On the modification of factorizations

Authors:
R. Fletcher and M. J. D. Powell

Journal:
Math. Comp. **28** (1974), 1067-1087

MSC:
Primary 65F30

MathSciNet review:
0359297

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Abstract | References | Similar Articles | Additional Information

Abstract: A number of methods are compared for modifying an factorization of a positive definite matrix *A* when a matrix of rank one is added to *A*. Both the efficiency and error propagation characteristics are discussed, and it is shown that a suitable method must be chosen with care. An error analysis of some of the algorithms is given which has novel features. A worked example which also illustrates error growth is presented. Extensions to the methods are desribed which enable them to be used advantageously in representing and updating positive semidefinite matrices.

**[1]**John M. Bennett,*Triangular factors of modified matrices*, Numer. Math.**7**(1965), 217–221. MR**0177503****[2]**W. M. GENTLEMAN,*Error Analysis of QR Decompositions by Givens Transformations*. (Author's Manuscript, 1973.)**[3]**W. Morven Gentleman,*Least squares computations by Givens transformations without square roots*, J. Inst. Math. Appl.**12**(1973), 329–336. MR**0329233****[4]**P. E. GILL, G. H. GOLUB, W. MURRAY & M. A. SAUNDERS,*Methods for Modifying Matrix Factorizations*, NPL Report NAC 29, 1972.**[5]**P. E. Gill and W. Murray,*Quasi-Newton methods for unconstrained optimization*, J. Inst. Math. Appl.**9**(1972), 91–108. MR**0300410****[6]**Donald Goldfarb,*Extension of Davidon’s variable metric method to maximization under linear inequality and equality constraints*, SIAM J. Appl. Math.**17**(1969), 739–764. MR**0290799****[7]**J. B. Rosen,*The gradient projection method for nonlinear programming. I. Linear constraints*, J. Soc. Indust. Appl. Math.**8**(1960), 181–217. MR**0112750****[8]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0359297-1

Keywords:
Matrix updating,
error analysis,
updating a triangular factorization,
quasi-Newton method,
positive definite matrix,
rank one matrix,
rank one correction

Article copyright:
© Copyright 1974
American Mathematical Society