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The Solution of Simultaneous Equations

Published online by Cambridge University Press:  01 January 2025

P. S. Dwyer
P. S. Dwyer*
Affiliation:
University of Michigan
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Abstract

This paper is an attempt to integrate the various methods which have been developed for the numerical solution of simultaneous linear equations. It is demonstrated that many of the common methods, including the Doolittle Method, are variations of the method of “single division.” The most useful variation of this method, in case symmetry is present, appears to be the Abbreviated Doolittle method. The method of multiplication and subtraction likewise can be abbreviated in various ways of which the most satisfactory form appears to be the new Compact method. These methods are then applied to such problems as the solution of related equations, the solution of groups of equations, and the evaluation of the inverse of a matrix.

Information

Type
Original Paper
Information
Psychometrika , Volume 6 , Issue 2 , April 1941 , pp. 101 - 129
Copyright
Copyright © 1941 The Psychometric Society

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References

Salisbury, F. S. A simplified method of computing multiple correlation constants. J. educ. Psych., 1929, 20, 44–52.CrossRefGoogle Scholar
Griffin, H. D. On partial correlation versus partial regression for obtaining the multiple regression equations. J. educ. Psych., 1931, 22, 35–44.CrossRefGoogle Scholar
Thompson, G. H. Some points of mathematical technique in the factorial analysis of ability. J. educ. Psych., 1936, 27, 37–54.CrossRefGoogle Scholar
Wren, F. L. The calculation of partial and multiple coefficients of regression and correlation. J. educ. Psych., 1938, 29, 695–700.CrossRefGoogle Scholar
Huntington, E. V. Curve-fitting by the method of least squares and the method of moments. Handbook of mathematical statistics. Chap. IV. New York: Houghton Mifflin Co.Google Scholar
Brolyer, C. R. Another short method for calculating β weights and the resulting multipleR on a computing machine. J. gen. Psych., 1933, 8, 278–281.CrossRefGoogle Scholar
Chauncey, Henr. A method for solving simultaneous equations, with particular reference to multiple correlation problems. Harvard educ. Rev., 1939, 9, 63–68.Google Scholar
Deming, H. G. A systematic method for the solution of simultaneous linear equations. Amer. math. Monthly, 1928, 35, 360–363.CrossRefGoogle Scholar
Aitken, A. C. Studies in practical mathematics. I. The evaluation, with applications, of a certain triple product matrix. Proc. Royal Soc., Edinburgh, 1937, 57, 172–181.CrossRefGoogle Scholar
Doolittle, M. H. Method employed in the solution of normal equations and the adjustment of a triangulation. U. S. Coast and Geodetic Survey Report, 1878, 115–120.Google Scholar
Waugh, F. V. A simplified method of determining multiple regression constants. J. Amer. Stat. Assoc., 1935, 30, 694–700.CrossRefGoogle Scholar
Kurtz, A. K. The use of the Doolittle method in obtaining related multiple correlation coefficients. Psychometrika, 1936, 1, 45–51.CrossRefGoogle Scholar
Tolley, H. R. and Ezekial, Mordeca. The Doolittle method for solving multiple correlation equations versus the Kelley-Salisbury iteration method. J. Amer. Stat. Assoc., 1927, 22, 497–500.Google Scholar
Kelley, T. L. and McNemar, O. Doolittle versus Kelley-Salisbury methods for computing multiple regression coefficients. J. Amer. Stat. Assoc., 1929, 24, 164–169.CrossRefGoogle Scholar
Rider, P. R. An introduction to modern statistical methods, New York: John Wiley and Sons, 1939.Google Scholar
Fisher, R. A. Statistical methods for research workers Fifth Edition, London: Oliver and Boyd, 1934.Google Scholar
Dwyer, P. S. The simultaneous computation of groups of regression equations and associated multiple correlation coefficients. Annals math. Stat., 1937, 8, 224–231.CrossRefGoogle Scholar
Horst, Pau. A short method for solving for a coefficient of multiple correlation. Annals math. Stat., 1932, 3, 40–44.CrossRefGoogle Scholar
Horst, Pau. A general method for evaluating multiple regression constants. J. Amer. Stat. Assoc., 1932, 27, 270–278.CrossRefGoogle Scholar
Dunlap, J. W. Workbook in statistical method, New York: Prentice Hall, Inc., 1939.Google Scholar
Tucker, L. R. A method for finding the inverse of a matrix. Psychometrika, 1938, 3, 189–197.CrossRefGoogle Scholar
Gauss, C. F. Supplementum theoriae combinationis observationum erroribus minimis obnoxiae. Werke, Band 15, 1873.Google Scholar
Schott, C. A.The solution of normal equations by indirect elimination. U. S. Coast and Geodetic Survey Report, 1855, 255–264.Google Scholar
Carr, G. S. A synopsis of elementary results in pure mathematics. London: Francis Hodgson. 1886, 42–42.Google Scholar
Merriman, M. Text book in method of least squares. New York: John Wiley and Sons. 1884, 51–59.Google Scholar
Whittaker, E. T. and Robinson, G. The calculus of observations. London: Blackie and Son. 1924, 231–239.Google Scholar
Wren, F. L. Neo-Sylvester contractions and the solution of systems of linear equations. Bull. Amer. math Soc., 1937, 43, 823–834.CrossRefGoogle Scholar
Wherry, R. J. A modification of the Doolittle method: A logarithmic solution. J. educ. Psych., 1932, 23, 455–459.CrossRefGoogle Scholar