What is nonlinear-solvers?

A guide for choosing and configuring numerical methods for solving equations, optimization problems, and least-squares problems. It covers methods such as Newton, BFGS, Broyden, and Levenberg–Marquardt.

Use it to choose solvers, configure line searches or trust regions, assess Jacobians, and investigate convergence using Python and NumPy.

It helps select a suitable solver and diagnose why a calculation is converging slowly, failing, or becoming unstable.

nonlinear-solvers — Skill for Claude Code by HeshamFS

Install

Getting it into your agent

One page per mod, every tool's command on it. A separate URL per tool would split the same page into five that compete with each other.

Any agent

npx skills add HeshamFS/materials-simulation-skills --skill nonlinear-solvers

Clone the repo

git clone --depth 1 https://github.com/HeshamFS/materials-simulation-skills

Made for: Claude Code.

Or install core-numerical, the plugin that ships this one along with the rest of its 8 skills.

Wrote this? Show the measurements

A badge with what this costs and how it scanned, read live from this page, so it follows the numbers instead of freezing them. Markdown for a README, HTML for a documentation site or a project page.

README.md

[![agentmods](https://agentmods.dev/badge/skills/heshamfs/materials-simulation-skills/nonlinear-solvers/github.svg)](https://agentmods.dev/skills/heshamfs/materials-simulation-skills/nonlinear-solvers)

Your own site

<a href="https://agentmods.dev/skills/heshamfs/materials-simulation-skills/nonlinear-solvers"><img src="https://agentmods.dev/badge/skills/heshamfs/materials-simulation-skills/nonlinear-solvers/github.svg" alt="Measured on agentmods" height="20"></a>

Or the 80×15 button, for a site that already has a row of RSS and ATOM ones. Only the verdict fits; the numbers stay here.

Your own site · 80×15

<a href="https://agentmods.dev/skills/heshamfs/materials-simulation-skills/nonlinear-solvers"><img src="https://agentmods.dev/badge/skills/heshamfs/materials-simulation-skills/nonlinear-solvers.svg" alt="Reviewed on agentmods" width="80" height="20"></a>

Per session 153 Skills are progressive disclosure: only the name and description are preloaded; the body loads when the skill is used.

When invoked 4,223 The whole file, excluding the scripts and references it only reads on demand.

Security scan A 0 findings. A grade says what 26 rules found in the file — not that it is safe.

Origin original No closer match found in the catalogue.

Token cost

What it costs to keep this loaded

Counted locally with the o200k_base tokenizer, which is exact for GPT models; Claude uses its own tokenizer and its counts differ. Treat this as one consistent yardstick across the catalogue rather than a bill. Prices are per million input tokens.

Model	Per session	Once invoked
Fable 5.1	$0.00153	$0.04223
Opus 5.5	$0.00061	$0.01689
Sonnet 5.5	$0.00031	$0.00845
Haiku 4.5	$0.00015	$0.00422

Measured yesterday against content hash c470fca1f9e1, method: parsed. Prices are Anthropic first-party input rates as of 2026-10-07, from the pricing page.

Security

Grade A, and why

nonlinear-solvers scanned grade A with 0 findings against 26 rules in 11 categories — prompt injection, anti-refusal, data exfiltration, privilege escalation, supply chain, agent snooping, system-prompt leakage, SSRF and excessive agency — measured yesterday.

The scan reads SKILL.md. This mod also ships 6 executable files (scripts/convergence_analyzer.py, scripts/globalization_advisor.py, scripts/jacobian_diagnostics.py, …), listed below but not scanned — reading those needs a real analyzer, not pattern matching.

A static scan of the body, not an audit. Every finding is printed with the line that produced it so you can judge whether it matters here. A mod is markdown that instructs an agent; that is exactly why what it instructs is worth reading.

Nothing flagged

None of the 26 patterns this scan looks for appear in this file: no shell pipes, no recursive deletes, no credential paths, no hidden text, no instruction-override or anti-refusal phrasing, no agent-config snooping. That is not a guarantee, it is the absence of the things that are checkable.

skills/core-numerical/nonlinear-solvers/SKILL.md · 277 lines

How it starts

The opening of the file, as written. The whole thing — 277 lines — stays where its author put it; the contents beside it link to each section on GitHub.

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`

Read the full file on GitHub · 277 lines

Files

What ships with it

12 files beside SKILL.md in the same directory: the scripts, references and assets a skill reads on demand. Not counted in the per-session cost; read them before you install if any of them is executable.

Changes

What this file has done since we first saw it

Hashed on every crawl. A supply-chain change to an agent config is a question of when, not whether, so the history is kept rather than the latest state alone.

yesterday Changed · +4 lines c470fca1f9e1
1mo ago First seen · 273 lines · 153 tokens per session scan A 1da26bfcf978

Subscribe to this mod's changes

nonlinear-solvers is a skill published in the GitHub repository HeshamFS/materials-simulation-skills (82 stars, last pushed 2d ago), licensed Apache-2.0. It adds 153 tokens to every session and 4,223 once invoked, about $0.0006 per session on Opus 5.5. A static security scan graded it A with 0 findings. No closer match exists in the catalogue, so it is treated as the original; first seen 2026-08-30.

LLM Scientific agent-skills agents cli-tools computational-science skills

personal-genomics

Analyse personal DNA / genome files for pharmacogenomics, disease risk, carrier status, ancestry and traits. Use whenever the user mentions DNA or genome analysis, a raw genome file, gene names, drug-gene interactions, or wants to combine multiple DNA sources.

sammcj/agentic-coding · 56 tokens

web-components

Architecture and coding rules for single-page applications using web components, BCE layering, lit-html templating, unidirectional Redux-style state management (reduction.js, standards-based), and standards-based client-side routing (Navigation API + URLPattern). Web standards and web platform first, minimal external…

AdamBien/airails · 160 tokens

sbce

Spec-driven BCE workflow where one capability spec equals one business component (same name) and the spec is the boundary contract. Invoked as /sbce new|apply (or by intent), it drives declare → converge; the stack's own test loop is the oracle for "done". new accepts a BC name or a natural-language feature…

AdamBien/airails · 215 tokens

javascript-conventions

Generic, composable JavaScript (ECMAScript) code conventions — modules, declarations, functions, collections, asynchrony, errors, JSDoc types, naming, and the Baseline lookup that decides which language and standard-library features may be used. Technology-neutral within the JavaScript world; meant to be composed with…

AdamBien/airails · 200 tokens

search-math-results

Find program-conditioned math results, constructions, examples, counterexamples, analogies, and background references. Use when the current active program needs repair, mutation, analogy, a program shift, or carefully gated obstruction search.

frenzymath/Danus · 48 tokens

web-system-tests

The web realization of the system-tests contract — browser-driven system tests with Playwright against a running frontend. Owns the project layout (tests/), the Playwright configuration (baseURL plus webServer, cross-engine projects), the role- and label-based selector policy, code coverage as an opt-in second run…

AdamBien/airails · 233 tokens