---
name: numerical-investigation
description: Complete guide for numerical mathematical investigations. Conducts structured interview to understand background, designs implementation and tests, creates detailed plan, then executes code, testing, and reporting.
license: Apache-2.0
metadata:
  version: "1.0"
  category: math
---

# Mathematical Numerical Investigation

Guide users through complete numerical investigations from problem understanding to final report.

## When to Use

- User wants to numerically test mathematical behavior
- Checking asymptotic limits or convergence
- Verifying conjectures computationally
- Extending analysis (e.g., n=1 to n=2)
- Parameter sweeps or existence checks

## Three-Phase Workflow

### Phase 1: Interview - Understand Background

Ask clarifying questions to understand the problem:

**1. Core Question**
- What mathematical behavior are you testing?
- What's the main conjecture or hypothesis?

**2. Mathematical Setup**
- What objects, maps, equations are involved?
- What are the key parameters?
- What varies, what's fixed?

**3. Scenarios to Test**
- What specific cases should be checked?
- Expected outcomes vs possible failures?
- Edge cases or special values?

**4. Background Reading**
- Any notes, papers, or code to read first?
- Previous results to build on?
- Existing codebase to understand?

**5. Success Criteria**
- What does "success" look like numerically?
- What would be concerning?
- Quantitative thresholds?

**Interview style**: Conversational, use user's terminology, ask follow-ups.

### Phase 2: Design - Create Investigation Plan

After understanding the problem, design the investigation:

**1. Reading (if needed)**
- Read provided materials
- Explore existing code
- Identify reusable components

**2. Implementation Strategy**
- Language: Python, Julia, Mathematica?
- Libraries needed
- Reuse vs rewrite existing code

**3. Test Design**
- What to compute
- Parameter ranges and sampling
- Initial guesses, edge cases
- Numerical tolerances

**4. Output/Report Design**
- Figures to generate
- Data formats
- Report structure

**5. Code Organization**
- Directory structure
- Configuration files
- Naming conventions

**Then**: Create detailed plan, iterate with user until approval.

### Phase 3: Execute - Implement and Report

After plan approval:

**1. Implementation**
- Write code according to plan
- Set up directory structure
- Create configuration

**2. Testing**
- Run initial tests
- Verify convergence/correctness
- Check edge cases

**3. Full Run**
- Execute complete investigation
- Generate all outputs
- Collect data

**4. Report Generation**
- Create figures
- Write summary
- Interpret results
- Deliver to user

Use agents/subtasks for long computations, parallel testing, visualizations.

## Resources

- `references/interview-guide.md` - Detailed Phase 1 questions
- `references/design-guide.md` - Phase 2 decision framework
- `references/plan-template.md` - Plan structure template
- `references/execution-checklist.md` - Phase 3 checklist