AMAT 12253: Numerical Methods I

Course Code     : AMAT 12253

Title            : Numerical Methods I

Pre-requisites        : AMAT 11223

Learning Outcomes:

At the end of the course, the student should be able to;

1. explain the issues of accuracy in floating-point representation
2. find roots of non-linear equations using appropriate numerical methods
3. identify the error in a given data set
4. apply appropriate interpolation and curve fitting techniques for a given application
5. solve a system of equations using direct methods
6. describe the limitations, advantages, and disadvantages of numerical methods.

Course Contents: 

Introduction: 

• Floating point number system, error in numerical computation, strategies for minimizing round-off errors, Ill conditioning, condition number, the notion of an algorithm.
  
  

Solution of equations with one variable: 

• Numerical solution of nonlinear equations using Bisection method, False Position method, Fixed-Point iteration method, Newton-Raphson method, Secant method and modified secant method, error analysis for iterative methods.
  
  

Difference Operators: 

• Forward, Backward, Central and Averaging operators, symbolic relations of difference operators, difference table and error propagation, difference equations, factorial polynomials.
  
  

Interpolation: 

• Collocation polynomial and its properties, Newton’s Forward and Backward difference formulae, Gauss’s Central Difference Formula, interpolation with unevenly spaced points: Lagrange’s and Newton’s interpolation, Spline Interpolation: Linear, Quadratic and Cubic Spline Interpolation.

      Least-square curve fitting techniques: 

• Linear functions: normal equations, coefficient of determination, non- linear functions: exponential model, power model, Saturation Growth Rate model.
  
  

Solution of System of Linear Equations (Direct Methods): 

• Matrix inversion, Naïve Gauss elimination, Gaussian elimination with partial pivoting, Ill conditioning Matrices, Operation counts, Matrix Decomposition Techniques: LU and QR Factorizations.

Method of Teaching and Learning : A combination of lectures and tutorial discussions

Assessment : Based on tutorials, tests and end of course examination

Recommended Reading :

1. Burden, R.L., Faires, J.D, Burden, A.M. (10th Ed., 2015). Numerical Analysis, Cengage Learning.
2. Sastry, S.S. (5th Ed., 2012). Introductory Methods of Numerical Analysis, Prentice-Hall India.
3. Kreyszig, E. (10th Ed., 2010). Advanced Engineering Mathematics, John Wiley.
4. Sauer, T. (2012). Numerical Analysis, Pearson.
5. Epperson, J.F. (2013). An Introduction to Numerical Methods and Analysis, Wiley.
6. Faul, A.C. (2016). A Concise Introduction to Numerical Analysis, Chapman and Hall/CRC.