linear-solvers

Linear Solvers

Goal

Provide a universal workflow to select a solver, assess conditioning, and diagnose convergence for linear systems arising in numerical simulations.

Requirements

• Python 3.8+
• NumPy, SciPy (for matrix operations)
• See individual scripts for dependencies

Inputs to Gather

| Input | Description | Example |

|-------|-------------|---------|

| Matrix size | Dimension of system | n = 1000000 |

| Sparsity | Fraction of nonzeros | 0.01% |

| Symmetry | Is A = Aᵀ? | yes |

| Definiteness | Is A positive definite? | yes (SPD) |

| Conditioning | Estimated condition number | 10⁶ |

Decision Guidance

Solver Selection Flowchart

Is matrix small (n < 5000) and dense?
├── YES → Use direct solver (LU, Cholesky)
└── NO → Is matrix symmetric?
    ├── YES → Is it positive definite?
    │   ├── YES → Use CG with AMG/IC preconditioner
    │   └── NO → Use MINRES
    └── NO → Is it nearly symmetric?
        ├── YES → Use BiCGSTAB
        └── NO → Use GMRES with ILU/AMG

Quick Reference

| Matrix Type | Solver | Preconditioner |

|-------------|--------|----------------|

| SPD, sparse | CG | AMG, IC |

| Symmetric indefinite | MINRES | ILU |

| Nonsymmetric | GMRES, BiCGSTAB | ILU, AMG |

| Dense | LU, Cholesky | None |

| Saddle point | Schur complement, Uzawa | Block preconditioner |

Script Outputs (JSON Fields)

| Script | Key Outputs |

|--------|-------------|

| scripts/solver_selector.py | recommended, alternatives, notes |

| scripts/convergence_diagnostics.py | rate, stagnation, recommended_action |

| scripts/sparsity_stats.py | nnz, density, bandwidth, symmetry |

| scripts/preconditioner_advisor.py | suggested, notes |

| scripts/scaling_equilibration.py | row_scale, col_scale, notes |

| scripts/residual_norms.py | residual_norms, relative_norms, converged |

Workflow

1. Characterize matrix - symmetry, definiteness, sparsity
2. Analyze sparsity - Run scripts/sparsity_stats.py
3. Select solver - Run scripts/solver_selector.py
4. Choose preconditioner - Run scripts/preconditioner_advisor.py
5. Apply scaling - If ill-conditioned, use scripts/scaling_equilibration.py
6. Monitor convergence - Use scripts/convergence_diagnostics.py
7. Diagnose issues - Check residual history with scripts/residual_norms.py

Conversational Workflow Example

User: My GMRES solver is stagnating after 50 iterations. The residual drops to 1e-3 then stops improving.

Agent workflow:

1. Diagnose convergence:

   python3 scripts/convergence_diagnostics.py --residuals 1,0.1,0.01,0.005,0.003,0.002,0.002,0.002 --json

1. Check for preconditioning advice:

   python3 scripts/preconditioner_advisor.py --matrix-type nonsymmetric --sparse --stagnation --json

1. Recommend: Increase restart parameter, try ILU(k) with higher k, or switch to AMG.

Pre-Solve Checklist

• [ ] Confirm matrix symmetry/definiteness
• [ ] Decide direct vs iterative based on size and sparsity
• [ ] Set residual tolerance relative to physics scale
• [ ] Choose preconditioner appropriate to matrix structure
• [ ] Apply scaling/equilibration if needed
• [ ] Track convergence and adjust if stagnation occurs

CLI Examples

# Analyze sparsity pattern
python3 scripts/sparsity_stats.py --matrix A.npy --json

# Select solver for SPD sparse system
python3 scripts/solver_selector.py --symmetric --positive-definite --sparse --size 1000000 --json

# Get preconditioner recommendation
python3 scripts/preconditioner_advisor.py --matrix-type spd --sparse --json

# Diagnose convergence from residual history
python3 scripts/convergence_diagnostics.py --residuals 1,0.2,0.05,0.01 --json

# Apply scaling
python3 scripts/scaling_equilibration.py --matrix A.npy --symmetric --json

# Compute residual norms
python3 scripts/residual_norms.py --residual 1,0.1,0.01 --rhs 1,0,0 --json

Error Handling

| Error | Cause | Resolution |

|-------|-------|------------|

| Matrix file not found | Invalid path | Check file exists |

| Matrix must be square | Non-square input | Verify matrix dimensions |

| Residuals must be positive | Invalid residual data | Check input format |

Interpretation Guidance

Convergence Rate

| Rate | Meaning | Action |

|------|---------|--------|

| < 0.1 | Excellent | Current setup optimal |

| 0.1 - 0.5 | Good | Acceptable for most problems |

| 0.5 - 0.9 | Slow | Consider better preconditioner |

| > 0.9 | Stagnation | Change solver or preconditioner |

Stagnation Diagnosis

| Pattern | Likely Cause | Fix |

|---------|--------------|-----|

| Flat residual | Poor preconditioner | Improve preconditioner |

| Oscillating | Near-singular or indefinite | Check matrix, try different solver |

| Very slow decay | Ill-conditioned | Apply scaling, use AMG |

Limitations

• Large dense matrices: Direct solvers may run out of memory
• Highly indefinite: Standard preconditioners may fail
• Saddle-point: Requires specialized block preconditioners

References

• references/solver_decision_tree.md - Selection logic
• references/preconditioner_catalog.md - Preconditioner options
• references/convergence_patterns.md - Diagnosing failures
• references/scaling_guidelines.md - Equilibration guidance

Version History

• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 6 solver analysis scripts