Gaussian elimination is stable for the inverse of a diagonally dominant matrix

Abstract:

Let $B\in M_n({\mathbf {C}})$ be a row diagonally dominant matrix, i.e., \[ \sigma _i |b_{ii}| = \sum _{\substack {j=1 j\ne i }}^n |b_{ij}|, \quad i = 1,\ldots ,n, \] where $0 \le \sigma _i < 1,\ i= 1,\ldots ,n,$ with $\sigma = \max _{1\le i \le n} \sigma _i.$ We show that no pivoting is necessary when Gaussian elimination is applied to $A = B^{-1}.$ Moreover, the growth factor for $A$ does not exceed $1 + \sigma .$ The same results are true with row diagonal dominance being replaced by column diagonal dominance.