  We present a numerical implementation of the gauge-invariant time-dependent
configuration interaction singles (TDCIS) method [Appl. Sci. 8, 433 (2018)] for
three-dimensional atoms. In our implementation, orbital-like quantity called
channel orbital [Phys. Rev. A 74, 043420 (2006)] is propagated instead of
configuration-interaction (CI) coefficients, which removes a computational
bottleneck of explicitly calculating and storing numerous virtual orbitals.
Furthermore, besides its physical consistency, the gauge-invariant formulation
allows to take advantages of the velocity gauge treatment of the laser-electron
interaction over the length gauge one in the simulation of high-field
phenomena. We apply the present implementation to high-harmonic generation from
helium and neon atoms, and numerically confirms the gauge invariance and
demonstrates the effectiveness of the rotated velocity gauge treatment.
