  We propose a verified computation method for eigenvalues in a region and the
corresponding eigenvectors of generalized Hermitian eigenvalue problems. The
proposed method uses complex moments to extract the eigencomponents of interest
from a random matrix and uses the Rayleigh$\unicode{x2013}$Ritz procedure to
project a given eigenvalue problem into a reduced eigenvalue problem. The
complex moment is given by contour integral and approximated using numerical
quadrature. We split the error in the complex moment into the truncation error
of the quadrature and rounding errors and evaluate each. This idea for error
evaluation inherits our previous Hankel matrix approach, whereas the proposed
method enables verification of eigenvectors and requires half the number of
quadrature points for the previous approach to reduce the truncation error to
the same order. Moreover, the Rayleigh$\unicode{x2013}$Ritz procedure approach
forms a transformation matrix that enables verification of the eigenvectors.
Numerical experiments show that the proposed method is faster than previous
methods while maintaining verification performance and works even for nearly
singular matrix pencils and in the presence of multiple and nearly multiple
eigenvalues.
