Bulletin of the American Mathematical Society

Solving linear algebraic equations can be interesting

George E. Forsythe

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Bull. Amer. Math. Soc. Volume 59, Number 4 (1953), 299-329.

Dates
First available in Project Euclid: 4 July 2007

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http://projecteuclid.org/euclid.bams/1183518018

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MR0056372

Zentralblatt MATH identifier
0050.34603

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Forsythe, George E. Solving linear algebraic equations can be interesting. Bulletin of the American Mathematical Society 59 (1953), no. 4, 299--329. http://projecteuclid.org/euclid.bams/1183518018.


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References

  • 1. A. A. Abramov, On a method of acceleration of iterative processes (Russian) Doklady Akademii Nauk SSSR. vol. 74 (1950) pp. 1051-1052.
  • 2. Shmuel Agmon, The relaxation method for linear inequalities, NAML Report 52-27, National Bureau of Standards, Los Angeles, 1951, multilithed typescript, 23 pp.
  • 3. A. C. Aitken, On Bernoulli's numerical solution of algebraic equations, Proceedings of the Royal Society of Edinburgh vol. 46 (1926) pp. 289-305.
  • 4. A. C. Aitken, Studies in practical mathematics. V. On the iterative solution of a system of linear equations, Proceedings of the Royal Society of Edinburgh, Section A vol. 63 (1950) pp. 52-60. D. G. Aronson, see [26].
  • 5. T. Banachiewicz, Méthode de résolution numérique des équations linéaires, du calcul des déterminants et des inverses, et de réduction des formes quadratiques, Bulletin International de l'Académie Polonaise des Sciences et des Lettres. Classe des Sciences. Mathématiques et Naturelles. Série A. Sciences Mathématiques (1938) pp. 393-404 Also in Cracow observatory reprint 22.
  • 6. V. Bargmann, D. Montgomery, and J. von Neumann, Solution of linear systems of high order, Report prepared for the Bureau of Ordnance (Contract NORD-9596), October 25, 1946, multigraphed, bound, 86 pp.
  • 7. Commandant Benoit, Note sur une méthode de résolution des équations normales provenant de l'application de la méthode des moindres carrés à un système d'équations linéaires en nombre inférieur à celui des inconnues.-Application de la méthode à la résolution d'un système défini d'équations linéaires (Procédé du Commandant Cholesky), Bull. Géodésique no. 2, 1924, pp. 67-77.
  • 8. M. D. Bingham, A new method for obtaining the inverse matrix, Journal of the American Statistical Association vol. 36 (1941) pp. 530-534.
  • 9. M. Š. Birman, Some estimates for the method of steepest descent (Russian), Uspehi Matematičeskih Nauk (N.S.) vol. 5 (1950) no. 3, pp. 152-155. Trans, by C. D. Benster in NBS Report 2007, National Bureau of Standards, Los Angeles, August, 1952.
  • 10. A. N. Black and R. V. Southwell, Relaxation methods applied to engineering problems. II. Basic theory, with application to surveying and to electrical networks and an extension to gyrostatic systems, Proc. Roy. Soc. London. Ser. A. vol. 164 (1938) pp. 447-467.
  • 11. O. Blumenthal, Über die Genauigkeit der Wurzeln linearer Gleichungen, Zeitschrift für Mathematik und Physik vol. 62 (1914) pp. 359-362.
  • 12. E. Bodewig, Bericht über die verschiedenen Methoden zur Lösung eines Systems linearer Gleichungen mit reellen Koeffizienten. I, II, III, IV, V, Neder. Akad. Wetensch., Proc. vol. 50 (1947) pp. 930-941, 1104-1116, 1285-1295, and vol. 51 (1948) pp. 53-64 and 211-219 = Koninklijke Nederlandsche Akademie van Wetenschappen. Indagationes Mathematicae vol. 9 (1947) pp. 441-452, 518-530, 611-621, and vol. 10 (1948) pp. 24-35 and 82-90.
  • 13. H. Boltz, Entwicklungsverfahren zur Ausgleichung geodätischer Netze nach der Methode der kleinsten Quadrate, Veröffentlichen des Preussischen Geodätischen Instituts, N. F. no. 90, Berlin, 1923.
