An efficient memory gradient method for extreme M-eigenvalues of elastic type tensors

M-eigenvalues of fourth order hierarchically symmetric tensors play a significant role in nonlinear elastic material analysis and quantum entanglement problems. This paper focuses on computing extreme M-eigenvalues for such tensors. To achieve this, we first reformulate the M-eigenvalue problem as a sequence of unconstrained optimization problems by introducing a shift parameter. Subsequently, we develop a memory gradient method specifically designed to approximate these extreme M-eigenvalues. Under this framework, we establish the global convergence of the proposed method. Finally, comprehensive numerical experiments demonstrate the efficacy and stability of our approach.