  High-index saddle dynamics provides an effective means to compute the
any-index saddle points and construct the solution landscape. In this paper we
prove error estimates for Euler discretization of high-index saddle dynamics
with respect to the time step size, which remains untreated in the literature.
We overcome the main difficulties that lie in the strong nonlinearity of the
saddle dynamics and the orthonormalization procedure in the numerical scheme
that is uncommon in standard discretization of differential equations. The
derived methods are further extended to study the generalized high-index saddle
dynamics for non-gradient systems and provide theoretical support for the
accuracy of numerical implementations.
