  The Sinc-Nystr\"{o}m method is a high-order numerical method based on Sinc
basis functions for discretizing evolutionary differential equations in time.
But in this method we have to solve all the time steps in one-shot (i.e.
all-at-once), which results in a large-scale nonsymmetric dense system that is
expensive to handle. In this paper, we propose and analyze preconditioner for
such dense system arising from both the parabolic and hyperbolic PDEs. The
proposed preconditioner is a low-rank perturbation of the original matrix and
has two advantages. First, we show that the eigenvalues of the preconditioned
system are highly clustered with some uniform bounds which are independent of
the mesh parameters. Second, the preconditioner can be used parallel for all
the Sinc time points via a block diagonalization procedure. Such a parallel
potential owes to the fact that the eigenvector matrix of the diagonalization
is well conditioned. In particular, we show that the condition number of the
eigenvector matrix only mildly grows as the number of Sinc time points
increases, and thus the roundoff error arising from the diagonalization
procedure is controllable. The effectiveness of our proposed PinT
preconditioners is verified by the observed mesh-independent convergence rates
of the preconditioned GMRES in reported numerical examples.
