  The ill-posed analytic continuation problem for Green's functions or
self-energies can be done using the Pad\'e rational polynomial approximation.
However, to extract accurate results from this approximation, high precision
input data of the Matsubara Green's function are needed. The calculation of the
Matsubara Green's function generally involves a Matsubara frequency summation
which cannot be evaluated analytically. Numerical summation is requisite but it
converges slowly with the increase of the Matsubara frequency. Here we show
that this slow convergence problem can be significantly improved by utilizing
the Pad\'e decomposition approach to replace the Matsubara frequency summation
by a Pad\'e frequency summation, and high precision input data can be obtained
to successfully perform the Pad\'e analytic continuation.