  • 14. A. L. Cauchy, Méthode générale pour la résolution des systèmes d'équations simultanées, C. R. Acad. Sci. Paris vol. 25 (1847) pp. 536-538.
  • 15. Lamberto Cesari, Sulla risoluzioni dei sistemi di equazioni lineari per approssimazioni successive, Extract from Rassegna delle poste, dei telegrafi e dei telefoni vol. 4, 1937, 37 pp. A. R. Collar, see [39].
  • 16. L. Collatz, Graphische und numerische Verfahren, pp. 1-92 of Alwin Walther, Applied mathematics. Part I, FIAT Review of German Science 1939-1946, Office of Military Government, Wiesbaden, 1948, 307 pp.
  • 17. L. Collatz, Über die Konvergenzkriterien bei Iterationsverfahren für lineare Gleichungssysteme, Math. Zeit. vol. 53 (1950) pp. 149-161.
  • 18. John H. Curtiss, Sampling methods applied to differential and difference equations, pp. 87-109 of Proceedings, seminar on scientific computation, November, 1949, New York, International Business Machines Corp., 1950, 109 pp.
  • 19. A. de la Garza, An iterative method for solving systems of linear equations, Report K-731, Carbide and Carbon Chemical Division, Union Carbide and Carbon Corporation, K-25 Plant, Oak Ridge, Tennessee, February 15, 1951, mimeographed, 15 pp. W. J. Duncan, see [39].
  • 20. Paul S. Dwyer, Linear computations, New York, Wiley, 1951, 344 pp.
  • 21. Eckert-Mauchly Division of Remington-Rand Corporation, Matrix algebra programs for the UNIVAC, Philadelphia, 1951, typescript, 11 pp.
  • 22. Engineering Research Associates, Highspeed computing devices, New York, McGraw-Hill, 1950, 451 pp. (Supervisors: C. B. Tompkins and J. H. Wakelin; editor: W. W. Stifler, Jr.)
  • 23. V. N. Faddeeva, Vyčislitel'nye metody lineĭnoĭ algebry (Computational methods of linear algebra) (Russian), Moscow-Leningrad, 1950, 240 pp. (A translation is being prepared at the National Bureau of Standards, Los Angeles.)
  • 24. Donald A. Flanders and George Shortley, Numerical determination of fundamental modes, Journal of Applied Physics vol. 21 (1950) pp. 1326-1332.
  • 25. A. I. and G. E. Forsythe, Punched-card experiments with accelerated gradient methods for linear equations, NBS Report 1643, National Bureau of Standards, Los Angeles, March 1952, multilithed typescript, 29 pp. To appear in the National Bureau of Standards Applied Mathematics Series.
  • 26. George E. Forsythe, Theory of selected methods of finite matrix inversion and decomposition, Lecture notes by D. G. Aronson and K. E. Iverson, INA Report 52-5, National Bureau of Standards, Los Angeles, 1951, multilithed typescript, 93 pp.
  • 27. George E. Forsythe, Tentative classification of methods and bibliography on solving systems of linear equations, INA Report 52-7, National Bureau of Standards, Los Angeles, 1951, multilithed typescript, 78 pp. To appear in [87].
  • 28. George E. Forsythe and Richard A. Leibler, Matrix inversion by a Monte Carlo method, Mathematical Tables and Other Aids to Computation vol. 4 (1950) pp. 127-129 and vol. 5 (1951) p. 55.
  • 29. G. E. Forsythe and T. S. Motzkin, Asymptotic properties of the optimum gradient method, Bull. Amer. Math. Soc. Abstract 57-3-231.
  • 30. G. E. Forsythe and T. S. Motzkin, Acceleration of the optimum gradient method. Preliminary report, Bull. Amer. Math. Soc. Abstract 57-4-392.
  • 31. G. E. Forsythe and T. S. Motzkin, An extension of Gauss' transformation for improving the condition of systems of linear equations, Mathematical Tables and Other Aids to Computation vol. 6 (1952) pp. 9-17.
  • 32. L. Fox, A short account of relaxation methods, The Quarterly Journal of Mechanics and Applied Mathematics vol. 1 (1948) pp. 253-280.
  • 33. L. Fox, Practical solution of linear equations and inversion of matrices, manuscript, about 60 pp. To appear in the National Bureau of Standards Applied Mathematics Series.
