Unit 2954: Numerical Methods

1. Introduction to Numerical Methods

• Definition and importance of numerical methods
• Approximation techniques and algorithms
• Applications in science and engineering

2. Error Analysis in Numerical Methods

• Types of errors
  • Round-off errors
  • Truncation errors
• Sources of errors in computations
• Error propagation and stability
• Techniques to analyze and minimize errors

3. Solutions of Nonlinear Equations

• Problem statement and challenges
• Bisection Method
  • Algorithm and convergence
  • Advantages and limitations
• Newton-Raphson Method
  • Derivation and iterative formula
  • Convergence criteria
• Secant Method
  • Algorithm and comparison with Newton-Raphson
• Practical examples and implementation

4. Interpolation and Curve Fitting

• Purpose and applications
• Lagrange Interpolation
  • Formula and properties
  • Computational aspects
• Newton's Divided Difference Interpolation
  • Algorithm and efficiency
• Least Squares Curve Fitting
  • Linear and polynomial fitting
  • Error minimization techniques
• Comparing interpolation methods

5. Numerical Differentiation and Integration

• Finite Difference Methods for Derivatives
  • Forward, backward, and central difference formulas
• Numerical Integration Techniques
  • Trapezoidal Rule
  • Simpson's Rule
  • Higher-order methods
• Error estimation in numerical differentiation and integration

6. Solving Systems of Linear Equations

• Importance in numerical computations
• Direct Methods
  • Gaussian Elimination
  • LU Decomposition
• Iterative Methods
  • Jacobi Method
  • Gauss-Seidel Method
  • Convergence considerations
• Practical applications and examples

7. Eigenvalues and Eigenvectors

• Definitions and significance
• Power Method
  • Algorithm and convergence
• QR Algorithm
  • Steps and computational complexity
• Applications in stability analysis and matrix computations

8. Numerical Solutions of Ordinary Differential Equations (ODEs)

• Initial Value Problems (IVPs)
• Euler's Method
  • Algorithm and error analysis
• Runge-Kutta Methods
  • RK4 method details
  • Accuracy and stability
• Finite Difference Methods for ODEs
• Practical implementation and examples

9. Boundary Value Problems (BVPs)

• Difference between IVPs and BVPs
• Shooting Method
  • Concept and implementation
• Finite Difference Method for BVPs
  • Discretization and solution
• Spectral Methods
  • Basis functions and convergence
• Applications in physics and engineering

10. Optimization Techniques

• Introduction to numerical optimization
• Gradient Descent
  • Algorithm and convergence
• Newton's Method for Optimization
  • Derivative requirements and efficiency
• Genetic Algorithms
  • Basics and applications
• Case studies and practical examples