Modification methods for inverting matrices and solving systems of linear algebraic equations
HTML articles powered by AMS MathViewer
- by D. Goldfarb PDF
- Math. Comp. 26 (1972), 829-852 Request permission
Abstract:
Modification methods for inverting matrices and solving systems of linear algebraic equations are developed from Broyden’s rank-one modification formula. Several algorithms are presented that take as few, or nearly as few, arithmetic operations as Gaussian elimination and are well suited for the handling of data. The effect of rounding errors is discussed briefly. Some of these algorithms are essentially equivalent to, or “compact” forms of, such known methods as Sherman and Morrison’s modification method, Hestenes’ biorthogonalization method, Gauss-Jordan elimination, Aitken’s below-the-line elimination method, Purcell’s vector method, and its equivalent, Pietrzykowski’s projection method, and the bordering method. These methods are thus shown to be directly related to each other. Iterative methods and methods for inverting symmetric matrices are also given, as are the results of some computational experiments.References
-
A. C. Aitken, “Studies in practical mathematics. I. The evaluation with application of a certain triple product matrix,” Proc. Roy Soc. Edinburgh Sect. A, v. 57, 1936/37, pp. 172-181.
- M. S. Bartlett, An inverse matrix adjustment arising in discriminant analysis, Ann. Math. Statistics 22 (1951), 107–111. MR 40068, DOI 10.1214/aoms/1177729698
- E. Bodewig, Matrix calculus, North-Holland Publishing Co., Amsterdam, 1956. MR 0080363
- C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp. 19 (1965), 577–593. MR 198670, DOI 10.1090/S0025-5718-1965-0198670-6 P. D. Crout, “A short method for evaluating determinants and solving systems of linear equations with real or complex coefficients,” Trans. Amer. Inst. Elect. Engrs., v. 60, 1941, pp. 1235-1240. J. E. Dennis, Jr., On the Convergence of Broyden’s Method for Nonlinear Systems of Equations, Technical Report #69-48, Dept. of Comput. Sci., Cornell University, Ithaca, N.Y., 1969. M. H. Doolittle, Method Employed in the Solution of Normal Equations and the Adjustment of Triangulation, U.S. Coast Guard and Geodetic Survey Report, 1878, pp. 115-120.
- W. J. Duncan, Some devices for the solution of large sets of simultaneous linear equations, Philos. Mag. (7) 35 (1944), 660–670. MR 12914 P. S. Dwyer, Linear Computations, Wiley, New York, 1951. MR 13, 283. A. P. Eršov(Ershov), “On a method for inverting matrices,” Dokl. Akad. Nauk SSSR, v. 100, 1955, pp. 209-211. (Russian) MR 16, 1082.
- D. K. Faddeev and V. N. Faddeeva, Computational methods of linear algebra, W. H. Freeman and Co., San Francisco-London, 1963. Translated by Robert C. Williams. MR 0158519
- L. Fox, An introduction to numerical linear algebra, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1964. MR 0164436
- L. Fox, H. D. Huskey, and J. H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. Appl. Math. 1 (1948), 149–173. MR 26421, DOI 10.1093/qjmam/1.1.149 C. F. Gauss, “Supplementum theoriae combinationis observationum erroribus minimis obnoxiae,” Werke. Band IV, Gottingen, 1873, pp. 55-93.
- D. Goldfarb, Modification methods for inverting matrices and solving systems of linear algebraic equations, Math. Comp. 26 (1972), 829–852. MR 317527, DOI 10.1090/S0025-5718-1972-0317527-4
- J. Greenstadt, Variations on variable-metric methods. (With discussion), Math. Comp. 24 (1970), 1–22. MR 258248, DOI 10.1090/S0025-5718-1970-0258248-4
- Magnus R. Hestenes, Iterative computational methods, Comm. Pure Appl. Math. 8 (1955), 85–95. MR 68313, DOI 10.1002/cpa.3160080106
- Magnus R. Hestenes, Inversion of matrices by biorthogonalization and related results, J. Soc. Indust. Appl. Math. 6 (1958), 51–90. MR 92215
- Harold Hotelling, Some new methods in matrix calculation, Ann. Math. Statistics 14 (1943), 1–34. MR 7851, DOI 10.1214/aoms/1177731489
- Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 0175290 M. A. Jenkins, The Solution of Linear Systems of Equations and Linear Least Squares Problems in APL, IBM-New York Scientific Center, Technical Report #320-2989, 1970.
- V. V. Kljuev and N. I. Kokovkin-Ščerbak, On the minimization of the number of arithmetic operations for solving linear algebraic systems of equations, Ž. Vyčisl. Mat i Mat. Fiz. 5 (1965), 21–33 (Russian). MR 172453 G. Kron, Diakoptics, Macdonald, London, 1963.
- J. Morris, An escalator process for the solution of linear simultaneous equations, Philos. Mag. (7) 37 (1946), 106–120. MR 18423
- Joseph Morris, The Escalator Method in Engineering Vibration Problems, John Wiley & Sons, Inc., New York, N. Y., 1947. MR 0023613 T. Pietrzykowski, Projection Method, Zakladu Aparatów Matematycznych Polskiej Akad. Nauk. Praca A8 (as reported in Householder [20]).
- M. J. D. Powell, A theorem on rank one modifications to a matrix and its inverse, Comput. J. 12 (1969/70), 288–290. MR 245190, DOI 10.1093/comjnl/12.3.288
- M. J. D. Powell, A new algorithm for unconstrained optimization, Nonlinear Programming (Proc. Sympos., Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1970, pp. 31–65. MR 0272162
- Everett W. Purcell, The vector method of solving simultaneous linear equations, J. Math. Physics 32 (1953), 180–183. MR 0059065
- Jack Sherman, Computations relating to inverse matrices, Simultaneous linear equations and the determination of eigenvalues, National Bureau of Standards Applied Mathematics Series, No. 29, U.S. Government Printing Office, Washington, D.C., 1953, pp. 123–124. MR 0057026 J. Sherman & W. J. Morrison, “Adjustment of an inverse matrix corresponding to changes in a given column or row of the original matrix,” Ann. Math. Statist., v. 20, 1949, p. 621.
- Jack Sherman and Winifred J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statistics 21 (1950), 124–127. MR 35118, DOI 10.1214/aoms/1177729893
- Volker Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969), 354–356. MR 248973, DOI 10.1007/BF02165411
- Herbert S. Wilf, Matrix inversion by the annihilation of rank, J. Soc. Indust. Appl. Math. 7 (1959), 149–151. MR 107968
- Herbert S. Wilf, Matrix inversion by the method of rank annihilation, Mathematical methods for digital computers, Wiley, New York, 1960, pp. 73–77. MR 0117911
- J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456
- Max A. Woodbury, Inverting modified matrices, Princeton University, Princeton, N. J., 1950. Statistical Research Group, Memo. Rep. no. 42,. MR 0038136 G. Zielke, “Inversion of modified symmetric matrices,” J. Assoc. Comput. Mach., v. 15, 1968, pp. 402-408.
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Math. Comp. 26 (1972), 829-852
- MSC: Primary 65F10; Secondary 65F30
- DOI: https://doi.org/10.1090/S0025-5718-1972-0317527-4
- MathSciNet review: 0317527