  In previous works, we devised a differential operator for evaluating typical
integrals appearing in the Cachazo-He-Yuan (CHY) forms and in this paper we
further streamline this method. We observe that at tree level, the number of
free parameters controlling the differential operator depends solely on the
number of external lines, after solving the constraints arising from the
scattering equations. This allows us to construct a reduction matrix that
relates the parameters of a higher-order differential operator to those of a
lower-order one. The reduction matrix is theory-independent and can be obtained
by solving a set of explicitly given linear conditions. The repeated
application of such reduction matrices eventually transforms a given tree-level
CHY-like integral to a prepared form. We also provide analytic expressions for
the parameters associated with any such prepared form at tree level. We finally
give a compact expression for the multidimensional residue for any CHY-like
integral in terms of the reduction matrices. We adopt a dual basis projector
which leads to the CHY-like representation for the non-local
Bern-Carrasco-Johansson (BCJ) numerators at tree level in Yang-Mills theory.
These BCJ numerators are efficiently computed by the improved method involving
the reduction matrix.
