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 3.10+
• 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 dense and small enough to factor in memory (dense float64
storage n²·8 bytes < ~2 GB, i.e. n ≲ 16000)?
├── YES → Use direct solver (Cholesky/LDLᵀ/LU by symmetry)
└── 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	SPD preconditioner (SSOR, symmetric block-diagonal, or AMG on SPD part)
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, asymptotic_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 --ill-conditioned --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

convergence_diagnostics.py reports two rates: rate (mean of all per-iteration residual ratios over the full history) and asymptotic_rate (mean over a short trailing window). The stagnation flag is driven by asymptotic_rate (> 0.95), because stagnation is a tail property — early fast drops can hide a flat tail. Read asymptotic_rate when judging the regime below:

Asymptotic rate	Meaning	Action
< 0.1	Excellent	Current setup optimal
0.1 - 0.5	Good	Acceptable for most problems
0.5 - 0.95	Slow	Consider better preconditioner
> 0.95	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

Verification checklist

Do not trust a solve until each of these is satisfied with a recorded value, not a "looks fine":

• Recorded asymptotic_rate from convergence_diagnostics.py and confirmed it is below the 0.95 stagnation threshold (and ideally < 0.5); a low whole-history rate alone does not rule out a flat tail.
• Checked the relative residual from residual_norms.py against the physics-scaled --rel-tol (default 1e-6), not just the absolute norm; for unscaled RHS use --require-both so an undersized rhs cannot fake convergence.
• Confirmed solver_selector.py recommended matches the actual matrix properties recorded from sparsity_stats.py (symmetry, and definiteness if known) — e.g. CG only when symmetric AND positive-definite, MINRES for symmetric-indefinite, GMRES/BiCGSTAB for nonsymmetric.
• For systems flagged ill-conditioned, ran scaling_equilibration.py and recorded row_scale_max/row_scale_min and col_scale_max/col_scale_min; for symmetric matrices used --symmetric (D A D) so symmetry is preserved, and applied row_scale THEN col_scale for nonsymmetric two-sided scaling.
• Reviewed sparsity_stats.py notes/zero_rows/zero_cols from scaling_equilibration.py — any zero row or column means the system is structurally singular and the scale-of-1 fallback is not a fix.
• Confirmed the preconditioner from preconditioner_advisor.py is admissible for the chosen Krylov method — in particular a MINRES preconditioner must be SPD (an indefinite incomplete LDLᵀ is invalid).

Common pitfalls & rationalizations

Tempting shortcut	Why it's wrong / what to do
"The mean rate is low, so it converged."	rate is the whole-history mean and is dominated by early fast drops; stagnation is a tail property. Read asymptotic_rate and confirm it is below 0.95.
"The absolute residual is tiny, so we're done."	A small absolute norm can be meaningless if the RHS is large or unscaled. Check the relative_norms / relative_value against a physics-scaled --rel-tol.
"It's symmetric, so just use CG."	CG requires symmetric AND positive-definite. A symmetric-indefinite matrix needs MINRES (with an SPD preconditioner); using CG can break down or stall. Confirm definiteness before selecting.
"Large system, so factor it directly."	solver_selector.py gates dense direct solvers on dense float64 storage (n²·8 bytes < ~2 GB, n ≈ 16384); above that a dense Cholesky/LU is infeasible and you must route to an iterative method.
"Scaling is just dividing each row by its max."	One-sided row scaling does not equilibrate. For nonsymmetric matrices derive col_scale from the row-scaled matrix and apply both; for symmetric matrices use the symmetric D A D scale or you destroy symmetry.
"GMRES stagnates, so add more iterations."	A flat tail means the preconditioner or restart length is the problem, not iteration count. Strengthen the preconditioner (higher ILU fill / AMG), increase the restart parameter, or switch methods.

Security

Input Validation

• All numeric inputs (residuals, tolerances, matrix entries) are validated as finite numbers
• Comma-separated residual/vector inputs are capped at 100,000 entries
• The solver_selector.py --size parameter is bounded at 10 billion
• --matrix-type is validated against a fixed allowlist (spd, symmetric-indefinite, nonsymmetric)
• Boolean flags (--symmetric, --positive-definite, --sparse, --ill-conditioned) are type-safe argparse flags

File Access

• sparsity_stats.py and scaling_equilibration.py read a single matrix file (.npy format) specified by --matrix
• np.load() is called with allow_pickle=False to prevent arbitrary code execution via crafted .npy files
• Matrix files are rejected if they exceed 500 MB before any parsing occurs
• Matrix dimension limits (100,000 per dimension) prevent memory exhaustion
• All other scripts read no external files; inputs are provided via CLI arguments

Tool Restrictions

• Read: Used to inspect script source, references, and matrix files
• Write: Used to save analysis results or solver recommendations; writes are scoped to the user's working directory
• Grep/Glob: Used to locate relevant files and search references
• The skill's allowed-tools excludes Bash to prevent the agent from executing arbitrary commands when processing untrusted matrix files or numeric inputs

Safety Measures

• No eval(), exec(), or dynamic code generation
• All subprocess calls use explicit argument lists (no shell=True)
• Reduced tool surface (no Bash) limits the agent to read/write operations only
• JSON output mode produces structured, parseable results without shell-interpretable content

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

• v1.2.0 (2026-06-23): Fixed asymptotic stagnation detection, dense-feasibility solver gating, saddle-point/small-dense direct-solver routing, equilibrating two-sided scaling, CG iteration-bound table, and doc/eval consistency
• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 6 solver analysis scripts