# Math Computational

> Solve numerical and symbolic math problems (root finding, integration, ODEs, symbolic manipulation) with stable algorithms and explicit error analysis.

- Skill: `quantumquirkxyz/math-computational` (Agent Skill)
- Install (CLI): `npx skillmds@latest add quantumquirkxyz/math-computational`
- Raw SKILL.md: https://api.skillmd.com/api/skills/quantumquirkxyz/math-computational/raw
- Safety review: pending
- Works with: Claude Code, Claude.ai, OpenAI Codex
- Category: Integrations & APIs
- Author: quantumquirkxyz (https://skillmd.com/u/quantumquirkxyz)
- Updated: 2026-09-17
- Page: https://skillmd.com/skills/quantumquirkxyz/math-computational

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## Contract

- **Input:** a numerical or symbolic mathematical problem.
- **Output:** Markdown artifact with algorithm choice, error analysis, computation, and verification.
- **Side effects:** none.
- **Dependencies:** none.
- **Stop condition:** artifact complete with error bounds and verification.
- **Risk:** low — computation only, no system changes.
- **Boundary:** may execute code for numerical work but only in-memory, no file writes.

# Computational Math

Solve a mathematical problem **numerically or symbolically** with an algorithm whose error, stability, and cost are explicit. Never ship a black-box answer.

## When to use

- The user wants a numerical result, symbolic simplification, or to choose between algorithms.
- A formula or expression needs to be evaluated, integrated, differentiated, or solved.
- An engineering / physics / quant problem needs a numerical kernel.

## Process

### 1. Frame the problem

Identify: input → output, the mathematical object (scalar / vector / function / matrix / ODE / PDE / integral), and the regime (smoothness, scale, boundary conditions). State whether an analytical or numerical answer is expected.

**Completion criterion:** problem restated as a clean mapping with regime noted.

### 2. Choose the algorithm

Pick the algorithm that matches the regime:

- **Root finding:** bisection (safe), Newton's (fast, needs derivative + good init), secant.
- **Integration:** Simpson / Gauss (smooth, 1D), Monte Carlo (high-D), adaptive quadrature (singularities).
- **ODEs:** explicit Euler (cheap, low order), RK4 (default), implicit (stiff), symplectic (Hamiltonian).
- **Linear systems:** direct (LU, Cholesky) vs iterative (CG, GMRES).
- **Symbolic:** CAS (SymPy / Mathematica) with manual simplification on top.

Name the algorithm and one reason it fits.

**Completion criterion:** algorithm named, with a one-line justification referencing the regime.

### 3. Estimate error and stability

For numerical work, state:

- **Local truncation error** and **global error** order.
- **Conditioning** — well-posed or ill-posed? Sensitivity to input noise.
- **Stability** — bounded-input-bounded-output? Stiffness concerns?
- **Cost** — O(n²)? O(n³)? Memory?

For symbolic work, state the simplification rules used and any non-canonical choices.

**Completion criterion:** error, stability, and cost all addressed for the chosen algorithm.

### 4. Implement / compute

Either run code (Python with NumPy/SciPy/SymPy) or hand-compute if tractable. For code, include:

- The algorithm in named functions, not one giant cell.
- Input validation (NaN, inf, dimension mismatch).
- Sanity asserts on outputs (sign, magnitude, symmetry).

**Completion criterion:** output produced and reproducible from the code shown.

### 5. Verify

Cross-check by at least one independent path:

- Different algorithm at lower fidelity.
- Asymptotic / limiting case (small/large parameter).
- Comparison against a known analytic solution or benchmark.
- Dimensional / sign / magnitude check.

**Completion criterion:** independent verification done; discrepancy explained or fixed.

### 6. Deliver

Markdown artifact with: problem, algorithm choice, error/stability/cost, code or computation, verification, and final answer with stated uncertainty (e.g. "to 6 significant digits").

**Completion criterion:** artifact contains algorithm, error bounds, verification, and a final answer whose precision is justified.