The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although in specific cases the loss of precision in GEPP due to roundoff errors can be very significant, empirical evidence strongly suggests that for a {\it typical} square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random $n\times n$ standard Gaussian coefficient matrix $A$, the {\it growth factor} of the Gaussian Elimination with Partial Pivoting is at most polynomially large in $n$ with probability close to one. This implies that with probability close to one the number of bits of precision sufficient to solve $Ax = b$ to $m$ bits of accuracy using GEPP is $m+O(\log n)$, which improves an earlier estimate $m+O(\log^2 n)$ of Sankar, and which we conjecture to be optimal by the order of magnitude. We further provide tail estimates of the growth factor which can be used to support the empirical observation that GEPP is more stable than the Gaussian Elimination with no pivoting.

翻译：部分主元高斯消去法（GEPP）是求解线性方程组的经典算法。尽管在某些特定情况下，由舍入误差导致的GEPP精度损失可能非常显著，但经验证据强烈表明，对于{\it 典型}的方阵系数矩阵，GEPP在数值上是稳定的。我们通过证明：给定随机$n\times n$标准高斯系数矩阵$A$，部分主元高斯消去法的{\it 增长因子}在接近概率为1的情况下最多为$n$的多项式量级，从而为该现象提供了（部分）理论依据。这意味着，使用GEPP将$Ax = b$求解至$m$位精度时，所需精度位数以接近1的概率为$m+O(\log n)$，这改进了Sankar此前给出的$m+O(\log^2 n)$估计，并且我们推测该结果在量级上是最优的。此外，我们进一步给出了增长因子的尾部估计，可用于支持GEPP数值稳定性优于无主元高斯消去法的经验观察。