Skill ファイル

Nonlinear Solvers

Name: Nonlinear Solvers
Author: catlab-design

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.

catlab-design0 スター2026/03/02

職業
カテゴリ: 機械学習

スキル内容

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`

関連 Skill

Nonlinear Solvers | Skills Pool

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

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	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`

Analyze convergence:

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:

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.

# 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	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

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

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

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

Nonlinear Solvers

Goal

Requirements

Inputs to Gather

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

Continuous Learning V2

Continuous Learning V2

Continuous Learning V2

Continuous Learning

Continuous Learning

Pytorch Patterns