  The adaptive cubic regularization algorithm employing the inexact gradient
and Hessian is proposed on general Riemannian manifolds, together with the
iteration complexity to get an approximate second-order optimality under
certain assumptions on accuracies about the inexact gradient and Hessian. The
algorithm extends the inexact adaptive cubic regularization algorithm under
true gradient in [Math. Program., 184(1-2): 35-70, 2020] to more general cases
even in Euclidean settings. As an application, the algorithm is applied to
solve the joint diagonalization problem on the Stiefel manifold. Numerical
experiments illustrate that the algorithm performs better than the inexact
trust-region algorithm in [Advances of the neural information processing
systems, 31, 2018].
