1. Introduction to Numerical Methods

• Definition and importance
• Differences between analytical and numerical solutions
• Sources of errors in numerical computations

2. Basics of Computer Arithmetic

• Floating-point representation
• Round-off errors
• Truncation errors

3. Root Finding Methods

• Bisection method
• Newton-Raphson method
• Secant method
• Fixed-point iteration

4. Systems of Linear Equations

• Direct methods
  • Gaussian elimination
  • LU decomposition
• Iterative methods
  • Jacobi method
  • Gauss-Seidel method
  • Successive over-relaxation (SOR)

5. Interpolation and Curve Fitting

• Polynomial interpolation
  • Lagrange interpolation
  • Newton’s divided difference interpolation
• Spline interpolation
• Least squares curve fitting

6. Numerical Differentiation

• Finite difference methods
  • Forward difference
  • Backward difference
  • Central difference

7. Numerical Integration

• Trapezoidal rule
• Simpson’s rules (1/3 and 3/8)
• Gaussian quadrature

8. Ordinary Differential Equations (ODEs)

• Initial value problems
  • Euler’s method
  • Improved Euler’s method (Heun’s method)
  • Runge-Kutta methods
• Boundary value problems
  • Shooting method
  • Finite difference method

9. Partial Differential Equations (PDEs)

• Classification of PDEs
• Finite difference methods for PDEs
• Stability analysis

10. Eigenvalues and Eigenvectors

• Power method
• Deflation method
• QR algorithm

11. Optimization Methods

• Unconstrained optimization
  • Golden section search
  • Gradient descent
• Constrained optimization
  • Linear programming
  • Quadratic programming

12. Special Topics (as per interest)

• Fast Fourier Transforms (FFT)
• Monte Carlo methods
• Boundary element methods

13. Software and Tools for Numerical Computations

• Introduction to MATLAB, Python (NumPy, SciPy), or other relevant software/tools
• Implementing numerical algorithms using software

14. Practical Applications and Case Studies

• Real-world problems solved using numerical methods
• Modeling and simulation of physical systems

15. Advanced Topics (optional, for deeper dives)

• Multigrid methods
• Finite element methods
• Advanced optimization techniques