Iterative Linear System Solvers
Solve large linear systems \( Ax = b \) using iterative approximation methods. These methods are memory efficient and work well for sparse matrices.
Jacobi Method
Simple iterative method using simultaneous updates
Convergence:Conditionally convergent
Parallelization:Excellent
Features:
- Simultaneous updates
- Simple implementation
- Memory efficient
Update Rule:
x_i^{(k+1)} = (b_i - Σ a_{ij}x_j^{(k)}) / a_{ii}Convergence Condition:
Diagonally dominant matrices
Gauss Seidel
Improved Jacobi method using sequential updates
Convergence:Faster than Jacobi
Parallelization:Limited
Features:
- Sequential updates
- Faster convergence
- Uses latest values
Update Rule:
x_i^{(k+1)} = (b_i - Σ a_{ij}x_j^{(k+1)} - Σ a_{ij}x_j^{(k)}) / a_{ii}Convergence Condition:
Positive definite or diagonally dominant
Method Comparison
| Method | Memory | Convergence Rate | Parallelizable | Best For |
|---|---|---|---|---|
| Jacobi | Low | Linear | Yes | Large sparse systems |
| Gauss Seidel | Low | Faster Linear | No | Sequential processing |
About Iterative Methods
Iterative methods solve linear systems by generating a sequence of approximations that converge to the exact solution. They are particularly useful for:
- Large sparse systems: Memory efficient storage
- Approximate solutions: Can stop when desired accuracy reached
- Parallel computing: Some methods parallelize well
- Preconditioned systems: Can be combined with preconditioning