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Finding the symmetric rank-1 approximation to a given symmetric tensor is an important problem due to its wide applications and its close relationship to the Z-eigenpair of a tensor. In this paper, we propose a method based on the proximal alternating linearized minimization to directly solve the optimization problem. Global convergence of our algorithm is established. Numerical experiments show that our algorithm is very competitive in speed, accuracy and robustness compared to other state-of-the-art methods. [ABSTRACT FROM AUTHOR]