Abstract
Recently, several iterative algorithms have been proposed for the solution of multilinear system Axm−1=b in the sense that the tensor A of order m and dimension n is a structured one. In this paper, we continue to address this multilinear system in which the coefficient tensor A is an unstructured one given in tensor-train format, and present a new iterative method for it. The method we propose here is an accelerated version of the modified Levenberg–Marquardt method for nonlinear equations (Fan, 2012), which contains two-step line searches corresponding to the LM step and the approximate LM step equipped with a novel LM parameter. It is shown that this algorithm has cubic convergence under the local error bound condition which is weaker than nonsingularity, and the computational complexity of which is free from the so-called curse-of-dimensionality. The performed numerical examples show that our method is promising.
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