# Fast Converging Gauss-Seidel Iterative Algorithm for Massive MIMO Systems
## 1 Introduction  
Massive MIMO technology is an extension of conventional MIMO technology that allows it to service more customers concurrently on the same frequency. This improves data rate, spectrum efficiency, energy efficiency, channel capacity, and connection stability [1]. Through further evolution, it is more frequently employed in telecommunications and related industries, such as the 5G new radio system. Massive MIMO is an essential foundational technology for future wireless systems, and it can be used in concert with other technologies like intelligent reflecting surfaces, orthogonal frequency division multiplexing, artificial intelligence, etc. Navigation, transportation, healthcare, and other fields can all benefit from the integration of communication, positioning, and sensing functions made possible by massive MIMO enabling intelligent reflecting surfaces. New human-computer interactions are made possible via the integration of artificial intelligence and MIMO technology, which will also enable extremely dependable real-time industrial information transmission [2]. Massive MIMO technology has several advantages, but it also has many drawbacks. In a massive MIMO communication system, other antennas interfere with the signal during transmission, making it far more difficult to detect the intended signal [3]. At the receiver side, a detector appropriate for enormous multiple-input multiple-output must thus be constructed to strike a compromise between low complexity and excellent performance. The maximum likelihood [4] detection algorithm finds the signal closest to the transmitted signal by rounding through all the received signals, but the high complexity of thetraversal signal makes it difficult to apply the algorithm in practice. In order to mitigate the cost of computing, linear detection schemes are recommended to obtain near-optimal detection performance, commonly known as the zero-forced [5] algorithm and the minimum mean squared error (MMSE) [6] detection algorithm. A highly complex matrix inversion is used in the MMSE method, which is difficult to implement in practice. To avoid inverse operations in high-dimensional matrices, a few iterative algorithms have emerged based on the MMSE detection algorithm. Depending on the iterative approach, they can be classified into the following three categories. The first category is the iterative algorithm based on gradient search. The convergence speed of the steepest descent [7] iterative algorithm, influenced by the initial point, becomes slower when approaching the final estimated signal. The conjugate gradient (CG) [8] detection algorithm is offered to increase the speed of convergence, but a complex pre-processing process is required. The second category is repetitive methods that rely on series expansion, like the Neumann series-expansion algorithm [9], where the complexity is acceptable if the quantity of series-expansion terms is less than three, but high complexity arises if the quantity of series-expansion levels is larger than or equal to three. The third group is those that rely on iterative schemes to solve linear equations. The common ones are the Jacobi iterative method [10], the successive over relaxation (SOR) iterative algorithm [11], the Richardson iterative algorithm [12], the Gauss-Seidel (GS) [13] iterative algorithm, etc. Such detection algorithms reverse the matrices into solving a system of linear equations, which have a strong stepwise nature and high requirements for the filter matrix. There is a limit to what can be achieved using a single algorithm for detection [14]. Therefore, the combination of different detection algorithms in a complementary manner can further improve the detection capability. A hybrid iterative algorithm combining adaptive damped Jacobi and conjugate gradient was proposed in the literature [15], which requires several iterations to achieve near-optimal performance.