BIT Numerical Methods
bitsemester 3
Numerical Methods
Subject Code: BIT203
Course Title: Numerical Methods
Course No: BIT203
Nature of Course: Theory & Practical
Full Marks: 100
Pass Marks: 40
Credit Hours: 3
Course Description
This course covers different concepts of numerical techniques of solving non-linear equations, system of linear equations, integration and differentiation, and ordinary and partial differential equations.
Course Objective
The main objective of this course is to provide concepts of numerical techniques for solving different types of equations and developing algorithms for solving scientific problems.
Course Contents
| Unit | Teaching Methodology | Teaching Hours |
|---|---|---|
| Unit 1: Solution of Nonlinear Equations (7 Hrs.) | ||
| 1.1 Errors in Numerical Calculations (True Error, Relative Error, Approximate Error, Relative Approximate Error), Sources of Errors (Round-off Error, Truncation Error), Propagation of Errors, Review of Taylor's Theorem | Lecture + Laboratory Work | 1 |
| 1.2 Concept of Linear and Non-linear Equations, Solving Non-linear Equations: Trial and Error Method, Bisection Method, Newton Raphson Method, Secant Method, Fixed Point Method, False Position Method, Newton's Method for Calculating Multiple Roots, Evaluating Polynomials with Horner's Method, Derivation, Algorithm and Example of each method | 6 | |
| Unit 2: Interpolation and Regression (8 Hrs.) | ||
| 2.1 Concept of Interpolation and Extrapolation, Lagrange's Interpolation, Newton's Interpolation using divided differences, forward differences and backward differences, Derivation, Algorithm and Example of each method | Lecture + Laboratory Work | 5 |
| 2.2 Concept of Regression, Regression vs. Interpolation, Least Squares Methods, Linear Regression, Non-linear Regression: Exponential regression by linearization, and Polynomial, Derivation, Algorithm and Example of each method | 3 | |
| Unit 3: Numerical Differentiation and Integration (9 Hrs.) | ||
| 3.1 Concept of Differentiation, Differentiating Continuous Functions (Two-Point Forward and Backward Difference Formulae, Three-Point Formula), Differentiating Tabulated Functions by using Newton's Differences (Divided, Forward and Backward Difference Formulae), Maxima and minima of Tabulated Functions, Derivation, Algorithm and Example of each method | Lecture + Laboratory Work | 5 |
| 3.2 Concept of Integration, Newton-Cote's Quadrature Formulae: Trapezoidal rule, Multi-Segment Trapezoidal rule, Simpson's 1/3 rule, Multi-Segment Simpson's 1/3 rule, Simpson's 3/8 rule, Multi-Segment Simpson's 3/8 rule, Derivation, Algorithm and Example of each method | 4 | |
| Unit 4: Solving System of Linear Equations (8 Hrs.) | ||
| 4.1 Existence of Solutions, Properties of Matrices, Matrix Representation, Gaussian Elimination Method, Partial and Complete Pivoting with Gaussian Elimination Method, Gauss-Jordan method, Inverse of matrix using Gauss-Jordan method, Derivation, Algorithm and Example of each method | Lecture + Laboratory Work | 3 |
| 4.2 Concept of LU Decomposition, Matrix factorization and Solving System of Linear Equations by using Doolittle and Cholesky Algorithm, Derivation, Algorithm and Example of each method | 2 | |
| 4.3 Iterative Solutions of System of Linear Equations, Jacobi Iteration Method, Gauss-Seidel Method, Derivation, Algorithm and Example of each method | 2 | |
| 4.4 Eigen Values and Eigen Vectors Problems, Power Method, Algorithm and Example of the method | 1 | |
| Unit 5: Solution of Ordinary Differential Equations (8 Hrs.) | ||
| 5.1 Concept of Differential Equations, Initial Value Problem, Taylor Series Method, Euler's Method, Heun's Method, Runge-Kutta Methods, Derivation, Algorithm and Example of each method | Lecture + Laboratory Work | 5 |
| 5.2 Solving System of Ordinary Differential Equations, Solution of the Higher Order Equations, Boundary Value Problems, Shooting Method, Derivation, Algorithm and Example of each method | 3 | |
| Unit 6: Solution of Partial Differential Equations (5 Hrs.) | ||
| 6.1 Concept of Partial Differential Equations, Classification of PDE, Deriving Difference Equations, Solving Laplacian Equation and Poisson's Equation, Derivation, Algorithm and Example of each method | Lecture + Laboratory Work | 5 |
Text Books
- W. Chency and D. Kincaid, "Numerical Mathematics and Computing", 7th Edition, Brooks Cole Publisher.
- C.F. Gerald and P.O. Wheatley, "Applied Numerical Analysis", 9th Edition, Addison Wesley Publisher.
Reference Books
- W.H. Press, B.P. Flannery et al., "Numerical Recipes: Art of Scientific Computing", 3rd Edition, Cambridge Press.
- J. M. Mathews and K. Fink, "Numerical Methods using MATLAB", 4th Edition, Prentice Hall Publication.
- S. S. Sastry, "Introductory Methods of Numerical Analysis", 5th Edition, PHI Publication.
- E Balagurusamy, Numerical Methods, 1st Edition, McGraw Hill India.
- A. S. Saud, Numerical Methods with Practical Approach, Numerical Methods, KEC Publication and Distributors.
Lab Works
| Lab Sheet # | Chapter | Activities |
|---|---|---|
| Lab Sheet #1 | Chapter 1 | 1. Write programs to implement Bisection method, Newton Raphson method, Secant method, False Position method, and Fixed Point method for solving non-linear equations. 2. Write a program to implement Horner's method of polynomial evaluation. |
| Lab Sheet #2 | Chapter 2 | 1. Write programs for Lagrange interpolation, Newton's divided/forward/backward difference interpolation. 2. Write programs to implement linear and non-linear regression. |
| Lab Sheet #3 | Chapter 3 | 1. Write programs to implement Two-Point Forward Difference, Two-Point Backward Difference and Three-Point algorithm to compute derivatives of continuous functions. 2. Write programs to implement Newton's Divided, Forward and Backward Difference algorithms to compute derivatives of discrete functions. 3. Write a program to compute maxima and minima of tabulated functions. 4. Write program to compute integral value of functions using trapezoidal rule, multi-segment trapezoidal rule, Simpson's 1/3 rule, multi-segment Simpson's 1/3 rule, Simpson's 3/8 rule and multi-segment Simpson's 3/8 rule. |
| Lab Sheet #4 | Chapter 4 | 1. Write programs to solve system of non-linear equations using Gauss-Elimination and Gauss Jordan method. 2. Write programs to factorize matrix using Doolittle and Cholesky method. 3. Write programs to solve system of non-linear equations using Jacobi Iteration and Gauss-Seidel method. 4. Write a program to find eigenvalue and eigenvector using power method. |
| Lab Sheet #5 | Chapter 5 | 1. Write programs to implement Taylor's method, Euler's method, Heun's method and RK method to solve ordinary differential equations. 2. Write a program to solve second order ODE. 3. Write a program to solve boundary value problem using shooting method. |
| Lab Sheet #6 | Chapter 6 | 1. Write programs to solve Laplacian Equation and Poisson's Equation. |