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Linear Solvers · FreedomIntelligence | DecimalAI

Linear Solvers

SkillScore52Maintained

linear-solvers

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.

scienceby FreedomIntelligence

Links this skill to your agents and keeps it current with the author’s updates.

Automatic

Coding pack

1 · One-time — install the CLI (docs ↗)

pip install decimalai

2 · Get the skill — writes .claude/skills/linear-solvers/SKILL.md

decimalai skills pull linear-solvers --out .claude/skills/

No CLI? Get the same file — curl · raw file

curl -fsSL --create-dirs -o .claude/skills/linear-solvers/SKILL.md https://app.decimal.ai/s/linear-solvers/SKILL.md

Raw file: /s/linear-solvers/SKILL.md ↗ · machine-readable summary at /s/linear-solvers.json ↗

In productionGlobal

how this skill behaves on real traffic · last 30 days

No production data for this skill yet — usage numbers appear here once agents start running it.

Anonymized aggregate across every org using this skill — proof from real production use, not a demo.

Does this skill help? · Benchmark: with vs. withoutsign in to run this on your agents

23 test cases from the publisher’s eval.yaml · each run twice on gemini-3.6-flash — with the skill injected, then without · last run 57d ago

Yes — +43% lift+19% tokens0% turns52% of whole cases pass with skill (12/23)

Test case	Without → With	Effect	Δ tokens	Δ turns
case-04	✗→✓	▲ Improved	—	—
case-20	✗→✓	▲ Improved	—	—
case-03	✗→✓	▲ Improved	—	—
case-02	✗→✓	▲ Improved	—	—
case-14	✗→✓	▲ Improved	—	—

Showing 5 of 23 cases

SkillSafety

✓ 3 of 3 passed

Code scan15 of 15 checks clean

0 critical · 0 warning · 0 info — scanner v4 — 7/30/2026

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IntentSafe

LLM judge · does the skill do what it claims

ContentClean

LLM judge · policy & harmful-content review

Source

Imported

FreedomIntelligence / OpenClaw-Medical-Skills

skills/linear-solvers/SKILL.md

View on GitHub ↗

Imported 8/1/2026Version v1Bundled 10 files · scripts

Skill contents

scripts/

references/

SKILL.md

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:

bash 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:

bash 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

bash
# 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

Illustrative Want to see what linear-solvers would put in front of your model? Preview the rendered body and check its trigger phrases — no LLM call, no signup.

Latest benchmark · publisher runVerified

7/27/2026, 06:02 UTC · 23 cases

Lift

+43%

52% vs 9%

Agent turns

1.0 vs 1.0

Tokens

+19%

2,918.68 vs 2,456.91

fail → pass · 10pass → pass · 2fail → fail · 11

Per-case results (23)

