We prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics, which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere. Due to the semi-implicit treatment and the novel computational procedure, the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations, and the computations of the state variable and directional vectors are coupled with the retraction, the vector transport and the orthonormalization procedure, which significantly complicates the analysis. We address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings.

翻译：我们证明了球面约束高指标鞍点动力学半隐式数值格式的误差估计，该动力学是寻找高维球面约束系统鞍点并构建解景观的有力工具。由于半隐式处理和新型计算流程，无法完全利用每个时间步数值解的标准正交性来简化推导，并且状态变量与方向向量的计算与收缩、向量传输及正交化过程相互耦合，这显著增加了分析的复杂性。我们针对这些问题提出了所提半隐式格式的误差估计证明，并通过数值实验验证了理论结果。