  • 34. L. Fox, Practical methods for the solution of linear equations and the inversion of matrices, Journal of the Royal Statistical Society Series B. vol. 12 (1950) pp. 120-136.
  • 35. L. Fox, H. D. Huskey, and J. H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, The Quarterly Journal of Mechanics and Applied Mathematics vol. 1 (1948) pp. 149-173.
  • 36. J. S. Frame, Machines for solving algebraic equations, Mathematical Tables and Other Aids to Computation vol. 1 (1945) pp. 337-353.
  • 37. [Stanley P. Frankel], Bibliography on computing machines, Analysis Laboratory, California Institute of Technology, Pasadena, October 1949, hectographed typescript, 44 pp. (Frankel's name not on copy.)
  • 38. Stanley P. Frankel, Convergence rates of iterative treatments of partial differential equations, Mathematical Tables and Other Aids to Computation vol. 4 (1950) pp. 65-75.
  • 38a. Ann D. Franklin and Eric V. Hankam, Bibliography on the use of IBM machines in science, statistics, and education, New York, International Business Machines Corporation, 1952, 50 pp. See also [62].
  • 39. R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementary matrices and some applications to dynamics and differential equations, Cambridge University Press, 1938, 416 pp.
  • 40. Konrad Friedrich and Werner Jenne, Geometrisch-anschauliche Auflösung linearer mit Nullkoeffizienten ausgestatteter Gleichungssysteme, Deutsche Akad. Wiss. Berlin, Veröff. Geodät. Inst. Potsdam, no. 5, 1951, 68 pp.
  • 41. C. F. Gauss, Letter to Gerling, December 26, 1823, Werke, Göttingen, vol. 9, pp. 278-281. Trans, by G. E. Forsythe, Mathematical Tables and Other Aids to Computation vol. 5 (1951) pp. 255-258.
  • 42. C. F. Gauss, Supplementum theoriae combinationis observationum erroribus minimis obnoxiae, 1826, Werke, Göttingen, vol. 4, pp. 55-93.
  • 43. M. K. Gavurin, Application of polynomials of best approximation to improving the convergence of iterative processes (Russian), Uspehi Matematičeskih Nauk (N.S.) vol. 5 (1950) no. 3, pp. 156-160. Trans. by C. D. Benster in NBS Report 2007, National Bureau of Standards, Los Angeles, August, 1952.
  • 44. Hilda Geiringer, On the solution of systems of linear equations by certain iteration methods, pp. 365-393 of Reissner Anniversary Volume, Contributions to applied mechanics, Ann Arbor, Mich., Edwards, 1949. Hilda Geiringer, see also [123].
  • 45. Herman H. Goldstine and John von Neumann, Numerical inverting of matrices of high order. II, Proceedings of the American Mathematical Society vol. 2 (1951) pp. 188-202. Part I is [124]. Eric V. Hankam, see [38a].
  • 46. D. R. Hartree, Experimental arithmetic, Eureka vol. 10 (1948) pp. 13-18.
  • 47. Harvard Computation Laboratory, A manual of operation for the automatic sequence controlled calculator, Harvard University Press, 1946, 561 pp.
  • 48. R. M. Hayes, Iterative methods of solving linear problems on Hilbert space, Ph.D. Thesis, University of California, Los Angeles, 1952, multilithed typescript, 62 pp. To appear in the National Bureau of Standards Applied Mathematics Series.
  • 49. A. Hertwig, Die Lösung linearer Gleichungen durch unendliche Reihen und ihre Anwendungen auf die Berechnung hochgradig statisch unbestimmter Systeme, pp. 37-59 of Festschrift für H. Müller-Breslau, Leipzig, 1912.
  • 50. M. R. Hestenes, Iterative methods for solving linear equations, NAML Report 52-9, National Bureau of Standards, Los Angeles, 1951, multilithed typescript, 19 pp.
  • 51. M. R. Hestenes, Urs Hochstrasser, and L. S. Wilson, Some numerical examples on solving systems of linear equations by the conjugate gradient method for non-symmetric systems of equations, National Bureau of Standards, Los Angeles, multilithed typescript, August 21, 1952, 34 pp.
  • 52. Magnus R. Hestenes and William Karush, A method of gradients for the calculation of the characteristic roots and vectors of a real symmetric matrix, Journal of Research of the National Bureau of Standards vol. 47 (1951) pp. 45-61.
