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Modification methods for inverting matrices and solving systems of linear algebraic equations. (English) Zbl 0268.65026


MSC:

65F05 Direct numerical methods for linear systems and matrix inversion
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[1] 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. · Zbl 0016.24102
[2] M. S. Bartlett, An inverse matrix adjustment arising in discriminant analysis, Ann. Math. Statistics 22 (1951), 107 – 111. · Zbl 0042.38203
[3] E. Bodewig, Matrix calculus, North-Holland Publishing Company, Amsterdam, 1956. · Zbl 0086.32501
[4] C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp. 19 (1965), 577 – 593. · Zbl 0131.13905
[5] 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.
[6] 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.
[7] 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.
[8] W. J. Duncan, Some devices for the solution of large sets of simultaneous linear equations, Philos. Mag. (7) 35 (1944), 660 – 670. · Zbl 0061.27010
[9] P. S. Dwyer, Linear Computations, Wiley, New York, 1951. MR 13, 283. · Zbl 0044.12804
[10] A. P. Eršov(Ershov), “On a method for inverting matrices,” Dokl. Akad. Nauk SSSR, v. 100, 1955, pp. 209-211. (Russian) MR 16, 1082.
[11] D. K. Faddeev and V. N. Faddeeva, Computational methods of linear algebra, Translated by Robert C. Williams, W. H. Freeman and Co., San Francisco-London, 1963. · Zbl 0112.07503
[12] L. Fox, An introduction to numerical linear algebra, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1964. · Zbl 0122.35701
[13] 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. · Zbl 0033.28503
[14] C. F. Gauss, “Supplementum theoriae combinationis observationum erroribus minimis obnoxiae,” Werke. Band IV, Gottingen, 1873, pp. 55-93.
[15] D. Goldfarb, Modification methods for inverting matrices and solving systems of linear algebraic equations, Math. Comp. 26 (1972), 829 – 852. · Zbl 0268.65026
[16] J. Greenstadt, Variations on variable-metric methods. (With discussion), Math. Comp. 24 (1970), 1 – 22. · Zbl 0204.49601
[17] Magnus R. Hestenes, Iterative computational methods, Comm. Pure Appl. Math. 8 (1955), 85 – 95. · Zbl 0066.10201
[18] Magnus R. Hestenes, Inversion of matrices by biorthogonalization and related results, J. Soc. Indust. Appl. Math. 6 (1958), 51 – 90. · Zbl 0085.33003
[19] Harold Hotelling, Some new methods in matrix calculation, Ann. Math. Statistics 14 (1943), 1 – 34. · Zbl 0061.27007
[20] Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. · Zbl 0329.65003
[21] 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.
[22] 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).
[23] G. Kron, Diakoptics, Macdonald, London, 1963.
[24] J. Morris, An escalator process for the solution of linear simultaneous equations, Philos. Mag. (7) 37 (1946), 106 – 120. · Zbl 0061.27101
[25] Joseph Morris, The Escalator Method in Engineering Vibration Problems, John Wiley & Sons, Inc., New York, N. Y., 1947. · Zbl 0029.14803
[26] T. Pietrzykowski, Projection Method, Zakladu Aparatów Matematycznych Polskiej Akad. Nauk. Praca A8 (as reported in Householder [20]). · Zbl 0142.11404
[27] M. J. D. Powell, A theorem on rank one modifications to a matrix and its inverse, Comput. J. 12 (1969/1970), 288 – 290. · Zbl 0182.48904
[28] 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. · Zbl 0228.90043
[29] Everett W. Purcell, The vector method of solving simultaneous linear equations, J. Math. Physics 32 (1953), 180 – 183. · Zbl 0051.35303
[30] 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. · Zbl 0052.34905
[31] 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.
[32] 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. · Zbl 0037.00901
[33] Volker Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969), 354 – 356. · Zbl 0185.40101
[34] Herbert S. Wilf, Matrix inversion by the annihilation of rank, J. Soc. Indust. Appl. Math. 7 (1959), 149 – 151. · Zbl 0086.10902
[35] Herbert S. Wilf, Matrix inversion by the method of rank annihilation, Mathematical methods for digital computers, Wiley, New York, 1960, pp. 73 – 77.
[36] J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. · Zbl 1041.65502
[37] Max A. Woodbury, Inverting modified matrices, Statistical Research Group, Memo. Rep. no. 42, Princeton University, Princeton, N. J., 1950.
[38] G. Zielke, “Inversion of modified symmetric matrices,” J. Assoc. Comput. Mach., v. 15, 1968, pp. 402-408. · Zbl 0162.46703
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