Unsymmetric Positive Definite Linear Systems

Abstract

Is it necessary to pivot when solving an unsymmetric positive definite linear system $Ax = b?$ Define $T = (A + A^{T})/2$ and $S=(A+AA^{T})/2$. It is shown that pivoting is unnecessary if the quantitity is $\Vert S T^{-1} S \Vert_{2}/ \Vert A \Vert_{2}$ is suitably small with respect to the working machine precision.