  Best rank-one approximation is one of the most fundamental tasks in tensor
computation. In order to fully exploit modern multi-core parallel computers, it
is necessary to develop decoupling algorithms for computing the best rank-one
approximation of higher-order tensors at large scales. In this paper, we first
build a bridge between the rank-one approximation of tensors and the
eigenvector-dependent nonlinear eigenvalue problem (NEPv), and then develop an
efficient decoupling algorithm, namely the higher-order self-consistent field
(HOSCF) algorithm, inspired by the famous self-consistent field (SCF) iteration
frequently used in computational chemistry. The convergence theory of the HOSCF
algorithm and an estimation of the convergence speed are further presented. In
addition, we propose an improved HOSCF (iHOSCF) algorithm that incorporates the
Rayleigh quotient iteration, which can significantly accelerate the convergence
of HOSCF. Numerical experiments show that the proposed algorithms can
efficiently converge to the best rank-one approximation of both synthetic and
real-world tensors and can scale with high parallel scalability on a modern
parallel computer.