  • 53. Magnus R. Hestenes and Marvin L. Stein, The solution of linear equations by minimization, NAML Report 52-45, National Bureau of Standards, Los Angeles, December 12, 1951, multilithed typescript, 35 pp.
  • 54. Magnus R. Hestenes and Eduard Stiefel, Method of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards vol. 49 (1952) pp. 409-436.
  • 55. Thomas J. Higgins, A survey of the approximate solution of two-dimensional physical problems by variational methods and finite difference procedures, pp. 169-198 of L. E. Grinter (editor), Numerical methods of analysis in engineering, New York, MacMillan, 1949, 207 pp.
  • 56. [Urs] Hochstrasser, Die Berechnung der Verschiebungen in einer parallelogrammförmigen Scheibe, Bericht S-17, Techn. Abteilung, Eidgenössisches Flugzeugwerk, Emmen, Switzerland, multigraphed typescript, 21 pp.
  • 57. Urs Hochstrasser, The solution of systems of linear equations by the conjugate gradient method for use on IBM equipment, National Bureau of Standards, Los Angeles, manuscript. Urs Hochstrasser, see also [51 ].
  • 58. Harold Hotelling, Some new methods in matrix calculation, Ann. Math. Statist. vol. 14 (1943) pp. 1-33 and 440-441.
  • 59. A. S. Householder, Principles of numerical analysis, New York, McGraw-Hill, to appear in 1953.
  • 60. A. S. Householder, Some numerical methods for solving systems of linear equations. Amer. Math. Monthly vol. 57 (1950) pp. 453-459.
  • 61. H. D. Huskey, Characteristics of the Institute for Numerical Analysis computer, Mathematical Tables and Other Aids to Computation vol. 4 (1950) pp. 103-108. H. D. Huskey, see also [35].
  • 62. International Business Machines Corp., Bibliography on the use of IBM machines in science, statistics, and education, New York, 1950. Apparently superseded by [38a]. K. E. Iverson, see [26].
  • 63. C. G. J. Jacobi, Ueber eine neue Auflösungsart der bei der Methode der kleinsten Quadrate vorkommenden lineären Gleichungen, Astronomische Nachrichten vol. 22 (1845) no. 523, pp. 297-306. (Also in Jacobi's Werke, vol. 3, p. 467?.)
  • 64. M. Janet, Sur les systèmes d'équations aux dérivées partielles, C. R. Acad. Sci. Paris vol. 170 (1920) pp. 1101-1103 and J. Math. Pures Appl. (8) vol. 3 (1920) pp. 65-151. Werner Jenne, see [40].
  • 65. Henry Jensen, An attempt at a systematic classification of some methods for the solution of normal equations, Meddelelse No, 18, Geodaetisk Institut, Copenhagen, 1944, 45 pp.
  • 66. Enno Jürgens, Zur Auflösung linearer Gleichungssysteme und numerischen Berechnung von Determinanten, Aachen, Palm (printer), 1886, 20 pp.
  • 67. S. Kaczmarz, Angenäherte Auflösung von Systemen linearer Gleichungen, Bulletin International de l'Académie Polonaise des Sciences et des Lettres. Classe des Sciences Mathématiques et Naturelles. Série A. Sciences Mathématiques (1937) pp. 355-357.
  • 68. L. V. Kantorovič, Functional analysis and applied mathematics (Russian), Uspehi Matematičeskih Nauk (N.S.) vol. 3 (1948) no. 6, pp. 89-185. (Trans. in NBS Report 1509, National Bureau of Standards, Los Angeles, March 7, 1952, multilithed typescript, 202 pp.)
  • 69. L. V. Kantorovič and V. I. Krylov, Približennye metody vysšego analiza (Approximate methods of higher analysis) (Russian), 3d ed., Moscow-Leningrad, 1950, 695 pp.
  • 70. W. Karush, Convergence of a method of solving linear problems, Proceedings of the American Mathematical Society vol. 3 (1952) pp. 839-851. W. Karush, see also [52].
  • 70a. M. A. Krasnosel'skiĭ and S. G. Kreĭn, The iterative process with minimal residuals (Russian), Matematičeskiĭ Sbornik (N.S.) vol. 31 (73) (1952) pp. 315-334. V. I. Krylov, see [69].
