Introduction to Numerical Analysis

This course introduces fundamental concepts and algorithms in Numerical Analysis, focusing on approximation theory, solving nonlinear equations, interpolation, numerical differentiation and integration, and error/stability analysis. Emphasis is placed on understanding accuracy, computational cost, and convergence behavior of numerical methods.

📘 Objective

Students will build both theoretical foundations and computational implementation skills using MATLAB.

📂 Lecture Notes (updated regularly) 👉 Here

🧠 Topics Covered

✅ Approximation of Series

  • Taylor series approximation
  • Positive series tests
    • Integral test
    • Ratio test (D’Alembert)
  • Alternating series tests
    • Leibniz test
    • Calabrese test
  • Error estimates and exercises

🔍 Solving Nonlinear Equations

  • Bisection method
  • Newton’s method
  • Secant method
  • Fixed-point iteration
  • Convergence analysis & stability

📈 Interpolation & Approximation

  • Linear, quadratic, and polynomial interpolation
  • Divided differences & Newton interpolation
  • Interpolation error analysis
  • Piecewise interpolation & cubic splines
  • Approximation theory foundations

🧮 Numerical Differentiation & Integration

  • Trapezoid / Midpoint / Simpson’s rules
  • Forward, backward, central difference formulas

🔢 Numerical Linear Algebra

  • Gaussian elimination & pivoting strategies
  • LU factorization
  • Iterative methods (Jacobi, Gauss-Seidel, basic CG ideas)
  • Conditioning and stability

🧾 Additional Topics (if time permits)

  • Numerical ODEs: Euler, Improved Euler, Runge–Kutta
  • Introduction to optimization & derivative-free methods
  • Numerical stability and sensitivity analysis

💻 Software

We will use:

  • MATLAB / MATLAB Grader
  • Google Colab / Jupyter Notebook

Coding exercises will reinforce theoretical concepts and numerical implementations.

📅 Tentative Weekly Schedule

Week Topic Notes
1-2 Calculus review - Differentiation, Taylor Series, Rate of Convergence Floating-point arithmetic & error analysis
3–6 Approximation of series Convergence tests & error bounds
8-9 Interpolation & approximation Splines, divided differences
10–12 Numerical differentiation & integration Composite rules, stability
13–14 Numerical linear algebra LU, conditioning, iterative methods
14+ Numerical ODEs / Special topics  

References