Gaussian Elimination for Solving Ax=b with Algorithm Implementation Analysis

Solving linear systems Ax = b is a classical problem in numerical computation. Depending on matrix characteristics, either direct methods or iterative methods can be employed. Here's an analysis of implementation approaches for several common methods:

### 1. Gaussian Elimination Gaussian elimination is a direct method for solving linear systems that transforms matrix A into an upper triangular matrix through elementary row operations, followed by back substitution. Key implementation steps include: Forward elimination: Transform the coefficient matrix into upper triangular form using row operations while simultaneously adjusting the right-hand side vector b. Back substitution: Starting from the last row, progressively solve for unknowns x_i. This method is suitable for small to medium-sized dense matrices but may be unstable for ill-conditioned matrices. Code implementation typically involves nested loops for elimination and a separate loop for back substitution.

### 2. Triangular Decomposition (LU Factorization) LU factorization decomposes matrix A into a lower triangular matrix L and an upper triangular matrix U, such that A = LU. The solution process involves: Decomposition phase: Obtain L and U through a modified Gaussian elimination process using Doolittle's or Crout's algorithm. Solution phase: First solve Ly = b (forward substitution), then solve Ux = y (back substitution). LU factorization is particularly efficient when solving for multiple right-hand sides b, as the decomposition needs to be performed only once. The implementation requires storing both L and U matrices.

### 3. Jacobi Iterative Method Jacobi iteration is a matrix splitting-based iterative method suitable for diagonally dominant matrices. The core algorithm decomposes A into diagonal matrix D, strictly lower triangular matrix L, and strictly upper triangular matrix U, then iteratively updates: [ x^{(k+1)} = D^{-1}(b - (L + U)x^{(k)}) ] Jacobi method implementation is straightforward but converges slowly and may not converge for non-diagonally dominant matrices. Code implementation involves calculating the inverse of D (which is simple since D is diagonal) and matrix-vector operations.

### 4. Gauss-Seidel (GS) Iterative Method GS iteration improves upon Jacobi by immediately using updated components x_1^{(k+1)}, dots, x_{i-1}^{(k+1)} when computing x_i^{(k+1)}. The iteration formula is: [ x^{(k+1)} = (D + L)^{-1}(b - Ux^{(k)}) ] GS typically converges faster than Jacobi but remains sensitive to matrix condition number. Implementation advantage: requires only one vector storage since updates occur in-place, making it more memory-efficient than Jacobi.

### 5. Successive Over-Relaxation (SOR) Method SOR accelerates GS convergence by introducing a relaxation factor ω to control step size: [ x^{(k+1)} = (1 - ω)x^{(k)} + ω(D + ωL)^{-1}(b - Ux^{(k)}) ] When ω = 1, SOR reduces to GS method. Proper choice of ω (typically 1 < ω < 2) can significantly improve convergence rates, though optimal ω requires empirical or theoretical estimation. Implementation includes an additional weighting step compared to GS.

### Method Selection Guidelines Direct methods (Gaussian elimination, LU factorization) are suitable for small-to-medium problems or scenarios requiring high precision. Iterative methods (Jacobi, GS, SOR) are appropriate for large sparse matrices, especially when matrices exhibit diagonal dominance. In terms of convergence rates: SOR > GS > Jacobi, but practical performance depends on parameter selection (such as ω for SOR). For code implementation, consider matrix structure, memory requirements, and convergence criteria when choosing methods.