  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.
