Numerical Computing Course Outline - University of Sargodha

Course Title: Numerical Computing

Course Code: CS-3941
Course Structure: Lectures: 3 / Labs: 0
Credit Hours: 3
Prerequisites: None

Course Objectives:

On completion of this course, students will be able to demonstrate programming proficiency
using structured programming techniques to implement numerical methods for solutions using
computer-based programming techniques using Mat Lab for all methods. The course must serve
the purpose of scientific software development for science and engineering problems.

Course Syllabus:

Mathematical Preliminaries and Error Analysis. Solutions of Equations in One Variable.   Interpolation and Polynomial Approximation. Numerical Differentiation and Integration.Gaussian Quadrature. Multiple Integrals. Improper Integrals. Intial-Value Problems for Ordinary

Differential Equations. Euler's Method. Higher-Order Taylor Methods. Runge-Kutta Methods.
Error Control and the Runge-Kutta-Fehlberg Method. Multistep Methods. Variable Step-Size
Multistep Methods. Extrapolation Methods. Higher-Order Equations and Systems of Differential
Equations. Stability. Stiff Differential Equations. Direct Methods for Solving Linear Systems.
Iterative Techniques in Matrix Algebra. Approximation Theory. Approximating Eigenvalues.
Numerical Solutions of Nonlinear Systems of Equations. Boundary-Value Problems for Ordinary
Differential Equations. Numerical Solutions to Partial Differential Equations.

Course Outline:

1.     Mathematical Preliminaries and Error Analysis: Round-off Errors and Computer
Arithmetic. Algorithms and Convergence. [TB1: Ch. 1]

2.      Solutions of Equations in One Variable: The Bisection Method. Fixed-Point Iteration.
Newton's Method and its Extensions. Error Analysis for Iterative Methods. Accelerating
Convergence. Zeros of Polynomials and Muller's Method. [TBI: Ch. 2]

3.      Interpolation and Polynomial Approximation: Interpolation and the Lagrange
Polynomial. Data Approximation and Neville's Method. Divided Differences. Hermite
Interpolation. Cubic Spline Interpolation. Parametric Curves. [TBI: Ch. 3]

4.      Numerical Differentiation and Integration: Numerical Differentiation. Richardson's
Extrapolation. Elements of Numerical Integration. Composite Numerical Integration.
Romberg Integration. Adaptive Quadrature Methods. Gaussian Quadrature. Multiple
Integrals. Improper Integrals. [TBI: Ch. 4]

5.      Intial-Value Problems for Ordinary Differential Equations: Elementary Theory of Initial-
Value Problems. Euler's Method. Higher-Order Taylor Methods. Runge-Kutta Methods.
Error Control and the Runge-Kutta-Fehlberg Method. Multistep Methods. Variable Step-
Size Multistep Methods. Extrapolation Methods. Higher-Order Equations and Systems of
Differential Equations. Stability. Stiff Differential Equations. [TBI: Ch. 5]

6.      Direct Methods for Solving Linear Systems: Linear Systems of Equations. Pivoting
Strategies. Linear Algebra and Matrix Inversion. The Determinant of a Matrix. Matrix
Factorization. Special Types of Matrices. [TBI: Ch. 6]

7.      Iterative Techniques in Matrix Algebra: Norms of Vectors and Matrices. Eigenvalues and
Eigenvectors. The Jacobi and Gauss-Siedel Iterative Techniques. Iterative Techniques for
Solving Linear Systems. The Conjugate Gradient Method. [TBI: Ch. 7]

8.      Approximation Theory: Discrete Least Squares Approximation. Orthogonal Polynomials
and Least Squares Approximation. Rational Function Approximation. Trigonometric
Polynomial Approximation. [TBI: Ch. 8]

9.      Approximating Eigenvalues: Linear Algebra and Eigenvalues. Orthogonal Matrices and
Similarity Transformations. The Power Method. Householder's Method. The QR
Algorithm. [TBI: Ch. 9]

10. Numerical Solutions of Nonlinear Systems of Equations: Fixed Points for Functions of
Several Variables. Newton's Method. Quasi-Newton Methods. Homotopy and
Continuation Methods. [TB1: Ch. 10]

11.  Boundary-Value Problems for Ordinary Differential Equations: The Linear Shooting
Method. The Shooting Method for Nonlinear Problems. Finite-Difference Methods for
Linear Problems. Finite-Difference Methods for Nonlinear Problems. [TB1: Ch. 11]

12. Numerical Solutions to Partial Differential Equations: Elliptic Partial Differential

Equations. Parabolic Partial Differential Equations. Hyperbolic Partial Differential
Equations. [TB1: Ch. 12]

Textbook(s):

•         Numerical Analysis by Richard L. Burden and J. Douglas Faires, 9^th Edition, Cengage
Learning (2011). ISBN-10: 0538733519

•         Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms by
Anne Greenbaum & Timothy P. Chartier, Princeton University Press (April 1, 2012).
ISBN-10: 0691151229 [ For MatLab implementation]

Reference Material:

•          A First Course in Numerical Analysis by Anthony and Philip Rabinowitz , Dover
Publications; 2^nd Edition (February 6, 2001). ISBN-10: 048641454X

•         Numerical Methods in Scientific Computing by Germund Dahlquist and Ake Bjorck,
Society for Industrial and Applied Mathematics (2008). ISBN-10: 0898716446

•         Numerical Methods for Scientific Computing by J.H. Heinbockel, Create Space
Independent Publishing Platform (2006). ISBN-10: 1412031532

•         Numerical Methods for Scientists and Engineers (Dover Books on Mathematics) by R.
W. Hamming, Dover Publications; 2^nd Edition (1987). ISBN-10: 0486652416

Note: This content is obtained from official documents of University of Sargodha and applied on BS Computer Science for Main Campus, Sub Campuses, and Affiliated Colleges.