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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Methods for modifying matrix factorizations
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by P. E. Gill, G. H. Golub, W. Murray and M. A. Saunders PDF
Math. Comp. 28 (1974), 505-535 Request permission


In recent years, several algorithms have appeared for modifying the factors of a matrix following a rank-one change. These methods have always been given in the context of specific applications and this has probably inhibited their use over a wider field. In this report, several methods are described for modifying Cholesky factors. Some of these have been published previously while others appear for the first time. In addition, a new algorithm is presented for modifying the complete orthogonal factorization of a general matrix, from which the conventional QR factors are obtained as a special case. A uniform notation has been used and emphasis has been placed on illustrating the similarity between different methods.
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  • $\ast$ 5. P. E. Gill & W. Murray, A Numerically Stable Form of the Simplex Algorithm, National Physical Laboratory, DNAM Report Number 87, Teddington, England, 1970.
  • P. E. Gill and W. Murray, Quasi-Newton methods for unconstrained optimization, J. Inst. Math. Appl. 9 (1972), 91–108. MR 300410
  • $\ast$ 7. P. E. Gill & W. Murray, Two Methods For the Solution of Linearly Constrained and Unconstrained Optimization Problems, National Physical Laboratory, DNAC Report Number 25, Teddington, England, 1972.
  • G. H. Golub and M. A. Saunders, Linear least squares and quadratic programming, Integer and nonlinear programming, North-Holland, Amsterdam, 1970, pp. 229–256. MR 0437049
  • $\ast$ 9. G. H. Golub & P. H. Styan, Numerical Computations for Univariate Linear Models, Computer Science Department Report Number CS-236-71, Stanford University, Stanford, California, 1971.
  • Richard J. Hanson and Charles L. Lawson, Extensions and applications of the Householder algorithm for solving linear least squares problems, Math. Comp. 23 (1969), 787–812. MR 258258, DOI 10.1090/S0025-5718-1969-0258258-9
  • R. S. Martin, G. Peters & J. H. Wilkinson, "The QR algorithm for real Hessenberg matrices," Handbook for Automatic Computation, Vol. 2. Edited by J. H. Wilkinson and C. Reinsch, Springer-Verlag, Berlin and New York, 1971, pp. 359-371. $\ast$ 12. W. Murray, An Algorithm for Indefinite Quadratic Programming, National Physical Laboratory, DNAC Report Number 1, Teddington, England, 1971. G. Peters & J. H. Wilkinson, "The least squares problem and pseudo-inverses," Comput. J., v. 13, 1970, pp. 309-316. $\ast$ 14. M. A. Saunders, Large-Scale Linear Programming Using the Cholesky Factorization, Computer Science Department Report Number CS-72-252, Stanford University, Stanford, California, 1972. $\ast$ 15. M. A. Saunders, Product Form of the Cholesky Factorization for Large-Scale Linear Programming, Computer Science Department Report Number CS-72-301, Stanford University, Stanford, Calif., 1972.
  • J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Math. Comp. 28 (1974), 505-535
  • MSC: Primary 65F05
  • DOI:
  • MathSciNet review: 0343558