Sor: Convert Msor To

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 ).

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3. MSOR Generalization

MSOR allows a different relaxation parameter ( \omega_i ) for each equation (or for blocks). The iteration becomes:

[ x_i^(k+1) = (1-\omega_i) x_i^(k) + \frac\omega_ia_ii \left( b_i - \sum_j=1^i-1 a_ij x_j^(k+1) - \sum_j=i+1^n a_ij x_j^(k) \right) ]

If ( \omega_i = \omega ) for all ( i ), MSOR collapses to standard SOR. convert msor to sor

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Why Convert MSOR to SOR?

You might need to convert MSOR to SOR for several practical reasons:

1. Simplicity: SOR has only one parameter to tune. MSOR requires tuning two (or more) parameters, which is complex.
2. Convergence Guarantees: For symmetric positive definite (SPD) matrices, SOR converges if ( 0 < \omega < 2 ). MSOR's convergence is trickier and may diverge even for SPD matrices if ( \omega_1 ) and ( \omega_2 ) are not chosen carefully.
3. Software Compatibility: Many legacy or standard libraries (like some implementations in LAPACK or custom engineering solvers) only support SOR.
4. Performance Analysis: When debugging slow convergence, stripping MSOR back to SOR (with ( \omega_1 = \omega_2 )) helps isolate whether the issue is the problem itself or the parameter splitting.

🔄 Feature: “Relaxation Traceability Map”

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:

• Detect symmetry & positive definiteness in the matrix to decide if MSOR’s two parameters ( \omega_1, \omega_2 ) collapse to a single ( \omega ) in SOR.
• Generate an interactive plot showing convergence rate before/after conversion.
• Output a “recommended ω” for SOR based on the spectral radius of the MSOR iteration matrix.
• Add an optional consistency check: “Does MSOR reduce to standard SOR?” — if yes, the tool flags identical parameters.

Procedure (algorithmic steps)

1. Parse MSOR input into a list L of tokens (operations/variables) with metadata:

   • Unique identifier
   • Dependencies: set of predecessors that must occur before this token
   • Side-effects (reads/writes) and resource conflicts
   • Original SOR rank if available (used for tie-breaking)

2. Build a directed acyclic graph (DAG):

   • Nodes = tokens
   • Edges u -> v if token u must precede v (from dependencies or write-read/write-write conflicts)

3. Validate DAG:

   • If a cycle exists, report error (cyclic dependency). Attempt to break by:
     • Reporting conflicting tokens and required human resolution, or
     • If allowed, insert explicit synchronization/atomic operations to serialize the conflict (policy decision).

4. Topological sort to produce SOR:

   • Use Kahn’s algorithm or DFS topological sort.
   • For deterministic ordering among multiple available nodes, apply tie-breakers in this order: a. Native SOR rank (if provided) b. Lexicographic order of token identifiers c. Original MSOR relative order (stable sort)

5. Post-processing:

   • Validate produced sequence against original semantics:
     • Re-run dependency checks to ensure all constraints satisfied.
     • Optionally run lightweight static checks or unit tests.
   • If side-effect semantics change (e.g., changes in observable timing), annotate or flag for review.

6. Output:

   • Emit SOR sequence as a list.
   • Provide a mapping table from original MSOR positions to SOR positions (for traceability).
   • If any cycles or unresolved conflicts were found, include a short error report listing problematic tokens and suggested remediation.