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Nonlinear Solvers · FreedomIntelligence | DecimalAI

Nonlinear Solvers

SkillScore70Maintained

nonlinear-solvers

Select and configure nonlinear solvers for f(x)=0 or min F(x). Use for Newton methods, quasi-Newton (BFGS, L-BFGS), Broyden, Anderson acceleration, diagnosing convergence issues, choosing line search vs trust region, and analyzing Jacobian quality.

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/nonlinear-solvers/SKILL.md

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

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

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

Raw file: /s/nonlinear-solvers/SKILL.md ↗ · machine-readable summary at /s/nonlinear-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 58d ago

Yes — +57% lift · at higher cost+44% tokens0% turns70% of whole cases pass with skill (16/23)

Test case	Without → With	Effect	Δ tokens	Δ turns
case-01	✗→✓	▲ Improved	—	—
case-18	✗→✓	▲ Improved	—	—
case-20	✗→✓	▲ Improved	—	—
case-03	✗→✓	▲ Improved	—	—
case-11	✗→✓	▲ 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/nonlinear-solvers/SKILL.md

View on GitHub ↗

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

Skill contents

scripts/

references/

SKILL.md

Nonlinear Solvers

Goal

Provide a universal workflow to select a nonlinear solver, configure globalization strategies, and diagnose convergence for root-finding, optimization, and least-squares problems.

Requirements

Python 3.8+
NumPy (for Jacobian diagnostics)
SciPy (optional, for advanced analysis)

Inputs to Gather

| Input | Description | Example | |-------|-------------|---------| | Problem type | Root-finding, optimization, least-squares | root-finding | | Problem size | Number of unknowns | n = 10000 | | Jacobian availability | Analytic, finite-diff, unavailable | analytic | | Jacobian cost | Cheap or expensive to compute | expensive | | Constraints | None, bounds, equality, inequality | none | | Smoothness | Is objective/residual smooth? | yes | | Residual history | Sequence of residual norms | 1,0.1,0.01,... |

Decision Guidance

Solver Selection Flowchart

Is Jacobian available and cheap?
├── YES → Problem size?
│   ├── Small (n < 1000) → Newton (full)
│   └── Large (n ≥ 1000) → Newton-Krylov
└── NO → Is objective smooth?
    ├── YES → Memory limited?
    │   ├── YES → L-BFGS or Broyden
    │   └── NO → BFGS
    └── NO → Anderson acceleration or Picard

Quick Reference

| Problem Type | First Choice | Alternative | Globalization | |--------------|--------------|-------------|---------------| | Small root-finding | Newton | Broyden | Line search | | Large root-finding | Newton-Krylov | Anderson | Trust region | | Optimization | L-BFGS | BFGS | Wolfe line search | | Least-squares | Levenberg-Marquardt | Gauss-Newton | Trust region | | Bound constrained | L-BFGS-B | Trust-region reflective | Projected |

Script Outputs (JSON Fields)

| Script | Key Outputs | |--------|-------------| | scripts/solver_selector.py | recommended, alternatives, notes | | scripts/convergence_analyzer.py | converged, convergence_type, estimated_rate, diagnosis | | scripts/jacobian_diagnostics.py | condition_number, jacobian_quality, rank_deficient | | scripts/globalization_advisor.py | strategy, line_search_type, trust_region_type, parameters | | scripts/residual_monitor.py | patterns_detected, alerts, recommendations | | scripts/step_quality.py | ratio, step_quality, accept_step, trust_radius_action |

Workflow

Characterize problem - Identify type, size, Jacobian availability
Select solver - Run scripts/solver_selector.py
Choose globalization - Run scripts/globalization_advisor.py
Analyze Jacobian - If available, run scripts/jacobian_diagnostics.py
Monitor residuals - During solve, use scripts/residual_monitor.py
Analyze convergence - Run scripts/convergence_analyzer.py
Evaluate steps - For trust region, use scripts/step_quality.py

Conversational Workflow Example

User: My Newton solver for a phase-field simulation is converging very slowly. After 50 iterations, the residual only dropped from 1 to 0.1.

Agent workflow:

Analyze convergence:

bash python3 scripts/convergence_analyzer.py --residuals 1,0.8,0.6,0.5,0.4,0.3,0.2,0.15,0.12,0.1 --json

Check globalization strategy:

bash python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality ill-conditioned --previous-failures 0 --json

Recommend: Switch to trust region with Levenberg-Marquardt regularization, or use Newton-Krylov with better preconditioning.

