Numerical investigation

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