What does PGS mean in UNCLASSIFIED

Working Principle

PGS method iteratively updates the solution vector by using the values from the previous iteration. At each iteration, the method updates each component of the solution vector using the following formula:

x_i^{(k+1)} = (1 - ω) * x_i^{(k)} + ω * (b_i - ∑_{j=1, j ≠ i}^{n} a_{ij} x_j^{(k+1)}) / a_{ii}

where:

• x_i^{(k)} is the i-th component of the solution vector at the k-th iteration
• ω is a relaxation factor that controls the rate of convergence
• b_i is the i-th component of the right-hand side vector
• a_{ij} is the (i, j)-th entry of the coefficient matrix

Convergence

PGS method converges under certain conditions. If the coefficient matrix is symmetric and positive definite, and the relaxation factor ω is chosen appropriately, the method will converge to the solution of the linear system. The optimal value of the relaxation factor depends on the problem and can be determined empirically.

Advantages

• Simplicity: PGS method is easy to implement and requires minimal memory.
• Robustness: It is relatively robust to changes in the coefficient matrix and the right-hand side vector.
• Parallelizable: The method can be easily parallelized, making it suitable for large-scale problems.

Disadvantages

• Slow convergence: PGS method can be slower to converge compared to other iterative methods.
• Memory requirements: The method requires storing the entire coefficient matrix in memory, which can be a problem for large-scale problems.