# Advanced Topics Category Template

**Audience:** Expert users, algorithm developers, performance engineers
**Tone:** Technical, concise, assumes competence
**Emphasis:** Customization, advanced features, internals, edge cases

---

# 📘 {Technical Topic}

> {Concise technical description}

⏱️ **{30-45} minutes** | 📊 **Level: ●●● Advanced** | ⚙️ **{Technical area}**

---

## 🎯 Learning Objectives

After completing this advanced tutorial:
- ✓ {Advanced technical skill 1}
- ✓ {Advanced technical skill 2}
- ✓ {Advanced technical skill 3}
- ✓ {Implementation/customization skill}

---

## 🗺️ Prerequisites

**Required knowledge:**
- ✓ Strong Python and NumPy proficiency
- ✓ Understanding of {fundamental concept}
- ✓ Completed [{intermediate tutorial}](link)
- ✓ Familiarity with JAX {if applicable}

**Required notebooks:**
- [{Core tutorial 1}](link)
- [{Core tutorial 2}](link)

---

## 💡 When You Need This

**Use cases:**
- {Specialized scenario 1}
- {Edge case scenario 2}
- {Performance-critical scenario 3}

**This is NOT needed if:**
- You're solving standard curve fitting problems → Use [Core Tutorials](../02_core_tutorials/)
- You're looking for domain examples → See [Gallery](../04_gallery/)

---

## ⚡ Quick Example

```python
{Advanced code example showcasing the technique}
```

---

## 🔧 Setup

**IMPORTANT:** Always include this configuration cell first (before any imports):

```python
# Configure matplotlib for inline plotting in VS Code/Jupyter
# MUST come before importing matplotlib
%matplotlib inline
```

**Then add your imports:**

```python
import numpy as np
import jax.numpy as jnp
from nlsq import curve_fit
# ... other imports
```

---

## 📖 Technical Deep Dive

### 1. Theoretical Background

{Mathematical/algorithmic foundation}

**Key concepts:**
- **{Advanced concept 1}:** {Technical explanation}
- **{Advanced concept 2}:** {Technical explanation}

**Algorithm:**
{Pseudocode or detailed description}

### 2. Implementation Details

{Low-level implementation guidance}

**Architecture:**
{System design or algorithm structure}

**Key components:**
```python
{Core implementation code with detailed comments}
```

### 3. Customization Points

**Extension mechanisms:**
- **Custom {component}:** {How to implement}
- **Override {method}:** {When and how}

**Example customization:**
```python
{Complete custom implementation example}
```

### 4. Performance Characteristics

**Computational complexity:**
- Time: O({complexity})
- Space: O({complexity})

**Benchmarks:**
{Detailed performance data}

**Optimization strategies:**
1. {Advanced optimization 1}
2. {Advanced optimization 2}

### 5. Edge Cases and Limitations

**Known limitations:**
- {Limitation 1}: {Workaround}
- {Limitation 2}: {Workaround}

**Numerical stability:**
{Discussion of numerical considerations}

---

## 🔧 API Reference

**Main functions/classes:**

**`{FunctionName}({params})`**
- **Purpose:** {What it does}
- **Parameters:**
  - `param1` ({type}): {Description}
  - `param2` ({type}, optional): {Description, default}
- **Returns:** {Return type and description}
- **Raises:** {Exception types and when}

**Configuration options:**
```python
config = {
    'option1': {value},  # {Description and impact}
    'option2': {value},  # {Description and impact}
}
```

---

## 💡 Advanced Patterns

**Pattern 1: {Name}**
```python
{Code showing advanced usage pattern}
```
**When to use:** {Scenario}
**Trade-offs:** {Pros and cons}

**Pattern 2: {Name}**
```python
{Another advanced pattern}
```
**When to use:** {Scenario}

---

## ⚠️ Common Advanced Issues

**Issue 1: {Technical problem}**
- **Symptoms:** {How to recognize}
- **Root cause:** {Technical explanation}
- **Debug approach:**
  ```python
  {Diagnostic code}
  ```
- **Solution:** {Fix with explanation}

**Issue 2: {Technical problem}**
- **Cause:** {Explanation}
- **Solution:** {Advanced fix}

---

## 🔬 Research Applications

**Published implementations:**
- {Paper 1}: [{Title}](DOI) - {Application}
- {Paper 2}: [{Title}](DOI) - {Application}

**Novel extensions:**
- {Extension 1}: {Description and potential}
- {Extension 2}: {Description and potential}

---

## 🔗 Related Advanced Topics

**Build on this:**
- [{Advanced topic 1}](link) - {Relationship}
- [{Advanced topic 2}](link) - {Relationship}

**Alternative approaches:**
- [{Alternative}](link) - {When to prefer}

**References:**
- [Source code](link to GitHub)
- [Research paper](DOI)
- [Technical documentation](link)

---

## ❓ Expert Q&A

**Q: How does this compare to {academic algorithm/approach}?**
A: {Detailed comparison with theoretical justification}

**Q: Can I combine this with {other advanced feature}?**
A: {Technical answer with example}

**Q: What are the theoretical guarantees?**
A: {Convergence properties, complexity guarantees, etc.}

---

## 📚 Technical Glossary

**{Advanced term 1}:** {Rigorous definition}
**{Algorithm abbreviation}:** {Full name, reference}
**{Mathematical concept}:** {Definition with equations}

[Complete glossary](../../docs/glossary.md)
