  The main goal of this work is to determine which entire functions preserve
nonnegativity of matrices of a fixed order $n$ -- i.e., to characterize entire
functions $f$ with the property that $f(A)$ is entrywise nonnegative for every
entrywise nonnegative matrix $A$ of size $n\times n$. Towards this goal, we
present a complete characterization of functions preserving nonnegativity of
(block) upper-triangular matrices and those preserving nonnegativity of
circulant matrices. We also derive necessary conditions and sufficient
conditions for entire functions that preserve nonnegativity of symmetric
matrices. We also show that some of these latter conditions characterize the
even or odd functions that preserve nonnegativity of symmetric matrices.
