*Quart. Jour. Mech. Appl. Math.*, v. 1, 1948, p. 253-280. MR**0029270 (10:574c)****[3]**Ludwig Seidel, ``Über ein Verfahren, die Gleichungen, auf welche die Methode der kleinsten Quadrate führt, sowie lineare Gleichungen überhaupt, durch successive Annäherung aufzulösen,''*Akad. Wiss., Munich, mat.-nat. Abt. Abhandlungen*, v. 11, 1874, p. 81-108. Although Seidel's process is frequently called the Gauss-Seidel process, I know of no place where Gauss mentions it.**[4]**The letter is in Gauss's*Werke*, v. 9, 1903, p. 278-281. It is probably the letter referred to in the footnote, p. 257, of E. T. Whittaker and G. Robinson,*The Calculus of Observations*, 1st edit., London and Glasgow, 1924, as a source of their example of relaxation (p. 257-8).**[5]**Words within brackets are translations of inserts by L. Krüger, who prepared volume 9 of Gauss's*Werke*.**[7]**The symmetric treatment of the unknowns is an essential idea in this and other letters. Here Gauss mentions only its advantage as a device which sets up a column-sum check to detect errors. In later letters (cf., e.g., Gauss to Gerling, 19 January 1840,*Werke*, vol. 9, p. 250-3) Gauss is convinced that the symmetric treatment of all unknowns yields normal equations whose iterative solution converges significantly faster. The trick is later described by: Christian Ludwig Gerling (recipient of the letter),*Die Ausgleichungsrechnung der practischen Geometrie*. Hamburg and Gotha, 1843 (p. 157-8, p. 163, p. 386, p. 390); by R. Dedekind, ``Gauss in seiner Vorlesung über die Methode der kleinsten Quadrate,''*Festschrift zur Feier des 150-jährigen Bestehen der königlichen Gesellschaft der Wissenschaften zu Göttingen*. Berlin, 1901 (pp. 45-59) and*Gesammelte Mathematische Werke*. V. 2, 1931, p. 293-306; and by R. Zurmühl,*Matrizen*, Berlin, 1950, p. 280-282. (Zurmühl is wrong, however, in stating that the trick will improve the convergence for all badly conditioned systems of equations.) For some discussion of when and why the trick may be expected to improve the convergence of iterative processes for solving linear equations, see George E. Forsythe and Theodore S. Motzkin, ``An extension of Gauss's transformation for improving the condition of systems of linear equations,'' multilithed typescript at the National Bureau of Standards, Los Angeles.**[8]**Gauss is here using a method which relaxers recommend: liquidating ``that residual . . . which requires the largest 'displacement' `` (Fox,*op. cit.*, p. 256).**[9]**Study of the table shows the algorithm to be precisely the relaxation method described by Fox (*op. cit.*, p. 255-6). In comparing Gauss's and Fox's presentations, we note that Gauss uses the method mentioned in note 8, and that he ``liquidates the residuals'' only approximately at each stage, as recommended by Fox (*op. cit.*, p. 257-8) to save unnecessary arithmetic. Gauss does not, however, propose ``under-relaxation'' or ``over-relaxation,'' as Fox does. On the other hand, Gauss's trick mentioned in note 7 is not mentioned by Fox, although it is extremely helpful in many common problems.

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DOI:
https://doi.org/10.1090/S0025-5718-51-99414-8

Article copyright:
© Copyright 1951
American Mathematical Society