What is linear-solvers?

A guide for choosing methods to solve linear equations written as Ax=b, often produced by scientific and engineering simulations. It covers direct methods, iterative methods, matrix properties, and preconditioners that can improve convergence.

Use it to select and configure LU, Cholesky, CG, GMRES, BiCGSTAB, or MINRES solvers. It also helps assess matrices, choose AMG, ILU, or IC preconditioners, and apply scaling.

It helps avoid choosing a solver that uses too much memory or converges poorly for the matrix at hand. The guidance connects matrix size, sparsity, symmetry, and conditioning to solver choices.

linear-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 linear-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/linear-solvers/github.svg)](https://agentmods.dev/skills/heshamfs/materials-simulation-skills/linear-solvers)

Your own site

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

Per session 140 Skills are progressive disclosure: only the name and description are preloaded; the body loads when the skill is used.

When invoked 3,287 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	$0.00140	$0.03287
Opus 5.5	$0.00056	$0.01315
Sonnet 5.5	$0.00028	$0.00657
Haiku 4.5	$0.00014	$0.00329

Measured 2d ago against content hash 5d339bc2a9e6, method: parsed. Prices are Anthropic first-party input rates as of 2026-10-07, from the pricing page.

Security

Grade A, and why

linear-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 2d ago.

The scan reads SKILL.md. This mod also ships 6 executable files (scripts/convergence_diagnostics.py, scripts/preconditioner_advisor.py, scripts/residual_norms.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/linear-solvers/SKILL.md · 253 lines

How it starts

The opening of the file, as written. The whole thing — 253 lines — stays where its author put it; the contents beside it link to each section on GitHub.

Linear Solvers

Goal

Provide a universal workflow to select a solver, assess conditioning, and diagnose convergence for linear systems arising in numerical simulations.

Requirements

Python 3.10+
NumPy, SciPy (for matrix operations)
See individual scripts for dependencies

Inputs to Gather

Input	Description	Example
Matrix size	Dimension of system	`n = 1000000`
Sparsity	Fraction of nonzeros	`0.01%`
Symmetry	Is A = Aᵀ?	`yes`
Definiteness	Is A positive definite?	`yes (SPD)`
Conditioning	Estimated condition number	`10⁶`

Decision Guidance

Solver Selection Flowchart

Is matrix dense and small enough to factor in memory (dense float64
storage n²·8 bytes < ~2 GB, i.e. n ≲ 16000)?
├── YES → Use direct solver (Cholesky/LDLᵀ/LU by symmetry)
└── NO → Is matrix symmetric?
    ├── YES → Is it positive definite?
    │   ├── YES → Use CG with AMG/IC preconditioner
    │   └── NO → Use MINRES
    └── NO → Is it nearly symmetric?
        ├── YES → Use BiCGSTAB
        └── NO → Use GMRES with ILU/AMG

Quick Reference

Matrix Type	Solver	Preconditioner
SPD, sparse	CG	AMG, IC
Symmetric indefinite	MINRES	SPD preconditioner (SSOR, symmetric block-diagonal, or AMG on SPD part)
Nonsymmetric	GMRES, BiCGSTAB	ILU, AMG
Dense	LU, Cholesky	None
Saddle point	Schur complement, Uzawa	Block preconditioner

Script Outputs (JSON Fields)

Script	Key Outputs
`scripts/solver_selector.py`	`recommended`, `alternatives`, `notes`
`scripts/convergence_diagnostics.py`	`rate`, `asymptotic_rate`, `stagnation`, `recommended_action`
`scripts/sparsity_stats.py`	`nnz`, `density`, `bandwidth`, `symmetry`
`scripts/preconditioner_advisor.py`	`suggested`, `notes`
`scripts/scaling_equilibration.py`	`row_scale`, `col_scale`, `notes`
`scripts/residual_norms.py`	`residual_norms`, `relative_norms`, `converged`

Read the full file on GitHub · 253 lines

Files

What ships with it

13 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.

2d ago Changed · +4 lines 5d339bc2a9e6
1mo ago First seen · 249 lines · 140 tokens per session scan A 32a2ed1027a8

Subscribe to this mod's changes

linear-solvers is a skill published in the GitHub repository HeshamFS/materials-simulation-skills (83 stars, last pushed 3d ago), licensed Apache-2.0. It adds 140 tokens to every session and 3,287 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.

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