1---2name: nonlinear-solvers3description: Select and configure nonlinear solvers for root-finding f(x)=0, optimization min F(x), and least-squares problems — choose among Newton, Newton-Krylov, quasi-Newton (BFGS, L-BFGS), Broyden, Anderson acceleration, and Levenberg-Marquardt methods, configure line search or trust-region globalization, diagnose convergence rate (quadratic, linear, stagnated), and assess Jacobian quality and conditioning. Use when a Newton solver converges slowly or diverges, choosing between line search and trust region, debugging nonlinear iteration failures in FEM or phase-field codes, or selecting a solver for large-scale unconstrained optimization, even if the user only says "my Newton iterations aren't converging."4---56# Nonlinear Solvers78## Goal910Provide a universal workflow to select a nonlinear solver, configure globalization strategies, and diagnose convergence for root-finding, optimization, and least-squares problems.1112## Requirements1314- Python 3.10+15- NumPy (for Jacobian diagnostics)16- SciPy (optional, for advanced analysis)1718## Inputs to Gather1920| Input | Description | Example |21|-------|-------------|---------|22| Problem type | Root-finding, optimization, least-squares | `root-finding` |23| Problem size | Number of unknowns | `n = 10000` |24| Jacobian availability | Analytic, finite-diff, unavailable | `analytic` |25| Jacobian cost | Cheap or expensive to compute | `expensive` |26| Constraints | None, bounds, equality, inequality | `none` |27| Smoothness | Is objective/residual smooth? | `yes` |28| Residual history | Sequence of residual norms | `1,0.1,0.01,...` |2930## Decision Guidance3132### Solver Selection Flowchart3334```35Is Jacobian available and cheap?36├── YES → Problem size?37│ ├── Small (n < 1000) → Newton (full)38│ └── Large (n ≥ 1000) → Newton-Krylov39└── NO → Is objective smooth?40 ├── YES → Memory limited?41 │ ├── YES → L-BFGS or Broyden42 │ └── NO → BFGS43 └── NO → Anderson acceleration or Picard44```4546### Quick Reference4748| Problem Type | First Choice | Alternative | Globalization |49|--------------|--------------|-------------|---------------|50| Small root-finding | Newton | Broyden | Line search |51| Large root-finding | Newton-Krylov | Anderson | Trust region |52| Optimization | L-BFGS | BFGS | Wolfe line search |53| Least-squares | Levenberg-Marquardt | Gauss-Newton | Trust region |54| Bound constrained | L-BFGS-B | Trust-region reflective | Projected |5556## Script Outputs (JSON Fields)5758| Script | Key Outputs |59|--------|-------------|60| `scripts/solver_selector.py` | `recommended`, `alternatives`, `notes` |61| `scripts/convergence_analyzer.py` | `converged`, `convergence_type`, `estimated_rate`, `diagnosis` |62| `scripts/jacobian_diagnostics.py` | `condition_number`, `jacobian_quality`, `rank_deficient` |63| `scripts/globalization_advisor.py` | `strategy`, `line_search_type`, `trust_region_type`, `parameters` |64| `scripts/residual_monitor.py` | `patterns_detected`, `alerts`, `recommendations` |65| `scripts/step_quality.py` | `ratio`, `step_quality`, `accept_step`, `trust_radius_action` |6667## Workflow68691. **Characterize problem** - Identify type, size, Jacobian availability702. **Select solver** - Run `scripts/solver_selector.py`713. **Choose globalization** - Run `scripts/globalization_advisor.py`724. **Analyze Jacobian** - If available, run `scripts/jacobian_diagnostics.py`735. **Monitor residuals** - During solve, use `scripts/residual_monitor.py`746. **Analyze convergence** - Run `scripts/convergence_analyzer.py`757. **Evaluate steps** - For trust region, use `scripts/step_quality.py`7677## Conversational Workflow Example7879**User**: My Newton solver for a phase-field simulation is converging very slowly. After 50 iterations, the residual only dropped from 1 to 0.1.8081**Agent workflow**:821. Analyze convergence:83 ```bash84 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 --json85 ```862. Check globalization strategy:87 ```bash88 python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality ill-conditioned --previous-failures 0 --json89 ```903. Recommend: Switch to trust region with Levenberg-Marquardt regularization, or use Newton-Krylov with better preconditioning.9192## Pre-Solve Checklist9394- [ ] Confirm problem type (root-finding, optimization, least-squares)95- [ ] Assess Jacobian availability and cost96- [ ] Check initial guess quality97- [ ] Set appropriate tolerances98- [ ] Choose globalization strategy99- [ ] Prepare to monitor convergence100101## CLI Examples102103```bash104# Select solver for large unconstrained optimization105python3 scripts/solver_selector.py --size 50000 --smooth --memory-limited --json106107# Select solver for a small nonlinear least-squares (data-fitting) problem108python3 scripts/solver_selector.py --problem-type least-squares --size 6 --jacobian-available --smooth --json109110# Analyze convergence from residual history111python3 scripts/convergence_analyzer.py --residuals 1,0.1,0.01,0.001,0.0001 --tolerance 1e-6 --json112113# Diagnose Jacobian quality114python3 scripts/jacobian_diagnostics.py --matrix jacobian.txt --json115116# Get globalization recommendation117python3 scripts/globalization_advisor.py --problem-type optimization --jacobian-quality good --json118119# Globalization for a distant initial guess (favors trust region)120python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality good --far-from-solution --json121122# Monitor residual patterns123python3 scripts/residual_monitor.py --residuals 1,0.8,0.9,0.7,0.75,0.6 --target-tolerance 1e-8 --json124125# Evaluate step quality for trust region126python3 scripts/step_quality.py --predicted-reduction 0.5 --actual-reduction 0.4 --step-norm 0.8 --gradient-norm 1.0 --trust-radius 1.0 --json127```128129## Error Handling130131| Error | Cause | Resolution |132|-------|-------|------------|133| `problem_size must be positive` | Invalid size | Check problem dimension |134| `problem_size (...) exceeds maximum (...)