  • 71. A. G. Kuroš, A. I. Markuševič, and P. K. Raševskiĭ, Matematika v SSSR za tridcat' let 1917-1947 (Mathematics in the USSR in the thirty years 1917-1947) (Russian), Moscow-Leningrad, 1948, 1044 pp. (pp. 759-857 on numerical and graphical methods contain a survey and comprehensive bibliography of Russian work).
  • 72. Cornelius Lanczos, Solution of systems of linear equations by minimized iterations, Journal of Research of the National Bureau of Standards vol. 49 (1952) pp. 33-53.
  • 73. Cornelius Lanczos, Chebyshev polynomials in the solution of large-scale linear systems, pp. 124-183 of the Proceedings of the Association for Computing Machinery, meeting at Toronto September 8-10, 1952, Washington, D.C., Sauls Lithograph Co. 1953?, 160 pp.
  • 74. H. Liebmann, Die angenäherte Ermittlung harmonischer Funktionen und konformer Abbildung, Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Abteilung der Bayerischen Akademie der Wissenschaften zu München. Physikalisch-Mathematische Klasse vol. 47 (1918) pp. 385-416.
  • 75. A. T. Lonseth, The propagation of error in linear problems, Trans. Amer. Math. Soc. vol. 62 (1947) pp. 193-212.
  • 76. Samuel Lubkin, A method of summing infinite series, Journal of Research of the National Bureau of Standards vol. 48 (1952) pp. 228-254.
  • 77. L. A. Lyusternik, Remarks on the numerical solution of boundary problems for Laplace's equation and the calculation of characteristic values by the method of nets (Russian), Trudy Matematičeskogo instituta imeni V. A. Steklova vol. 20 (1947) pp. 49-64.
  • 78. C. C. MacDuffee, The theory of matrices, New York, Chelsea, 1946, 110 pp.
  • 79. Wladimir Markoff, Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen, Math. Ann. vol. 77 (1916) pp. 213-258. (Translation and condensation by J. Grossman of Russian article published in 1892.) A. I. Markuševič, see [71 ].
  • 80. S. N. Mergelyan, Uniform approximation of functions in the complex plane (Russian), Uspehi Matematičeskih Nauk (N.S.) vol. 7 (1952) no. 2, pp. 31-122. Trans, in Amer. Math. Soc. Translation T-115.
  • 81. William Edmund Milne, Numerical calculus, Princeton University Press, 1949, 393 pp. D. Montgomery, see [6].
  • 82. Theodore S. Motzkin, Bibliography on linear inequalities, linear programming, game strategy, economic behavior, and statistical decision functions, in preparation for probable issue by the National Bureau of Standards, Los Angeles. Theodore S. Motzkin, see also [29; 30; 31; 83].
  • 83. T. S. Motzkin and I. J. Schoenberg, On the relaxation method for linear inequalities, NBS Report 1881, National Bureau of Standards, Los Angeles, August, 1952, multilithed typescript, 21 pp.
  • 84. F. R. Moulton, On the solutions of linear equations having small determinants, Amer. Math. Monthly vol. 20 (1913) pp. 242-249.
  • 85. H. P. Mulholland, On the distribution of a convex even function of several independent rounding-off errors, Proceedings of the American Mathematical Society vol. 3 (1952) pp. 310-321.
  • 86. Francis J. Murray, The theory of mathematical machines, rev. ed., New York, King's Crown Press, 1948, 139 pp.
  • 87. National Bureau of Standards, Simultaneous linear equations and the determination of eigenvalues, Applied mathematics series, vol. 29, U.S. Govt. Printing Office, in press.
  • 88. P. A. Nekrasov, Determination of the unknowns by the method of least squares for very many unknowns (Russian), Maternatičeskiĭ Sbornik vol. 12 (1884) pp. 189-204.
  • 89. Alexandre Ostrowski, Sur la détermination des bornes inférieures pour une classe des déterminants, Bull. Sci. Math. (2) vol. 61 (1937) pp. 19-32.
  • 90. Alexandre Ostrowski, Sur la variation de la matrice inverse d'une matrice donnée, C. R. Acad. Sci. Paris vol. 231 (1950) pp. 1019-1021.
