Convert Msor To Sor Now

Produce a deterministic procedure that takes an MSOR sequence and produces the equivalent SOR sequence while preserving semantic dependencies and effects.

Converting MSOR to SOR is straightforward: replace the per-equation relaxation parameters ( \omega_i ) with a single constant ( \omega ). This reduces MSOR to standard SOR, simplifying the algorithm at the cost of flexibility. No further structural changes are required, as the iteration order and matrix splitting remain identical.

Here’s a step-by-step guide to convert the MSOR (Modified Successive Over-Relaxation) method into the standard SOR (Successive Over-Relaxation) method for solving linear systems ( Ax = b ).

Input MSOR tokens with dependencies:

Graph edges: C->B, A->D, B->D Topological sort (break ties by lexicographic): A, C, B, D Output SOR: A, C, B, D Mapping: MSOR positions -> SOR positions: A:1->1, B:2->3, C:3->2, D:4->4

If you don’t have a heuristic, run a small calibration loop:

def find_equivalent_sor(A, b, omega1, omega2, test_omegas=np.linspace(1.0, 1.9, 10)):
    x_msor = msor_solve(A, b, omega1, omega2, tol=1e-8)
    best_omega = 1.0
    best_error = float('inf')
    for omega in test_omegas:
        x_sor = sor_solve(A, b, omega, tol=1e-8)
        err = np.linalg.norm(x_sor - x_msor)
        if err < best_error:
            best_error = err
            best_omega = omega
    return best_omega
To convert MSOR to SOR, we unify the relaxation factor and remove the branch. convert msor to sor
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
When converting MSOR to SOR, this feature visually maps how each modified relaxation parameter ( \omega_\textMSOR ) is transformed into the standard ( \omega_\textSOR ) for a given linear system.
For example, it could:
If your MSOR implementation uses the same relaxation factor for all subgroups (( \omega_1 = \omega_2 = ... = \omega_n )), then your MSOR is already SOR. The conversion is simply a matter of renaming variables and removing subgroup logic. Produce a deterministic procedure that takes an MSOR
Before (MSOR with identical omegas):
if (i is in Group 1):
    omega = 1.5
else:
    omega = 1.5

After (SOR):
omega = 1.5  # Single global parameter

You might need to convert MSOR to SOR for several practical reasons: Here’s a step-by-step guide to convert the MSOR