  We propose the reduced-shifted Conjugate-Gradient (RSCG) method, which is
numerically efficient to calculate a matrix element of a Green's function
defined as a resolvent of a Hamiltonian operator, by solving linear equations
with a desired accuracy. This method does not calculate solution vectors of
linear equations but does directly calculate a matrix element of the resolvent.
The matrix elements with different frequencies are simultaneously obtained.
Thus, it is easy to calculate the exception value expressed as a Matsubara
summation of these elements. To illustrate a power of our method, we choose a
nano-structured superconducting system with a mean-field Bogoliubov-de Gennes
(BdG) approach. This method allows us to treat with the system with the
fabrication potential, where one can not effectively use the
kernel-polynomial-based method. We consider the d-wave nano-island
superconductor by simultaneously solving the linear equations with a large
number (~ 50000) of Matsubara frequencies.