  • 91. Alexandre Ostrowski, Determinants with dominant principal diagonal and the absolute convergence of linear iteration processes (text in German), NBS Report 1727, National Bureau of Standards, Washington, June 1, 1952, multilithed typescript, 49 pp.
  • 92. Alexandre Ostrowski, On the convergence of cyclic linear iterations for symmetric and nearly symmetric matrices. II, NBS Report 1759, National Bureau of Standards, Los Angeles, June 26, 1952, multilithed typescript, 5 pp.
  • 93. Alexandre Ostrowski, On the linear iteration procedures for symmetric matrices, NBS Report 1844, National Bureau of Standards, Los Angeles, August 4, 1952, multilithed typescript, 29 pp.
  • 94. Sam Pedis, Theory of matrices, Cambridge, Mass., Addison-Wesley, 1952, 237 pp.
  • 95. P. Pizzetti, Sulla compensazione delle osservazioni secondo il metodo dei minimi quadrati, nota I, II, Rendiconti della Reale Accademia Nazionale dei Lincei, Rome (4) vol. 3 (1887) pp. 230-235 and 288-293. Hilda Pollaczek-Geiringer, see [44] and [123]. P. K. Raševskiĭ, see [71 ].
  • 96. Edgar Reich, The solution of linear algebraic equations by successive approximations, Memorandum M-565, Servomechanisms Laboratory, Massachusetts Institute of Technology, Cambridge, August 5, 1948, hectographed, 36 pp.
  • 97. Edgar Reich, On the convergence of the classical iterative method of solving linear simultaneous equations, Ann. Math. Statist, vol. 20 (1949) pp. 448-451.
  • 98. L. F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philos. Trans. Roy. Soc. London. Ser. A vol. 210 (1910) pp. 307-357.
  • 99. Maria Sofia Roma, Sulla risoluzione numerica dei sistemi di equazioni algebriche lineari col metodo della ortogonalizzazione, La Riccera Scientifica vol. 20 (1950) pp. 1288-1290; Consiglio Nazionale delle Ricerche. Pubblicazioni dell'Istituto per le Applicazioni del Calcolo, no. 283. R. L. Rosenberg, see [113].
  • 100. J. Barkley Rosser, Rapidly converging iterative methods for solving linear equations. To appear in [87].
  • 101. Paul A. Samuelson, A convergent iterative process, Journal of Mathematics and Physics vol. 24 (1945) pp. 131-134.
  • 102. F. E. Satterthwaite, Error control in matrix calculation, Ann. Math. Statist, vol. 15 (1944) pp. 373-387. I. J. Schoenberg, see [83].
  • 103. E. Schröder, Über unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Ann. vol. 2 (1870) pp. 317-365.
  • 104. G. Schulz, Iterative Berechnung der reziproken Matrix, Zeitschrift für Angewandte Mathematik und Mechanik vol. 13 (1933) pp. 57-59.
  • 105. I. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. Reine Angew. Math. vol. 147 (1917) pp. 205-232.
  • 106. [Hans Schwerdtfeger], Bibliography on iteration [reproduced at National Bureau of Standards, Los Angeles, 1951], multilithed typescript, 13 pp. (Bracketed material not on copy.)
  • 107. Ludwig Seidel, Ueber ein Verfahren, die Gleichungen, auf welche die Methode der kleinsten Quadrate führt, sowie lineäre Gleichungen überhaupt, durch successive Annäherung aufzulösen, Abhandlungen der Bayerischen Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Abteilung vol. 11 (1874) no. 3, pp. 81-108.
  • 108. Daniel Shanks, An analogy between transients and mathematical sequences and some nonlinear sequence-to-sequence transforms suggested by it. Part I, NOLM 9994, Naval Ordnance Laboratory, Silver Spring, Maryland, July 26, 1949, multilithed typescript, 42 pp. George Shortley, see [24].
  • 109. R. V. Southwell, Relaxation methods in engineering science, a treatise on approximate computation, Oxford University Press, 1940, 252 pp.
  • 110. R. V. Southwell, Relaxation methods in theoretical physics, Oxford University Press, 1946, 248 pp. R. V. Southwell, see also [10].
  • 111. J. L. Stearn, Iterative solutions of normal equations, Bull. Géodésique (1951) pp. 331-339.