Pre-Solve Checklist

] Confirm problem type (root-finding, optimization, least-squares)
] Assess Jacobian availability and cost
] Check initial guess quality
] Set appropriate tolerances
] Choose globalization strategy
] Prepare to monitor convergence

CLI Examples

bash
# Select solver for large unconstrained optimization
python3 scripts/solver_selector.py --size 50000 --smooth --memory-limited --json

# Analyze convergence from residual history
python3 scripts/convergence_analyzer.py --residuals 1,0.1,0.01,0.001,0.0001 --tolerance 1e-6 --json

# Diagnose Jacobian quality
python3 scripts/jacobian_diagnostics.py --matrix jacobian.txt --json

# Get globalization recommendation
python3 scripts/globalization_advisor.py --problem-type optimization --jacobian-quality good --json

# Monitor residual patterns
python3 scripts/residual_monitor.py --residuals 1,0.8,0.9,0.7,0.75,0.6 --target-tolerance 1e-8 --json

# Evaluate step quality for trust region
python3 scripts/step_quality.py --predicted-reduction 0.5 --actual-reduction 0.4 --step-norm 0.8 --gradient-norm 1.0 --trust-radius 1.0 --json

Error Handling

| Error | Cause | Resolution | |-------|-------|------------| | problem_size must be positive | Invalid size | Check problem dimension | | constraint_type must be one of... | Unknown constraint | Use: none, bound, equality, inequality | | residuals must be non-negative | Invalid residual data | Check residual computation | | Matrix file not found | Invalid path | Verify Jacobian file exists |

Interpretation Guidance

Convergence Type

| Type | Meaning | Action | |------|---------|--------| | quadratic | Optimal Newton | Continue, near solution | | superlinear | Quasi-Newton working | Monitor for stagnation | | linear | Acceptable | May improve with preconditioner | | sublinear | Too slow | Change method or formulation | | stagnated | No progress | Check Jacobian, preconditioner | | diverged | Increasing residual | Add globalization, check Jacobian |

Jacobian Quality

| Quality | Condition Number | Action | |---------|------------------|--------| | good | < 10⁶ | Standard Newton works | | moderately-conditioned | 10⁶ - 10¹⁰ | Consider scaling | | ill-conditioned | > 10¹⁰ | Use regularization | | near-singular | ∞ | Reformulate or use LM |

Step Quality (Trust Region)

| Ratio ρ | Quality | Trust Radius | |---------|---------|--------------| | ρ < 0 | very_poor | Shrink aggressively | | ρ < 0.25 | marginal | Shrink | | 0.25 ≤ ρ < 0.75 | good | Maintain | | ρ ≥ 0.75 | excellent | Expand if at boundary |

Limitations

No global convergence guarantee: All methods may fail for pathological problems
Jacobian accuracy: Finite-difference Jacobian may be inaccurate near discontinuities
Large dense problems: May require specialized solvers not covered here
Constrained optimization: Complex constraints need SQP or interior point methods

References

references/solver_decision_tree.md - Problem-based solver selection
references/method_catalog.md - Method details and parameters
references/convergence_diagnostics.md - Diagnosing convergence issues
references/globalization_strategies.md - Line search and trust region

Version History

v1.0.0 : Initial release with 6 analysis scripts

Illustrative Want to see what nonlinear-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

