Numerical Methods

Course Title: Numerical Methods

Course No: CSIT.224

Nature of the Course: Theory + Lab

Semester: 4

Full Marks: 60 + 20 + 20

Pass Marks: 24 + 10 + 10

Credit Hours: 3

Course Description

This course introduces students to a variety of numerical methods and then applies these methods to solve a broad range of scientific problems. These problems include examples from physics as well as several other disciplines, including chemistry, mathematics, economics, and finance. Numerical techniques for solving problems expressed in terms of matrix, differential and integral equations will be developed.

Course Objectives

Understand and estimate errors due to round-off and truncation; understand error propagation and numerical instability; use bracketing and non-bracketing techniques to find approximate roots of non-linear equations and analyze the errors; perform data analysis using interpolation, extrapolation, and curve-fitting including quantification of the degree of fit; solve linear systems of equations using direct and iterative methods; calculate approximate derivatives and finite integrals; apply numerical techniques to solve ordinary differential equations.

Course Contents

1. Unit I: Mathematical Review and Errors

2 hrs

1.1. Mathematical Review: Taylor's Series, Mean Value Theorem, Asymptotic Notations

1.2. Errors in Numerical Computation: True Error, Relative Error, Approximate Error, Relative Approximate Error, Sources of Error: (Round off Error, Truncation Error)

1.3. Error Propagation, Floating Point Representation

2. Unit II: Solution of Nonlinear Equations

8 hrs

2.1. Nonlinear Equations Solution Approaches: Direct Analytical Method, Graphical Method, Trial and Error Method, Iterative Methods

2.2. Iterative Methods: Bisection Method, False Position Method, Newton-Raphson Method, Secant Method, Fixed Point Iteration Method and Proof of their Convergences

2.3. Synthetic Division, Remainder Theorem, Horner's Method for Polynomial Evaluation, Finding Multiple Roots

3. Unit III: Interpolation and Regression

8 hrs

3.1. Interpolation vs Extrapolation, Lagrange Interpolation, Newton's Divided Difference Interpolation

3.2. Interpolation with Equally Spaced Data: Newton's Forward Difference Interpolation, Newton's Backward Difference Interpolation

3.3. Spline Interpolation: What is Spline? Natural Cubic Splines

3.4. Regression vs Interpolation, Least Square Methods, Linear Regression

3.5. Non-Linear Regression: Polynomial Regression, Exponential Regression

4. Unit IV: Solving Systems of Linear Equations

8 hrs

4.1. System of Equations, Matrix Representation, Existence of Solution

4.2. Direct Methods for Solving System of Equations: Basic Gauss Elimination Method, Gauss-Elimination with Partial Pivoting, Gauss Jordan Method, Matrix Inversion

4.3. Matrix Factorization: LU Decomposition, Doolittle LU Decomposition, Cholesky's Method

4.4. Iterative Methods for Solving System of Equations: Jacobi Iteration Method, Gauss-Seidel Method

4.5. Ill-Conditioning, Eigenvalues and Eigenvectors, Power Method

5. Unit V: Numerical Differentiation

5 hrs

5.1. Numerical Differentiation: Introduction, Real Applications

5.2. Differentiating Continuous Functions: Forward Difference Formula, Backward Difference Formula, Central Difference Formula

5.3. Differentiating Discrete Functions: Derivatives using Newton's Divided Difference Formula, Derivatives using Newton's Forward Difference Formula

6. Unit VI: Numerical Integration

5 hrs

6.1. Numerical Integration: Introduction, Definite Integral Applications

6.2. Newton Cotes Integration Formulae, A General Quadrature Formula For Equally Spaced Arguments

6.3. Trapezoidal Rule, Composite (Multi-segment) Trapezoidal Rule, Simpson's 1/3 Rule, Composite (Multi-segment) Simpson's 1/3 Rule, Simpson's 3/8 Rule, Composite (Multi-segment) Simpson's 3/8 Rule

7. Unit VII: Solving Ordinary Differential Equations

6 hrs

7.1. Introduction: ODE vs PDE, Order, Degree and Solution of Differential Equations, Initial Value Problems and Boundary Value Problems

7.2. Solving Initial Value Problems: Picard's Method, Euler's Method, Heun's Method, Fourth Order Runge-Kutta Method

7.3. Solving System of ODEs and Higher Order ODEs by using any Existing Method

7.4. Solving Boundary Value Problems: Shooting Method, Finite Difference Method

8. Unit VIII: Solving Partial Differential Equations

3 hrs

8.1. Partial Differential Equations: Introduction, Categorization of PDEs: Elliptic, Parabolic and Hyperbolic PDEs

8.2. Deriving Difference Equations, Solving Laplace Equation, Solving Poisson's Equation

Laboratory Works

Students write programs and prepare lab sheets for each of the topics discussed in class. A minimum of 3 lab hours per week is required. The nature of programming problems can be decided by the instructor. A lab sheet of around 35 programming problems is recommended.

1.Error Analysis
2.Solution of Nonlinear Equations
3.Interpolation Methods
4.Regression Methods
5.Direct Methods for Linear Systems
6.Iterative Methods for Linear Systems and Eigenvalues
7.Numerical Differentiation
8.Numerical Integration
9.Solving Ordinary Differential Equations
10.Boundary Value Problems and Partial Differential Equations

Text Books

1.C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, 4th Edition, Addison Wesley Publishing Company, New York.
2.W.H. Press, B.P. Flannery et.al., Numerical Recipes in C, 1st Edition, Cambridge Press, 1988.

Reference Books

1.S.S. Shastry, Introductory Methods of Numerical Analysis, Fifth Edition, PHI Learning Pvt Limited, 2012.
2.Arjun Singh Saud, Bhupendra Singh Saud, Numerical Methods with Practical Approach, First Edition, Kriti Books and Publishers Pvt Limited, 2014.

Notes:

This syllabus follows the official CSIT curriculum of Far Western University. In case of any doubt or revision, the university's published syllabus shall be considered authoritative.

Numerical Method

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TabFlux . Numerical Methods . FWU . BSc. CSIT