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

2. Check for preconditioning advice:

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

3. 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