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 3.10+
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

# Select solver for a small nonlinear least-squares (data-fitting) problem
python3 scripts/solver_selector.py --problem-type least-squares --size 6 --jacobian-available --smooth --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

# Globalization for a distant initial guess (favors trust region)
python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality good --far-from-solution --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
`problem_size (...) exceeds maximum (...)`	Size above 10 billion cap	Re-check the unit/value
`constraint_type must be one of...`	Unknown constraint	Use: none, bound, equality, inequality
`problem_type must be one of...`	Unknown problem type	Use: root-finding, optimization, least-squares
`residuals must be non-negative`	Invalid residual data	Check residual computation
`residuals must be finite`	NaN/Inf in residual data	Sanitize residual history
`residual list length (...) exceeds limit (...)`	More than 100,000 entries	Downsample the history
`Matrix file not found`	Invalid path	Verify Jacobian file exists
`Matrix file exceeds size limit ...`	Matrix file too large	Use a smaller / sparser matrix

Interpretation Guidance

Convergence Type

Type	Meaning	Action
quadratic	Optimal Newton (order p ≈ 2)	Continue, near solution
superlinear	Ratios shrinking toward 0 (1 < p < 2); quasi-Newton working	Monitor for stagnation
linear	Constant contraction ratio (p ≈ 1); a small constant ratio is fast-linear, not superlinear	May improve with preconditioner
sublinear	Too slow (ratio → 1)	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

Verification checklist

Do not trust a "solved" claim until these concrete artifacts are recorded:

Logged the full residual norm history and ran
```
convergence_analyzer.py --residuals <history>
```
; recorded
```
convergence_type
```
and
```
estimated_rate
```
, and confirmed
```
converged: true
```
against the actual solver tolerance (not the default
```
1e-10
```
).
Confirmed the residual sequence is monotone-decreasing or fed it to
```
residual_monitor.py
```
; recorded
```
patterns_detected
```
and verified it does NOT include
```
diverging
```
,
```
oscillating
```
,
```
plateau
```
, or
```
slow_convergence
```
while still above tolerance.
If a Jacobian is available, ran
```
jacobian_diagnostics.py --matrix J.txt
```
and recorded
```
condition_number
```
and
```
jacobian_quality
```
; for an analytic Jacobian, passed
```
--finite-diff-matrix
```
and confirmed
```
finite_diff_error
```
is below ~1e-2 (no "Large discrepancy" note).
Checked
```
rank_deficient
```
from
```
jacobian_diagnostics.py
```
is
```
false
```
(or documented why a rank-deficient/near-singular Jacobian is expected and that Levenberg-Marquardt regularization is in use).
For a trust-region solve, evaluated accepted steps with
```
step_quality.py
```
and recorded the reduction
```
ratio
```
; confirmed accepted steps have
```
ratio >= 0.25
```
(not
```
very_poor
```
/
```
poor
```
) and that the
```
trust_radius_action
```
matches the recorded ρ.
Recorded the solver and globalization actually used and confirmed they match
```
solver_selector.py
```
and
```
globalization_advisor.py
```
recommendations for the stated problem type, size, and Jacobian quality (e.g., large/expensive-Jacobian → Newton-Krylov; least-squares → Levenberg-Marquardt trust region).
Re-confirmed convergence after any change to tolerance, initial guess, or preconditioner — the convergence type can flip (e.g., quadratic → linear/stagnated) and must be re-classified, not assumed.

