  Large kernel systems are prone to be ill-conditioned. Pivoted Cholesky
decomposition (PCD) render a stable and efficient solution to the systems
without a perturbation of regularization. This paper proposes a new PCD
algorithm by tuning Cross Approximation (CA) algorithm to kernel matrices which
merges the merits of PCD and CA, and proves as well as numerically exemplifies
that it solves large kernel systems two-order more efficiently than those
resorts to regularization. As a by-product, a diagonal-pivoted CA technique is
also shown efficient in eigen-decomposition of large covariance matrices in an
uncertainty quantification problem.
