An extension of Gauss’ transformation for improving the condition of systems of linear equations
HTML articles powered by AMS MathViewer
- by George E. Forsythe and Theodore S. Motzkin PDF
- Math. Comp. 6 (1952), 9-17 Request permission
Corrigendum: Math. Comp. 6 (1952), 126.
References
- N. Aronszajn, Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues. I. Operators in a Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 474–480. MR 27955, DOI 10.1073/pnas.34.10.474 N. Aronszajn, “Escalator and modified escalator methods” (unpublished manuscript, 1945, 16 pp.). E. Bodewig, “Bericht über die verschiedenen Methoden zur Lösung eines Systems linearen Gleichungen mit reellen Koeffizienten. III,” Akad. Wetensch., Amsterdam, Proc., v. 50, 1947, p. 1285-1295 = Indagationes Math., v. 9, 1947, p. 611-621. Lamberto Cesari, “Sulla risoluzioni dei sistemi di equazioni lineari per approssimazioni successive,” Reale Accad. dei Lincei, Classe scienze fis., mat., natur., Rendic., v. 25, s. 6a, 1937, p. 422-128. 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, p. 45-59. George E. Forsythe & Theodore S. Motzkin, “Acceleration of the optimum gradient method. Preliminary report,” [Abstract] Amer. Math. Soc., Bull., v. 57, 1951, p. 304-305.
- L. Fox, A short account of relaxation methods, Quart. J. Mech. Appl. Math. 1 (1948), 253–280. MR 29270, DOI 10.1093/qjmam/1.1.253
- L. Fox, Escalator methods for latent roots, Quart. J. Mech. Appl. Math. 5 (1952), 178–190. MR 48912, DOI 10.1093/qjmam/5.2.178 C. F. Gauss, “Letter to Gerling, 26 December 1823,” Werke v. 9, p. 278-281. For an annotated translation of Gauss’ letter by G. E. Forsythe, see MTAC, v. 5, p. 155-258. C. F. Gauss, “Letter to Gerling, 19 January 1840,” Werke v. 9, p. 250-253. Ernst A. Guillemin, Communications Networks. V. 2, New York, 1935, p. 187. C. G. J. Jacobi, “Ueber eine neue Auflösungsart der bei der Methode der kleinsten Quadrate vorkommenden linearen Gleichungen,” Astronomische Nachrichten, v. 22, 1845, no. 523, cols. 297-306.
- L. V. Kantorovič, Functional analysis and applied mathematics, Uspehi Matem. Nauk (N.S.) 3 (1948), no. 6(28), 89–185 (Russian). MR 0027947 Joseph Morris, The Escalator Method, New York, 1947, p. 111-112.
- Th. Motzkin, From among $n$ conjugate algebraic integers, $n-1$ can be approximately given, Bull. Amer. Math. Soc. 53 (1947), 156–162. MR 19653, DOI 10.1090/S0002-9904-1947-08772-3
- John von Neumann and H. H. Goldstine, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. 53 (1947), 1021–1099. MR 24235, DOI 10.1090/S0002-9904-1947-08909-6 Ludwig Seidel, “Ueber ein Verfahren, die Gleichungen, auf welche die Methode der kleinsten Quadrate führt, sowie lineäre Gleichungen überhaupt, durch successive Annäherung aufzulösen,” Akad. Wiss., Munich, mat.-nat. Abt., Abhandlungen, v. 11, no. 3, 1874, p. 81-108.
- John Todd, The condition of a certain matrix, Proc. Cambridge Philos. Soc. 46 (1950), 116–118. MR 33192, DOI 10.1017/S0305004100025536
- Ledyard R. Tucker, The determination of successive principal components without computation of tables of residual correlation coefficients, Psychometrika 9 (1944), 149–153. MR 10663, DOI 10.1007/BF02288719
- Alexander Weinstein, Separation theorems for the eigenvalues of partial differential equations, Reissner Anniversary Volume, Contributions to Applied Mechanics, J. W. Edwards, Ann Arbor, Michigan, 1948, pp. 404–414. MR 0029465 E. T. Whittaker & G. N. Watson, A Course of Modern Analysis. American edition, New York, 1943, p. 547. R. Zurmühl, Matrizen. Eine Darstellung für Ingenieure. Berlin, 1949, p. 280-282.
Additional Information
- © Copyright 1952 American Mathematical Society
- Journal: Math. Comp. 6 (1952), 9-17
- MSC: Primary 65.0X
- DOI: https://doi.org/10.1090/S0025-5718-1952-0048162-0
- MathSciNet review: 0048162