Common pitfalls & rationalizations

Tempting shortcut	Why it's wrong / what to do
"The residual ratio is a small constant (~0.1), so it's converging superlinearly."	A constant contraction ratio is linear, not superlinear — `convergence_analyzer.py` reports this as `linear` (annotated "fast linear"). Superlinear requires the ratio to tend to zero (order p > 1.2). Don't claim Newton-quality convergence from a flat ratio.
"It stopped without erroring, so the solver converged."	Run completion is not convergence. Check `converged` from `convergence_analyzer.py` / `residual_monitor.py` against the real tolerance; a `stagnated` or `plateau` result also "stops" but has not solved `f(x)=0` .
"Two iterations look like they're shrinking, so the rate is fine."	Order estimation needs at least 3 strictly decreasing positive residuals; with fewer, `convergence_analyzer.py` returns `unknown` /falls back to rate-only. Gather more iterations before quoting a convergence type.
"I coded the analytic Jacobian, so it must be right."	A wrong Jacobian still produces some step. Run `jacobian_diagnostics.py --finite-diff-matrix` and confirm `finite_diff_error` is small; a "Large discrepancy with finite-diff" note means the analytic Jacobian is buggy, which silently degrades Newton to linear convergence.
"Newton diverged, so I'll just shrink the global tolerance and call it close enough."	Divergence ( `convergence_type: diverged` , or `diverging` pattern) signals a bad step direction or far-from-solution start — add globalization. Run `globalization_advisor.py` (use `--far-from-solution` / report failures) and switch to a trust region or damped step instead of loosening the target.
"Trust-region step decreased the objective, so accept and expand the radius."	Acceptance and radius growth depend on the reduction ratio ρ, not just sign. `step_quality.py` only flags `expand` when ρ ≥ 0.75 and the step hit the boundary; a small positive ρ ( `marginal` ) means accept-but-shrink. Use the recorded `trust_radius_action` .
"The Jacobian is large and expensive, but full Newton is the gold standard, so I'll form it anyway."	For n ≥ 1000 or expensive Jacobians, `solver_selector.py` routes to matrix-free Newton-Krylov (JFNK) precisely because forming/factoring J is infeasible; use Jacobian-vector products plus a preconditioner instead.

Security

Input Validation

```
--size
```
(problem size) is validated as a positive integer, bounded at 10 billion
```
--residuals
```
are validated as finite non-negative numbers, capped at 100,000 entries
```
--tolerance
```
and
```
--target-tolerance
```
are validated as finite positive numbers
```
--problem-type
```
and
```
--constraint-type
```
are validated against fixed allowlists
```
--jacobian-quality
```
is validated against a fixed allowlist (
```
good
```
,
```
ill-conditioned
```
, etc.)

Step quality parameters (

predicted-reduction

actual-reduction

step-norm

gradient-norm

trust-radius

) are validated as finite numbers

File Access

```
jacobian_diagnostics.py
```
reads a single matrix file specified by
```
--matrix
```
; no directory traversal beyond the given path
Matrix files are size-limited and loaded with
```
allow_pickle=False
```
to prevent code execution
All other scripts read no external files; inputs are provided via CLI arguments
Scripts write only to stdout (JSON output)

Tool Restrictions

Read: Used to inspect script source, references, and user configuration files

Bash: Used to execute the six Python analysis scripts (

solver_selector.py

convergence_analyzer.py

jacobian_diagnostics.py

globalization_advisor.py

residual_monitor.py

step_quality.py

) with explicit argument lists

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

Safety Measures

No
```
eval()
```
,
```
exec()
```
, or dynamic code generation
All subprocess calls use explicit argument lists (no
```
shell=True
```
)
Matrix dimension limits prevent memory exhaustion when loading Jacobian files
Residual history analysis operates on bounded-length numeric arrays only

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

v1.2.2 (2026-06-24): Added a "Verification checklist" (evidence tied to each script's JSON outputs — convergence type/rate, residual patterns, Jacobian condition/finite-diff error, rank, trust-region step ratio, and solver/globalization agreement) and a "Common pitfalls & rationalizations" table covering constant-ratio-vs-superlinear, run-completion-vs-convergence, too-few-iterations, unverified analytic Jacobians, divergence handling, trust-region acceptance, and large/expensive-Jacobian routing
v1.2.0 (2026-06-23): Added
```
--problem-type
```
to
```
solver_selector.py
```
with a nonlinear least-squares path (Levenberg-Marquardt / Gauss-Newton); reordered solver selection so problem size dominates high-accuracy and routes large/expensive-Jacobian problems to Newton-Krylov; added
```
--far-from-solution
```
to
```
globalization_advisor.py
```
and surfaced Levenberg-Marquardt as the trust-region type for least-squares; corrected convergence classification so constant-ratio sequences are linear (not superlinear); RFC-8259-safe JSON (no
```
-Infinity
```
); input-validation hardening
v1.1.0 (2026-03-26): Optimized agent-discovery description, evaluation suite, security review docs, standardized metadata block, CHANGELOG
v1.0.0: Initial release with 6 analysis scripts

nonlinear-solvers

NPX Install

Tags

SKILL.md Content

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)

Verification checklist

Common pitfalls & rationalizations

Security

Input Validation

File Access

Tool Restrictions

Safety Measures

Limitations

References

Version History