Structured condition numbers for some matrix factorizations of structured matrices

Using the modified matrix-vector approach and the differential calculus, we study the structured condition numbers for LU, Cholesky and QR factorizations of some structured matrices that can be represented by sets of parameters. The obtained explicit expressions of these structured condition numbers are very general, which are applicable to most of linear and non-linear structured matrices, and include the popular normwise, mixed and componentwise condition numbers as special cases. More specific explicit expressions of the structured condition numbers for linear structured matrices are also provided. We compare the structured condition numbers with the corresponding unstructured ones in theory and experiment. Numerical results show that, for non-linear structured matrices, the structured condition numbers can be much smaller than the unstructured ones. In addition, we also test the applications of structured condition numbers in estimating the first-order perturbation bounds of matrix factorizations using numerical examples.