` | Size above 10 billion cap | Re-check the unit/value |135| `constraint_type must be one of...` | Unknown constraint | Use: none, bound, equality, inequality |136| `problem_type must be one of...` | Unknown problem type | Use: root-finding, optimization, least-squares |137| `residuals must be non-negative` | Invalid residual data | Check residual computation |138| `residuals must be finite` | NaN/Inf in residual data | Sanitize residual history |139| `residual list length (...) exceeds limit (...)` | More than 100,000 entries | Downsample the history |140| `Matrix file not found` | Invalid path | Verify Jacobian file exists |141| `Matrix file exceeds size limit ...` | Matrix file too large | Use a smaller / sparser matrix |142143## Interpretation Guidance144145### Convergence Type146147| Type | Meaning | Action |148|------|---------|--------|149| quadratic | Optimal Newton (order p ≈ 2) | Continue, near solution |150| superlinear | Ratios shrinking toward 0 (1 < p < 2); quasi-Newton working | Monitor for stagnation |151| linear | Constant contraction ratio (p ≈ 1); a small constant ratio is fast-linear, not superlinear | May improve with preconditioner |152| sublinear | Too slow (ratio → 1) | Change method or formulation |153| stagnated | No progress | Check Jacobian, preconditioner |154| diverged | Increasing residual | Add globalization, check Jacobian |155156### Jacobian Quality157158| Quality | Condition Number | Action |159|---------|------------------|--------|160| good | < 10⁶ | Standard Newton works |161| moderately-conditioned | 10⁶ - 10¹⁰ | Consider scaling |162| ill-conditioned | > 10¹⁰ | Use regularization |163| near-singular | ∞ | Reformulate or use LM |164165### Step Quality (Trust Region)166167| Ratio ρ | Quality | Trust Radius |168|---------|---------|--------------|169| ρ < 0 | very_poor | Shrink aggressively |170| ρ < 0.25 | marginal | Shrink |171| 0.25 ≤ ρ < 0.75 | good | Maintain |172| ρ ≥ 0.75 | excellent | Expand if at boundary |173174## Verification checklist175176Do not trust a "solved" claim until these concrete artifacts are recorded:177178- [ ] 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`).179- [ ] 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.180- [ ] 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).181- [ ] 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).182- [ ] 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.183- [ ] 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).184- [ ] 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.185186## Common pitfalls & rationalizations187188| Tempting shortcut | Why it's wrong / what to do |189|-------------------|-----------------------------|190| "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. |191| "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`. |192| "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. |193| "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. |194| "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. |195| "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`. |196| "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. |197198## Security199200### Input Validation201- `--size` (problem size) is validated as a positive integer, bounded at 10 billion202- `--residuals` are validated as finite non-negative numbers, capped at 100,000 entries203- `--tolerance` and `--target-tolerance` are validated as finite positive numbers204- `--problem-type` and `--constraint-type` are validated against fixed allowlists205- `--jacobian-quality` is validated against a fixed allowlist (`good`, `ill-conditioned`, etc.)206- Step quality parameters (`predicted-reduction`, `actual-reduction`, `step-norm`, `gradient-norm`, `trust-radius`) are validated as finite numbers207208### File Access209- `jacobian_diagnostics.py` reads a single matrix file specified by `--matrix`; no directory traversal beyond the given path210- Matrix files are size-limited and loaded with `allow_pickle=False` to prevent code execution211- All other scripts read no external files; inputs are provided via CLI arguments212- Scripts write only to stdout (JSON output)213214### Tool Restrictions215- **Read**: Used to inspect script source, references, and user configuration files216- **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 lists217- **Write**: Used to save analysis results or solver recommendations; writes are scoped to the user's working directory218- **Grep/Glob**: Used to locate relevant files and search references219220### Safety Measures221- No `eval()`, `exec()`, or dynamic code generation222- All subprocess calls use explicit argument lists (no `shell=True`)223- Matrix dimension limits prevent memory exhaustion when loading Jacobian files224- Residual history analysis operates on bounded-length numeric arrays only225226## Limitations227228- **No global convergence guarantee**: All methods may fail for pathological problems229- **Jacobian accuracy**: Finite-difference Jacobian may be inaccurate near discontinuities230- **Large dense problems**: May require specialized solvers not covered here231- **Constrained optimization**: Complex constraints need SQP or interior point methods232233## References234235- `references/solver_decision_tree.md` - Problem-based solver selection236- `references/method_catalog.md` - Method details and parameters237- `references/convergence_diagnostics.md` - Diagnosing convergence issues238- `references/globalization_strategies.md` - Line search and trust region239240## Version History241242- **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 routing243- **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 hardening244- **v1.1.0** (2026-03-26): Optimized agent-discovery description, evaluation suite, security review docs, standardized metadata block, CHANGELOG245- **v1.0.0**: Initial release with 6 analysis scripts
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