  Extremely large-scale multiple-input-multiple-output (XL-MIMO) is a promising
technology for the future sixth-generation (6G) networks to achieve higher
performance. In practice, various linear precoding schemes, such as
zero-forcing (ZF) and regularized zero-forcing (RZF) precoding, are capable of
achieving both large spectral efficiency (SE) and low bit error rate (BER) in
traditional massive MIMO (mMIMO) systems. However, these methods are not
efficient in extremely large-scale regimes due to the inherent spatial
non-stationarity and high computational complexity. To address this problem, we
investigate a low-complexity precoding algorithm, e.g., randomized Kaczmarz
(rKA), taking into account the spatial non-stationary properties in XL-MIMO
systems. Furthermore, we propose a novel mode of randomization, i.e., sampling
without replacement rKA (SwoR-rKA), which enjoys a faster convergence speed
than the rKA algorithm. Besides, the closed-form expression of SE considering
the interference between subarrays in downlink XL-MIMO systems is derived.
Numerical results show that the complexity given by both rKA and SwoR-rKA
algorithms has 51.3% reduction than the traditional RZF algorithm with similar
SE performance. More importantly, our algorithms can effectively reduce the BER
when the transmitter has imperfect channel estimation.