7/26/2026, 11:57 UTC · 23 cases

Lift

+57%

70% vs 13%

Agent turns

1.0 vs 1.0

Tokens

+44%

3,197.5 vs 2,227.18

fail → pass · 13pass → pass · 3fail → fail · 7

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-01 I am solving an unconstrained root-finding system with 12,000 unknowns. The Jacobian is analytic but expensive to calculate, and the objective function is smooth. Please analyze these parameters and provide a structured solver output including the main recommended solver, alternative solver choices, explanatory notes, and a recommended globalization strategy.	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-18 I am integrating the residual monitoring script into an automated solver pipeline. What exact JSON output fields will the residual monitoring script return upon execution?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-20 I am setting up an automated CLI script call for globalization advice when tackling a root-finding problem with an ill-conditioned Jacobian after zero previous failures. What parameters should be passed to the globalization advisor script?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-04 I am simulating an initial value problem using an explicit adaptive Runge-Kutta (RK45) time-stepper. What adaptive step-size control safety factor and tolerance scaling power should I use for error control in explicit ODE solvers?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-07 We are optimizing a smooth, unconstrained loss function with 500 parameters where no analytic Jacobian is available and memory is not constrained. Which quasi-Newton solver should be selected as the primary option, what is the top alternative, and which line search rule should be used?	pass→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-03 I am solving a large sparse linear system Ax = b using GMRES, and the matrix condition number is 1e8. Which preconditioner algorithm (ILU(k) or Jacobi) should I choose and how should I set the drop tolerance for sparse linear solvers?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-08 We need to minimize a smooth objective with 2,000,000 parameters. We cannot store full Hessian approximations in memory, and the gradient must be approximated. What is the recommended primary solver, and what key output schema fields are generated when running the solver selection script?	pass→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-23 Evaluate a trust region step where predicted reduction was 0.40 and actual reduction was 0.20 (ratio rho = 0.50). How is the step quality categorized, and what trust radius adjustment action should be taken?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-11 I am tackling a non-smooth root-finding formulation where finite-difference Jacobian estimates fail completely due to localized non-differentiability. Which fixed-point or sequence acceleration methods are recommended for this non-smooth scenario?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-05 I have a multi-variable math expression string and need to generate symbolic derivative expressions using expression tree differentiation rules. How do I construct the symbolic AST traversal algorithm?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-02 I need to check the quality of a trust region step where the predicted reduction was 0.50, actual reduction was 0.08, step norm was 0.6, gradient norm was 1.2, and current trust radius was 1.0. Please compute and return the reduction ratio, step quality rating, step acceptance decision, and the action to take regarding trust radius adjustment.	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-10 I need to solve an optimization task with simple upper and lower boundary constraints on all 1,000 decision variables. Which bound-constrained solver is the primary choice, and what alternative is suited for this setup?	pass→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-17 In a trust region step, the predicted reduction was 0.80 and the actual reduction achieved was 0.72 (ratio = 0.90) with the step taking the full boundary of the current trust radius. What step quality rating applies, is the step accepted, and what trust radius action is recommended?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-12 A 100x100 Jacobian matrix extracted during a Newton iteration has a computed condition number of 4.5e4. How should this Jacobian quality be rated, and what action is required?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-09 I am fitting a non-linear curve-fitting model to 500 observational data points by minimizing residual sum-of-squares. Which method is the first choice for non-linear least-squares, what alternative is standard, and what globalization approach is paired with it?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-06 I have a non-linear system of 50 equations with a cheap, exact analytic Jacobian. I want to find a root starting from an initial guess without bounds or constraints. Which algorithm should be the primary choice, what is the best backup method, and what globalization tactic is standard?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-21 When passing problem dimension parameters to the solver selection tool, an error occurs because a negative integer -5 was provided for problem size. What error message does the script generate, and what is the resolution?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-19 We need to log diagnostic results from the convergence analyzer script into our database. Which four JSON fields are produced by this convergence analysis script?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-14 A numerical Jacobian matrix diagnosis returns an infinite condition number during solver execution. What condition quality category does this represent, and what resolution should be applied?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-22 A script invocation provides `--constraint linear_inequality` to the solver selector. What error message is produced, and what valid constraint types are listed in the error resolution?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-13 When analyzing a system Jacobian, the condition number evaluates to 5.2e11. What quality tier does this fall into, and what solver modification or action should be taken?	fail→fail	—	—	—	—	—	—	—	—	—	—	—	—
case-16 A trust region step yielded a predicted reduction of 0.40 but the actual function value increased, resulting in an actual reduction of -0.10. What is the computed ratio rho, how is step quality rated, and what action must be taken on the trust radius?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—
case-15 During a solver run, the residual history across consecutive steps reads: 1.0, 0.45, 0.12, 0.119, 0.119, 0.118, 0.118. How is this convergence pattern classified, and what diagnostic steps are indicated?	fail→pass	—	—	—	—	—	—	—	—	—	—	—	—

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 +57 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: 13%
Pass rate with the skill: 69.6%
Cases attempted: 23
Cases counted toward lift: 22
Cases that regressed: 0
Skill version measured: v1
Run completed: 7/26/2026, 11:57 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 run70

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+22% 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 58 days ago

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

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Nonlinear Solvers

SkillSafety

Source

Nonlinear 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 Type

Jacobian Quality

Step Quality (Trust Region)

Limitations

References

Version History

How this skill was measured

Related skills