Solving Linear Equations Using the Simultaneous Iterative Reconstruction Technique (SIRT) Method

The Simultaneous Iterative Reconstruction Technique (SIRT) is an iterative method for solving systems of linear equations. Compared to traditional direct solution methods, SIRT not only yields more accurate results but also operates with greater computational efficiency. In each iteration cycle, the method calculates the residual vector - the difference between the current solution and the actual values - and applies corrections to gradually converge toward the true solution. The core algorithm typically implements weighted updates across all equations simultaneously, ensuring stability through relaxation parameters. This approach demonstrates notable robustness, handling ill-conditioned systems and noise-contaminated data effectively. Consequently, SIRT finds extensive application in scientific computing and engineering design, particularly in tomography reconstruction and large-scale inverse problems where direct methods become computationally prohibitive. The implementation commonly involves matrix-vector multiplications and requires proper initialization of relaxation factors to balance convergence speed and solution stability.