  • 112. Marvin L. Stein, Gradient methods in the solution of systems of linear equations, Journal of Research of the National Bureau of Standards vol. 48 (1952) pp. 407-413. Marvin L. Stein, see also [53].
  • 113. P. Stein and R. L. Rosenberg, On the solution of linear simultaneous equations by iteration, J. London Math. Soc. vol. 23 (1948) pp. 111-118.
  • 114. E. Stiefel, Über einige Methoden der Relaxationsrechnung, Zeitschrift für Angewandte Mathematik und Physik vol. 3 (1952) pp. 1-33. E. Stiefel, see also [54]. W. W. Stifler, Jr., see [22].
  • 115. O. Taussky, Note on the condition of matrices, Mathematical Tables and Other Aids to Computation vol. 4 (1950) pp. 111-112.
  • 116. Olga Taussky, Bibliography on bounds for characteristic roots of finite matrices, NBS Report 1162, National Bureau of Standards, Washington, September, 1951, multilithed typescript, 10 pp.
  • 117. Olga Taussky and John Todd, Systems of equations, matrices and determinants, Mathematics Magazine vol. 26 (1952) pp. 9-20 and 71-88.
  • 118. G. Temple, The general theory of relaxation methods applied to linear systems, Proc. Roy. Soc. London Ser. A. vol. 169 (1939) pp. 476-500.
  • 119. John Todd, The condition of a certain matrix, Proc. Cambridge Philos. Soc. vol. 46 (1949) pp. 116-118. John Todd, see also [117].
  • 120. C. Tompkins, Projection methods in calculation of some linear problems, [1949?], multilithed typescript, 52 pp., part of Engineering Research Associates, Logistics papers, issue no. IV, Appendix I to Bimonthly Progress Report No. 17, Contract N6onr-240, Task Order I, Office of Naval Research, Project NR 047 010. C. Tompkins, see also [22].
  • 121. L. B. Tuckerman, On the mathematically significant figures in the solution of simultaneous linear equations, Ann. Math. Statist, vol. 12 (1941) pp. 307-316.
  • 122. A. M. Turing, Rounding-off errors in matrix processes, The Quarterly Journal of Mechanics and Applied Mathematics vol. 1 (1948) pp. 287-308.
  • 123. R. von Mises and Hilda Pollaczek-Geiringer, Praktische Verfahren der Gleichungsauflösung, Zeitschrift für Angewandte Mathematik und Mechanik vol. 9 (1929) pp. 58-77 and 152-164. J. von Neumann, see [6] and [45].
  • 124. John von Neumann and H. H. Goldstine, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. vol. 53 (1947) pp. 1021-1099. Part II is [45]. J. H. Wakelin, see [22].
  • 125. Wolfgang Wasow, A note on the inversion of matrices by random walks, Mathematical Tables and Other Aids to Computation vol. 6 (1952) pp. 78-81. J. H. Wilkinson, see [35]. L. S. Wilson, see [51].
  • 126. Helmut Wittmeyer, Einfluss der Änderung einer Matrix auf die Lösung des zugehörigen Gleichungssystems, sowie auf die characteristischen Zahlen und die Eigenvectoren, Zeitschrift für Angewandte Mathematik und Mechanik vol. 16 (1936) pp. 287-300.
  • 127. Helmut Wittmeyer, Über die Lösung von linearen Gleichungssystemen durch Iteration, Zeitschrift für Angewandte Mathematik und Mechanik vol. 16 (1936) pp. 301-310.
  • 128. Max A. Woodbury, Inverting modified matrices, Memorandum Report 42, Princeton, Statistical Research Group, June 14, 1950, hectographed, 4 pp.
  • 129. David M. Young, Jr., Iterative methods for solving partial difference equations of elliptic type, Ph. D. Thesis, Harvard University Mathematics Department, 1950, blueprint, c. 100 pp.
  • 130. S. I. Zuhovickiĭ, An algorithm for the solution of the Čebyšev approximation problem in the case of a finite system of incompatible linear equations (Russian), Doklady Akademii Nauk SSSR. vol. 79 (1951) pp. 561-564.
  • 131. Rudolf Zurmühl, Matrizen. Eine Darstellung für Ingenieure, Berlin-Göttingen-Heidelberg, Springer, 1950, 427 pp.