  In this paper, we consider two fundamental symmetric kernels in linear
algebra: the Cholesky factorization and the symmetric rank-$k$ update (SYRK),
with the classical three nested loops algorithms for these kernels. In
addition, we consider a machine model with a fast memory of size $S$ and an
unbounded slow memory. In this model, all computations must be performed on
operands in fast memory, and the goal is to minimize the amount of
communication between slow and fast memories. As the set of computations is
fixed by the choice of the algorithm, only the ordering of the computations
(the schedule) directly influences the volume of communications.We prove lower
bounds of $\frac{1}{3\sqrt{2}}\frac{N^3}{\sqrt{S}}$ for the communication
volume of the Cholesky factorization of an $N\times N$ symmetric positive
definite matrix, and of $\frac{1}{\sqrt{2}}\frac{N^2M}{\sqrt{S}}$ for the SYRK
computation of $\mat{A}\cdot\transpose{\mat{A}}$, where $\mathbf{A}$ is an
$N\times M$ matrix. Both bounds improve the best known lower bounds from the
literature by a factor $\sqrt{2}$.In addition, we present two out-of-core,
sequential algorithms with matching communication volume: \TBS for SYRK, with a
volume of $\frac{1}{\sqrt{2}}\frac{N^2M}{\sqrt{S}} + \bigo{NM\log N}$, and \LBC
for Cholesky, with a volume of $\frac{1}{3\sqrt{2}}\frac{N^3}{\sqrt{S}} +
\bigo{N^{5/2}}$. Both algorithms improve over the best known algorithms from
the literature by a factor $\sqrt{2}$, and prove that the leading terms in our
lower bounds cannot be improved further. This work shows that the operational
intensity of symmetric kernels like SYRK or Cholesky is intrinsically higher
(by a factor $\sqrt{2}$) than that of corresponding non-symmetric kernels (GEMM
and LU factorization).
