Sensitivity analyses for factorizations of sparse or structured matrices Original Research Article

For a unique factorization of a matrix B, the effect of sparsity or other structure on measuring the sensitivity of the factors of B to some change G in B is considered. In particular, norm-based analyses of the QR and Cholesky factorizations are examined. If B is structured but G is not, it is shown that the expressions for the condition numbers are identical to those when B is not structured, but because of the structure the condition numbers may be easier to estimate. If G is structured, whether B is or not, then the expressions for the condition numbers can change, and it is shown how to derive the new expressions. Cases where B and G have the same sparsity structure occur often: here, for the QR factorization an example shows the value of the new expression can be arbitrarily smaller, but for the Cholesky factorization of a tridiagonal matrix and perturbation the value of the new expression cannot be significantly different from the value of the old one. Thus taking account of sparsity can show the condition is much better than would be suggested by ignoring it, but only for some classes of problems, and perhaps only for some types of factorization. The generalization of these ideas to other factorizations is discussed.