  We analyze the semi-implicit scheme of high-index saddle dynamics, which
provides a powerful numerical method for finding the any-index saddle points
and constructing the solution landscape. Compared with the explicit schemes of
saddle dynamics, the semi-implicit discretization relaxes the step size and
accelerates the convergence, but the corresponding numerical analysis
encounters new difficulties compared to the explicit scheme. Specifically, the
orthonormal property of the eigenvectors at each time step could not be fully
employed due to the semi-implicit treatment, and computations of the
eigenvectors are coupled with the orthonormalization procedure, which further
complicates the numerical analysis. We address these issues to prove error
estimates of the semi-implicit scheme via, e.g. technical splittings and
multi-variable circulating induction procedure. We further analyze the
convergence rate of the generalized minimum residual solver for solving the
semi-implicit system. Extensive numerical experiments are carried out to
substantiate the efficiency and accuracy of the semi-implicit scheme in
constructing solution landscapes of complex systems.
