Course Title: Numerical Analysis
Course Overview:
Numerical Analysis is a course that focuses on numerical methods for solving mathematical problems that arise in various fields, including engineering, physics, and computer science. The course covers techniques for solving equations, interpolation and approximation, numerical integration and differentiation, and solving systems of linear equations. Students will learn theoretical foundations as well as practical implementation of numerical algorithms.
Course Objectives:
Understand the principles and limitations of numerical methods.
Learn techniques for solving equations numerically.
Study interpolation and approximation methods.
Explore numerical differentiation and integration techniques.
Gain proficiency in solving systems of linear equations numerically.
Apply numerical methods to real-world problems and simulations.
Course Outline:
Introduction to Numerical Analysis
Overview of numerical methods and their applications
Sources of error in numerical computations
Round-off error and truncation error
Solving Nonlinear Equations
Bisection method
Newton-Raphson method
Secant method
Convergence analysis and error estimation
Interpolation and Approximation
Polynomial interpolation (Lagrange interpolation, Newton’s divided differences)
Splines interpolation
Least squares approximation
Numerical Differentiation
Finite difference methods (forward, backward, central differences)
Richardson extrapolation
Higher-order differentiation methods
Numerical Integration
Trapezoidal rule
Simpson’s rule
Romberg integration
Gaussian quadrature
Systems of Linear Equations
Direct methods (Gaussian elimination, LU decomposition)
Iterative methods (Jacobi method, Gauss-Seidel method, Successive Over-Relaxation)
Matrix factorization techniques (Cholesky decomposition, QR decomposition)
Eigenvalue Problems
Power method for eigenvalues and eigenvectors
QR algorithm
Lanczos algorithm for symmetric matrices
Numerical Solutions of Ordinary Differential Equations (ODEs)
Euler’s method
Runge-Kutta methods (e.g., RK4)
Boundary value problems and shooting methods
Numerical Solutions of Partial Differential Equations (PDEs)
Finite difference methods for PDEs
Finite element methods
Numerical solutions for heat equation, wave equation, and Laplace equation
Applications of Numerical Analysis
Computational simulations and modeling
Optimization problems
Data fitting and regression analysis
Assessment Methods:
Problem-solving assignments and projects using numerical methods.
Quizzes and exams covering theoretical concepts and numerical algorithms.
Implementation of numerical algorithms in programming languages (e.g., MATLAB, Python).
Case studies and applications of numerical analysis to real-world problems.
Textbook:
“Numerical Analysis” by Richard L. Burden and J. Douglas Faires
References:
“Numerical Recipes: The Art of Scientific Computing” by William H. Press et al.
Online resources and tutorials on numerical methods and algorithms
Admission Open for this course
Contact Number: 03307615544