Convert Msor To Sor

To convert MSOR to SOR, we unify the relaxation factor and remove the branch.

def sor_solve(A, b, omega, tol=1e-6, max_iter=1000):
    n = len(b)
    x = np.zeros_like(b)
    for _ in range(max_iter):
        x_old = x.copy()
        for i in range(n):
            sigma = np.dot(A[i, :], x) - A[i, i] * x[i]
            x[i] = (1 - omega) * x[i] + (omega / A[i, i]) * (b[i] - sigma)
        if np.linalg.norm(x - x_old) < tol:
            break
    return x


Build a directed acyclic graph (DAG):



Validate DAG:





Topological sort to produce SOR:



Post-processing:


If side-effect semantics change (e.g., changes in observable timing), annotate or flag for review.



Output:



Successive Over-Relaxation (SOR) is a classic iterative method for solving linear systems ( Ax = b ). The Modified Successive Over-Relaxation (MSOR) method is a variant that uses different relaxation parameters for different equations or variable groups. Converting MSOR to SOR typically involves parameter unification and algorithmic reduction, making MSOR a special case or a generalized form of SOR. convert msor to sor

To convert an existing MSOR implementation to SOR, follow these steps:
| Step | Action |
|------|--------|
| 1 | Identify the array or function that stores ( \omega_i ) for ( i = 1, \dots, n ). |
| 2 | Replace all instances of ( \omega_i ) with a single global variable ( \omega ). |
| 3 | Remove any logic that updates ( \omega_i ) per iteration or per equation. |
| 4 | (Optional) Choose ( \omega ) in the optimal range ( (0, 2) ), typically ( \omega = 1 ) for Gauss-Seidel or an estimated spectral radius value. |
Algorithmic change in pseudocode:
MSOR version:
for i = 1 to n
    x_new[i] = (1 - omega[i]) * x_old[i] + (omega[i]/A[i][i]) * (b[i] - sum)
end
SOR version (after conversion):
omega_const = 1.5   (example)
for i = 1 to n
x_new[i] = (1 - omega_const) * x_old[i] + (omega_const/A[i][i]) * (b[i] - sum)
end