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You can take the average: [ \omega = \frac1n \sum_i=1^n \omega_i ] Or use the spectral radius-minimizing value for the matrix at hand.
In the world of numerical linear algebra, iterative methods are essential for solving large, sparse systems of linear equations, ( Ax = b ). Among the most famous classical iterative techniques are the Jacobi, Gauss-Seidel, and Successive Over-Relaxation (SOR) methods.
MSOR often has logic like:
However, you may have encountered a variant called the method. While it sounds more advanced, the "conversion" from MSOR to SOR is not a transformation of results but rather a conceptual and algorithmic simplification.
Set all ( \omega_i ) in your code to this single ( \omega ). The algorithm becomes: [ x_i^(k+1) = (1 - \omega) x_i^(k) + \frac\omegaa_ii \left( b_i - \sum_j < i a_ij x_j^(k+1) - \sum_j > i a_ij x_j^(k) \right) ]
for i in range(n): if i < n/2: w = 1.2 else: w = 1.8 x_new[i] = (1-w)*x_old[i] + w*(b[i] - sum(A[i][j]*x_new[j] for j<i) - sum(A[i][j]*x_old[j] for j>i)) / A[i][i]
You can take the average: [ \omega = \frac1n \sum_i=1^n \omega_i ] Or use the spectral radius-minimizing value for the matrix at hand.
In the world of numerical linear algebra, iterative methods are essential for solving large, sparse systems of linear equations, ( Ax = b ). Among the most famous classical iterative techniques are the Jacobi, Gauss-Seidel, and Successive Over-Relaxation (SOR) methods.
MSOR often has logic like:
However, you may have encountered a variant called the method. While it sounds more advanced, the "conversion" from MSOR to SOR is not a transformation of results but rather a conceptual and algorithmic simplification.
Set all ( \omega_i ) in your code to this single ( \omega ). The algorithm becomes: [ x_i^(k+1) = (1 - \omega) x_i^(k) + \frac\omegaa_ii \left( b_i - \sum_j < i a_ij x_j^(k+1) - \sum_j > i a_ij x_j^(k) \right) ]
for i in range(n): if i < n/2: w = 1.2 else: w = 1.8 x_new[i] = (1-w)*x_old[i] + w*(b[i] - sum(A[i][j]*x_new[j] for j<i) - sum(A[i][j]*x_old[j] for j>i)) / A[i][i]