  We develop and analyze numerical discretization to the constrained high-index
saddle dynamics, the dynamics searching for the high-index saddle points
confined on the high-dimensional unit sphere. Compared with the saddle dynamics
without constraints, the constrained high-index saddle dynamics has more
complex dynamical forms, and additional operations such as the retraction and
vector transport are required due to the constraint, which significantly
complicate the numerical scheme and the corresponding numerical analysis.
Furthermore, as the existing numerical analysis results usually depend on the
index of the saddle points implicitly, the proved numerical accuracy may be
reduced if the index is high in many applications, which indicates the lack of
robustness with respect to the index. To address these issues, we derive the
error estimates for numerical discretization of the constrained high-index
saddle dynamics on high-dimensional sphere, and then improve it by providing an
index-robust error analysis in an averaged norm by adjusting the relaxation
parameters. The developed results provide mathematical supports for the
accuracy of numerical computations.
