Numerical Methods

Overview

This textbook introduces numerical analysis and computational methods for solving equations, approximating functions, differentiating and integrating numerically, working with linear algebra, optimizing functions, and solving ordinary and partial differential equations. The book emphasizes accuracy, stability, and the practical use of algorithms in scientific computing.

Table of Contents

1. Chapter 1: What Numerical Methods Are
2. Chapter 2: Error, Floating Point, and Stability
3. Chapter 3: Numerical Methods Foundations
4. Chapter 4: Interpolation and Polynomial Approximation
5. Chapter 5: Splines, Curve Fitting, and Least Squares
6. Chapter 6: Numerical Differentiation
7. Chapter 7: Numerical Integration
8. Chapter 8: Taylor Series and Function Approximation
9. Chapter 9: Numerical Linear Algebra
10. Chapter 10: Numerical Eigenvalue Methods
11. Chapter 11: Numerical Optimization
12. Chapter 12: Numerical Methods for Ordinary Differential Equations
13. Chapter 13: Partial Differential Equations
14. Chapter 14: Numerical Methods Capstone
15. Back Matter

Learning Outcomes

• Explain how numerical methods are used to approximate mathematical solutions.
• Analyze error, stability, and floating-point limitations in computation.
• Use interpolation, splines, and curve fitting to model data.
• Apply numerical techniques for differentiation and integration.
• Work with Taylor series and function approximation methods.
• Solve linear systems and eigenvalue problems numerically.
• Use optimization methods to locate extrema of functions.
• Approximate solutions to ordinary and partial differential equations.
• Interpret computational results with attention to accuracy and efficiency.
• Apply numerical thinking to scientific and engineering problems.