Linear minimum mean square error (MMSE) detection achieves a good trade‐off between performance and complexity for massive multiple‐input multiple‐output (MIMO) systems. To avoid the high‐dimensional matrix inversion involved, MMSE detection can be transformed into an unconstrained optimization problem and then solved by efficient numerical algorithms in an iterative way. Three low‐complexity Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) quasi‐Newton methods are proposed to iteratively realize massive MIMO MMSE detection without matrix inversion. The complexity can be reduced from scriptOfalse(K3false)$\mathcal {O}(K^{3})$ to scriptOfalse(LK2false)$\mathcal {O}(LK^{2})$, where K and L denote the number of users and iterations, respectively. Leveraging the special properties of massive MIMO, the authors first explore a simplified BFGS method (named S‐BFGS) to alleviate the computational burden in the search direction. For lower complexity, BFGS method with the unit step size (named U‐BFGS) is presented subsequently. When the base station (BS)‐to‐user‐antenna ratio (BUAR) is large enough, the two proposed BFGS methods can be integrated (named U‐S‐BFGS) to further reduce complexity. In addition, an efficient initialization strategy is devised to accelerate convergence. Simulation results verify that the proposed detection scheme can achieve near‐MMSE performance with a small number of iterations L as low as 2 or 3.

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The computation of the Moore-Penrose inverse is widely encountered in science and engineering. Due to the parallel-processing nature and strong-learning ability, the neural network has become a promising approach to solving the Moore-Penrose inverse recently. However, almost all the existing neural networks for matrix inversion are based on the gradient-descent (GD) method, whose main drawbacks are slow convergence and sensitivity to learning parameters. Moreover, there is no unified neural network to compute the Moore-Penrose inverse for both the full-rank matrix and rank-deficient matrix. In this paper, an efficient second-order neural network model with the improved Newton's method is proposed to obtain the accurate Moore-Penrose inverse of an arbitrary matrix by one epoch without any learning parameter. Compared with the GD-based neural networks for Moore-Penrose inverse computation, the proposed model converges faster and has lower complexity. Furthermore, through in-depth derivation, the neural network for computing the Moore-Penrose inverse is well interpretable. Numerical studies and application to the random matrix inversion in multiple-input multiple-output detection are provided to validate the efficiency of the proposed model for solving the Moore-Penrose inverse.