Case	Status	Duration (ms)			Turns			Tokens			Tool calls
Case	Status	Without	With	Δ	Without	With	Δ	Without	With	Δ	Without	With	Δ
case-09 A conjugate gradient run achieves a residual reduction factor per iteration of 0.05. Evaluate this convergence rate classification and state whether adjustments are required.	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-08 During an iterative solve, the residual norm decreases across iterations with an estimated convergence rate factor per iteration of 0.75. According to standard numerical diagnostics guidelines, how is this rate interpreted and what action should be taken?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-04 I have a 50,000 x 50,000 sparse matrix from a structural vibration analysis. The matrix is strictly symmetric, but it is indefinite (has both positive and negative eigenvalues). Which iterative solver should be selected, and what preconditioner is appropriate?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-12 During a Krylov subspace solve, the residual norms bounce erratically up and down between 0.1 and 100 without decaying. What structural characteristic does this pattern indicate, and what fix is suggested?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-20 When running linear solver diagnostic utilities, what error condition occurs if a non-square matrix (e.g., 1000x800) is passed as input, and what is its prescribed cause and resolution?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-13 A high-dimensional linear solve exhibits a very slow, sluggish residual decay over hundreds of iterations. What matrix property typically causes very slow decay, and what pre-solve step is recommended?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-03 I am setting up a simulation with a sparse linear system of size N=1,000,000 where the matrix is verified to be real symmetric and positive definite (SPD). Suggest the primary recommended iterative solver and appropriate preconditioner for this system, along with the command-line script execution to verify this selection.	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-02 Before executing my linear system solve, I want to inspect matrix structural properties and calculate equilibration factors for the ill-conditioned system. Please perform an analysis that outputs the non-zero count, matrix density, bandwidth, symmetry determination, along with row_scale and col_scale vectors and notes.	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-14 Generate the exact command line invocation using the repository's solver selection utility to evaluate a 1,000,000 dimension SPD sparse matrix and output JSON results.	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-01 My iterative solver seems stuck—the residual norm history across recent iterations reads [0.8, 0.15, 0.04, 0.012, 0.008, 0.0075, 0.0074, 0.0075, 0.0074, 0.0075]. Please analyze this convergence trajectory and return a report detailing the calculated convergence rate, a boolean indicating if stagnation occurred, and the recommended action to address it.	pass→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-07 An incompressible Navier-Stokes formulation generates a block saddle-point linear system [A B; B^T 0]. What specialized solver approach and preconditioner structure are recommended for this class of systems?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-15 Provide the exact command line execution for recommending a preconditioner for a non-symmetric sparse matrix experiencing stagnation, requesting structured JSON output.	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-06 A CFD discretization produces a non-symmetric 200,000 x 200,000 sparse matrix with low sparsity. It shows no near-symmetry properties. What solver and preconditioner combination should be chosen for robust convergence?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-21 I am integrating a stiff system of ordinary differential equations dy/dt = f(t, y) using an explicit Runge-Kutta method (RK4), but the step size is forcing extremely tiny time steps due to numerical instability. What adaptive implicit ODE integration schemes (e.g. Radau, BDF) should be used in SciPy's solve_ivp for stiff differential equations?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-10 An iterative solver shows a residual reduction factor per iteration of 0.95 over 40 steps. What is the convergence classification for a rate exceeding 0.9, and what action is required?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-16 Provide the command line execution using the local utility script to compute residual norms given residual values 1, 0.1, 0.01 and RHS vector values 1, 0, 0 in JSON output format.	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-23 I need to compute all eigenvalues and eigenvectors for a dense 3000x3000 non-Hermitian matrix to analyze spectral stability. What LAPACK routines or NumPy/SciPy functions (e.g. scipy.linalg.eig) are suitable for dense non-symmetric eigenvalue problems, and what is the computational complexity class?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-18 What exact JSON output keys are produced by the script scripts/solver_selector.py upon successful analysis?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-19 List the exact JSON output keys expected from executing scripts/residual_norms.py.	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-05 We are solving a sequence of dense 500x500 linear systems resulting from boundary element discretizations. Memory is not a constraint for these small matrices. Recommend whether an iterative or direct solver should be used, and name the specific solver type.	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-11 A GMRES run displays a completely flat residual trajectory after iteration 15. What is the likely cause of a flat residual pattern and what is the prescribed fix?	pass→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-17 Outline the ordered 7-step numerical workflow for handling linear systems from initial matrix evaluation to final issue diagnosis.	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-22 I am solving a system of non-linear equations F(x) = 0 using Newton-Raphson iteration. How do line search criteria (such as Armijo or Wolfe conditions) guarantee global convergence when the standard Newton step leads to an increase in residual \|\|F(x)\|\|^2?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—

These cases were also screened by the skill’s intent / safety pass.See the Trust review ↗Reproduce this benchmark in Colab ↗

How this skill was measured

DecimalAI ran this skill against gemini-3.6-flash twice over the same eval suite — once with the skill loaded and once without — and compared the two runs case by case. 23 cases were attempted, and 22 counted toward the lift figure. The other 1 produced results that are not comparable between the two arms, so they are excluded from the headline rather than averaged into it. The headline lift of +43 percentage points is the difference between those two pass rates over the 22 comparable cases.

Model measured: gemini-3.6-flash
Graded by: gemini-3.6-flash
Grading method: judged
Verification: verified
Pass rate without the skill: 8.7%
Pass rate with the skill: 52.2%
Cases attempted: 23
Cases counted toward lift: 22
Cases that regressed: 0
Skill version measured: v1
Run completed: 7/27/2026, 06:02 UTC

The per-case answers from this run were removed by the retention sweep, so the case table below shows the verdicts without the text either arm produced. The counts above were recorded at the time and are unaffected. Answers are now kept for 180 days.

SkillScore blends 4 signals1 of 4 complete

BenchmarkPass rate WITH skill on a verified run52

Live evalNeeds ≥10 evaluated traces in 30 days

AI ratingNeeds ≥10 rated traces

AdoptionNeeds ≥10 serves across ≥2 orgs in 30 days

Efficiency · separate axis (at equal correctness)+0% turns+44% tokens

Shown for transparency; not credited unless a verified run shows no correctness regression.

Backed by 1 of 4 signals — each added signal makes the score more robust.

Verified on gemini-3.6-flash · re-tested 57 days ago

Other measured skills in the registry, with their headline benchmark lift.

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Linear Solvers

SkillSafety

Source

Linear Solvers

Goal

Requirements

Inputs to Gather

Decision Guidance

Solver Selection Flowchart

Quick Reference

Script Outputs (JSON Fields)

Workflow

Conversational Workflow Example

Pre-Solve Checklist

CLI Examples

Error Handling

Interpretation Guidance

Convergence Rate

Stagnation Diagnosis

Limitations

References

Version History

How this skill was measured

Related skills