  The growth factor in Gaussian elimination measures how large the entries of
an LU factorization can be relative to the entries of the original matrix. It
is a key parameter in error estimates, and one of the most fundamental topics
in numerical analysis. We produce an upper bound of $n^{0.2079 \ln n +0.91}$
for the growth factor in Gaussian elimination with complete pivoting -- the
first improvement upon Wilkinson's original 1961 bound of $2 \, n ^{0.25\ln n
+0.5}$.
