Iterative Methods for Solving Linear Systems
A Comparison of Iterative methods are numerical techniques used to solve linear systems of equations, particularly when the system is large and sparse. Unlike direct methods, which aim to find the exact solution in a finite number of steps, iterative methods start with an initial guess and refine it through successive approximations until a desired level of accuracy is reached.
领英推è
Here are some of the most common iterative methods:
- Jacobi Method Concept: The Jacobi method updates each component of the solution vector in parallel using the previous iterations values. Convergence: Converges if the matrix is strictly diagonally dominant. Pros: Simple to implement and parallelize. Cons: Often slower than other methods.
- Gauss-Seidel Method Concept: Similar to Jacobi, but uses updated values as soon as they are available. Convergence: Converges faster than Jacobi for many problems. Pros: Generally more efficient than Jacobi. Cons: Not as easily parallelizable as Jacobi.
- Successive Over-Relaxation (SOR) Method Concept: Extends Gauss-Seidel by introducing a relaxation parameter (ω). If ω = 1, it reduces to Gauss-Seidel. Convergence: Can converge faster than Gauss-Seidel if the optimal ω is chosen. Pros: Can significantly improve convergence speed. Cons: Requires tuning the relaxation parameter.
- Conjugate Gradient Method Concept: Generates a sequence of orthogonal search directions to minimize a quadratic function. Convergence: Converges in at most n iterations for an n x n system. Pros: Often very efficient for symmetric positive-definite matrices. Cons: More complex to implement than Jacobi or Gauss-Seidel.
- Generalized Minimal Residual (GMRES) Method Concept: Finds an approximate solution by minimizing the residual norm over a Krylov subspace. Convergence: Effective for general non-symmetric systems. Pros: Robust and efficient for a wide range of problems. Cons: Can be computationally expensive, especially for large iteration counts.