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

Abstract:

An algorithm is presented for updating the LU factors of an $n \times n$ 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 $2.6{n^2}$ compared with $8{n^2}$ (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 $n \geqslant 10$ updating triangular factors is advantageous.