Numerical Methods

1. Numerical Integration
   1. Basic Concepts
      1. Integration Overview
         1. Definition of Integration
         2. Importance in Calculus and Applied Mathematics
      2. Definite vs. Indefinite Integrals
         1. Definition and Properties
         2. Examples and Simple Calculations
      3. Analytic vs. Numerical Integration
         1. Understanding Analytic Solutions
         2. When to Use Numerical Integration
   2. Methods
      1. Trapezoidal Rule
         1. Description and Formula
         2. Step-by-Step Application
         3. Modifications and Extensions
      2. Simpson's Rule
         1. Simpson's 1/3 and 3/8 Rules
         2. Derivation and Understanding
         3. Applicability and Enhancements
      3. Gaussian Quadrature
         1. Concept of Weighted Sum and Orthogonal Polynomials
         2. Gauss-Legendre Quadrature
         3. Other Quadrature Rules (Gauss-Chebyshev, Gauss-Laguerre)
      4. Monte Carlo Integration
         1. Principles of Random Sampling
         2. Applications and Best Practices
         3. Variance Reduction Techniques
   3. Applications
      1. Area Estimation
         1. Calculating Area Under Curves
         2. Real-World Examples
      2. Solving Integral Equations
         1. Fundamental Concepts
         2. Techniques and Examples
      3. Physics and Engineering
         1. Applications in Electromagnetism
         2. Heat Transfer Calculations
         3. Structural Analysis
      4. Economics and Finance
         1. Risk Management and Option Pricing
         2. Cost and Revenue Estimation
      5. Biology and Medicine
         1. Modeling Biological Systems
         2. Pharmacokinetics and Dose-Response Relationships
   4. Error Estimation and Analysis
      1. Types of Errors in Numerical Integration
         1. Truncation Error
         2. Round-off Error
      2. Error Boundaries and Estimates
         1. Techniques for Bounding Errors
         2. Role of Function Smoothness
      3. Adaptive Integration Techniques
         1. Concept of Adaptive Quadrature
         2. Implementing Adaptive Methods
      4. Improving Accuracy
         1. Refinement and Mesh Selection
         2. High Precision Computation
   5. Advanced Topics
      1. Adaptive Integration Strategies
         1. Recursive Subdivision
         2. Automatic Error Control
      2. Higher-Dimensional Integration
         1. Methods for Multi-Dimensional Integrals
         2. Applications in Computational Science
      3. Integration of Discontinuous Functions
         1. Strategies for Handling Discontinuities
         2. Examples and Techniques
      4. Influence of Function Properties
         1. Impact of Continuity and Differentiability
         2. Selecting Suitable Methods Based on Function Behavior
   6. Practical Considerations
      1. Trade-off Between Accuracy and Computational Cost
      2. Choosing the Right Method for the Problem
      3. Software Implementations and Their Implications
      4. Parallel and Distributed Computing Approaches
      5. Handling Ill-Conditioned Problems