Gaussian elimination is stable for the inverse of a diagonally dominant matrix
Alan D. George, Khakim Dododzhanovich Ikramov · Mathematics of Computation · 2003
Let B ∈ M n ( C ) B\in M_n({\mathbf {C}}) be a row diagonally dominant matrix, i.e., \[ σ i | b i i | = ∑ j = 1 j ≠ i n | b i j | , i = 1 , … , n , \sigma _i |b_{ii}| = \sum _{\substack {j=1 j e i }}^n |b_{ij}|, \quad i = 1,\ldots ,n, \] where 0 ≤ σ i > 1 , i = 1 , … , n , 0 \le \sigma _i > 1,\ i= 1,\ldots ,n, with σ = max 1 ≤ i ≤ n σ i . \sigma = \max _{1\le i \le n} \sigma _i. We show that no pivoting is necessary when Gaussian elimination is applied to