Elementary Numerical Analysis with Python Implementation

When analytical solutions to complex mathematical equations are impossible to find, numerical approximation becomes the essential tool for engineers, data scientists, and scientific programmers. This text-based course guides you through the fundamental algorithms used to approximate functions and solve continuous mathematical problems. You will transition from understanding core mathematical proofs to reading and writing clean, modern Python implementations of these classic numerical methods. In this course, you will build a solid theoretical and practical foundation in numerical computation. You will learn how to analyze errors systematically, construct approximating polynomials, and implement stable mathematical algorithms from scratch. What you'll learn: - Understand the foundational concepts of numerical error, floating-point arithmetic, and approximation limits - Construct interpolating polynomials using Lagrange and Newton divided difference methods - Analyze the theoretical error bounds of polynomial approximations to ensure computational accuracy - Implement piecewise polynomial approximations and cubic spline interpolation for smooth curve fitting - Apply cubic Hermite interpolation to match both function values and derivative data - Write and test clean, structured Python code using type hints to implement numerical algorithms We begin with essential mathematical definitions, error analysis frameworks, and core approximation concepts. Next, we progress step-by-step through divided differences, Hermite interpolation, and spline methods, pairing each mathematical theory with clear, line-by-line algorithm explanations and Python code structures. This course is designed for beginners in numerical analysis, undergraduate students in STEM fields, and self-taught programmers looking to strengthen their mathematical computing skills. No advanced mathematical background beyond basic calculus is required, and only introductory familiarity with Python is assumed. Start learning the mathematical foundations of modern scientific computing today.