Convergence and Divergence of Gauss-Seidel Method : Numerical Methods (2024)

Описание: Convergence and Divergence of Gauss-Seidel Method : Numerical Methods (2024)
The Gauss-Seidel method is an iterative technique used to solve a system of linear equations of the form
𝐴𝑥=𝑏
Ax=b, where, A is a square matrix, 𝑥, is the vector of unknowns, and b is a known vector. This method is particularly useful when solving large, sparse systems because it can converge more efficiently than direct methods like Gaussian elimination.

Convergence Criteria
The convergence of the Gauss-Seidel method depends primarily on the properties of the matrix 𝐴. The following are common conditions that can ensure the method converges:

Diagonally Dominant Matrices: If the matrix 𝐴 is strictly diagonally dominant, then the Gauss-Seidel method will converge. A matrix is diagonally dominant if for every row 𝑖, the magnitude of the diagonal element is greater than the sum of the magnitudes of the off-diagonal elements.
Divergence
The Gauss-Seidel method may diverge if the matrix 𝐴 does not satisfy the necessary conditions for convergence. Divergence means that the iterative process does not move closer to the true solution, and instead, the values may oscillate or grow without bound.

Non-Diagonally Dominant or Ill-Conditioned Matrices: If the matrix 𝐴 is not diagonally dominant and is ill-conditioned (i.e., its condition number is large), the method can diverge. This is often the case for matrices where small perturbations in the elements lead to large changes in the solution.

Matrices with Eigenvalues Greater than 1: If the iteration matrix 𝐵 has eigenvalues (or a spectral radius) greater than 1, the iterative process amplifies errors rather than reducing them, causing divergence.

Oscillatory Behavior: For certain types of matrices, especially those with complex or negative eigenvalues, the Gauss-Seidel method may oscillate between values without converging to the true solution.

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