linear-solvers

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Select and configure linear solvers for systems Ax=b in dense and sparse problems. Use when choosing direct vs iterative methods, diagnosing convergence issues, estimating conditioning, selecting preconditioners, or debugging stagnation in GMRES/CG/BiCGSTAB.

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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

Characterize matrix - symmetry, definiteness, sparsity
Analyze sparsity - Run scripts/sparsity_stats.py
Select solver - Run scripts/solver_selector.py
Choose preconditioner - Run scripts/preconditioner_advisor.py
Apply scaling - If ill-conditioned, use scripts/scaling_equilibration.py
Monitor convergence - Use scripts/convergence_diagnostics.py
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:

Diagnose convergence:

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

Check for preconditioning advice:

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

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