Unit 2948: Numerical Analysis

1. Introduction to Numerical Analysis

• Definition and scope of numerical analysis
• Importance in scientific computing and applied mathematics
• Limitations and challenges of numerical methods
• Types of problems addressed by numerical techniques

2. Error Analysis in Numerical Methods

• Types of errors:
  • Round-off errors
  • Truncation errors
• Sources and propagation of errors
• Techniques for error estimation and control
• Stability and sensitivity analysis

3. Root Finding Methods

• Problem statement: finding roots of nonlinear equations
• Bisection Method
  • Algorithm and implementation
  • Convergence properties
• Newton's Method
  • Derivation and iterative scheme
  • Convergence rate and conditions
• Secant Method
  • Algorithm and comparison with Newton's method
• Practical considerations and examples

4. Interpolation and Approximation

• Interpolation concepts and applications
• Lagrange Interpolation
  • Formula and computation
• Newton's Divided Differences
  • Recursive formulation
• Piecewise interpolation (brief overview)
• Approximation methods
  • Least squares approximation
  • Applications in data fitting

5. Numerical Differentiation and Integration

• Numerical differentiation
  • Finite difference approximations (forward, backward, central)
  • Error analysis in differentiation
• Numerical integration
  • Trapezoidal Rule
  • Simpson's Rule
  • Error bounds and accuracy considerations

6. Solving Systems of Linear Equations

• Importance in numerical analysis
• Direct methods
  • Gaussian Elimination
  • LU Decomposition
• Iterative methods
  • Jacobi Method
  • Gauss-Seidel Method
• Convergence criteria and efficiency

7. Numerical Solutions of Ordinary Differential Equations (ODEs)

• Initial value problems overview
• Euler's Method
  • Algorithm and implementation
  • Stability and error analysis
• Runge-Kutta Methods
  • Classical 4th order Runge-Kutta
  • Comparison with Euler's method
• Finite Difference Method (brief introduction)

8. Eigenvalue Problems and Matrix Computations

• Importance of eigenvalues and eigenvectors
• Power Method
  • Algorithm and convergence
• QR Decomposition
  • Computation and applications
• Singular Value Decomposition (SVD)
  • Concept and significance
• Applications in engineering and data science