Mathematics – 4, Numerical Method

Course Details

• Lectures: 50 Hours
• Exercise: 0 Hours
• Year: 3
• Semester: 5
• Tutorial: 30 Hours
• Full Marks: 100

A. Course Objective

To provide students with basic knowledge on solution of linear and nonlinear equations, interpolation and approximation, differentiation and integration, and differential equations.

B. Course Details

1.0 Solution of Nonlinear equations (10 Hours)

• 1.1 Review of calculus, continuity, differentiability, intermediate value theorem, Taylor's theorem
• 1.2 Absolute, relative and round-off errors, error bounds for computational errors
• 1.3 Bisection method, its error bounds and convergence
• 1.4 Newton's method, secant method and their convergence properties
• 1.5 Fixed point iteration, its convergence properties, Steffensen's algorithm
• 1.6 Zeros of polynomials by Horner's method

2.0 Interpolation and approximation (10 Hours)

• 2.1 Taylor's polynomial approximation, Lagrange's interpolation
• 2.2 Newton's interpolation and divided differences
• 2.3 Interactive interpolation
• 2.4 Cubic spline interpolation
• 2.5 Least squares method of fitting continuous and discrete data or functions

3.0 Numerical differentiation and integration (5 Hours)

• 3.1 Numerical differentiation formulae
• 3.2 Newton-Cotes numerical integration formulae, composite numerical integration
• 3.3 Romberg integration algorithm
• 3.4 Gaussian integration formula

4.0 Linear algebraic equations (10 Hours)

• 4.1 Review of the properties of matrices
• 4.2 Matrix form of Gaussian elimination, pivoting strategies, ill-conditioning
• 4.3 Cholesky's and related algorithms for matrix factorization
• 4.4 Eigenvalues and eigenvectors and the power method

5.0 Solutions of Ordinary Differential equations (7 Hours)

• 5.1 Euler’s Method of Solving Differential Equations of First Order and related methods
• 5.2 Runge-Kutta methods
• 5.3 Extension to higher-order equations
• 5.4 Initial value problems
• 5.5 Boundary value problems

6.0 Solution of partial differential equations (3 Hours)

• 6.1 Introduction to solution of partial differential equations
• 6.2 Civil Engineering Examples

Recommended Books

1. S. Yakwitz and F. Szidarovszky, “An introduction to Numerical Computation” Brooks/Cole Publishing Co., 1985
2. W. Cheney and D. Kincaid, “Numerical Mathematics and Computation” Brooks/Cole Publishing Co., 1985
3. C.F. Gerald and P.O. Wheatly, “Applied Numerical Analysis” Addison-Wesley Publishing Co., New York
4. H.W. Press, B.P. Flannery et al., “Numerical Recipes in C” Cambridge University Press, 1988
5. Lecture Notes