A stable algorithm for updating triangular factors under a rank one change

Authors:
R. Fletcher and S. P. J. Matthews

Journal:
Math. Comp. **45** (1985), 471-485

MSC:
Primary 65F05

MathSciNet review:
804936

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Abstract: An algorithm is presented for updating the *LU* factors of an matrix *A*, when *A* is changed by a matrix of rank one. The algorithm is based on the repeated use of triangular operations, and stability is obtained by allowing both row and column pivoting. The cost of the algorithm is approximately proportional to the maximum permitted depth for the pivot search. For well-conditioned matrices a maximum depth of 3 is sufficient to ensure stability. For substantially rank deficient matrices it is theoretically possible that pivots of any depth may be required, but in practice we find that a value of 5 is adequate. We suggest a pivot strategy, based on minimizing a growth bound, which penalizes deep pivots and imposes a maximum depth of pivot through a default value.

On well-behaved problems the asymptotic cost of the update is observed to be approximately compared with (or worse) for updating orthogonal factors. Given the accuracy obtained by the new algorithm, we feel that there are many applications in which the lower cost of triangular factors can be exploited. Comparison with *ab initio* factorization indicates that for updating triangular factors is advantageous.

**[1]**John M. Bennett,*Triangular factors of modified matrices*, Numer. Math.**7**(1965), 217–221. MR**0177503****[2]**J. W. Daniel, W. B. Gragg, L. Kaufman, and G. W. Stewart,*Reorthogonalization and stable algorithms for updating the Gram-Schmidt 𝑄𝑅 factorization*, Math. Comp.**30**(1976), no. 136, 772–795. MR**0431641**, 10.1090/S0025-5718-1976-0431641-8**[3]**R. Fletcher and S. P. J. Matthews,*Stable modification of explicit 𝐿𝑈 factors for simplex updates*, Math. Programming**30**(1984), no. 3, 267–284. MR**769232**, 10.1007/BF02591933**[4]**P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders,*Methods for modifying matrix factorizations*, Math. Comp.**28**(1974), 505–535. MR**0343558**, 10.1090/S0025-5718-1974-0343558-6**[5]**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/S0025-5718-1985-0804936-9

Keywords:
Rank one update,
*LU* factors,
triangular factors,
stability

Article copyright:
© Copyright 1985
American Mathematical Society