Module No:    CZ1104
Subject:      Numerical Methods I
Prerequisite: CZ1102 or its equivalent
Lecturer:     Roger Tan Choon Ee 
             (email: mattance@nus.edu.sg; Rm: S14, #05-19; Tel: 874-2769)
Contact Hours:Lecture - 2 hours per week; 
              Tutorial - 1 hour per fortnight; 
              Laboratory - 2 hours per fortnight

Aims: This course is designed to give you a knowledge of the broad areas 
      in which numerical methods are employed and an awareness of recent 
      trends. It introduces you to the theoretical basis of numerical 
      techniques and the associated error analysis in selected areas. 
      
      A student satisfactorily completing this module should possess:

     (i)   a knowledge of various numerical techniques required to control 
           error growth, both inherent and induced;
     (ii)  an ability to implement selected algorithms by means of computer 
           programs and/or standard software packages and analyse results 
           where possible;
     (iii) an ability to report work undertaken on a given problem in a clear and 
           comprehensive manner.

SYLLABUS

1. Computer Arithmetic and Computational Errors (4 hours)
Number systems, floating and fixed point representation, chopping and rounding modes.
sources of errors, round-off error, truncation error, propagation of error, error due to
cancellation of two nearly equal numbers. Numerical instability in some algorithms.

2. Solution of Linear Systems of Equations (4 hours)
Gaussian elimination, partial and complete pivoting, scaling. LU-decomposition (Crout's and
Doolittle's schemes), Choleski's decomposition for symmetric matrices.
Thomas algorithm for tridiagonal systems.

3. Interpolation (5 hours)
Polynomial interpolation, Lagrange's interpolation formula, truncation error,
Runge's example. Divided differences, Newton divided difference interpolation formula,
Newton-Gregory forward and backward formulas.

4. Least Squares Approximation (3 hours)
Least squares approximation for discrete data, least squares solution of
overdetermined linear systems, normal equations. Least squares approximation for
continuous functions.

5. Numerical Integration (5 hours)
Midpoint rule, Trapezoidal rule and Simpson's rule with error fomulas.
Newton-Cotes formulas, composite rules and errors. Richardson's
extrapolation and Romberg integration. Gaussian quadrature formulas

6. Solution of Nonlinear Equations (5 hours)
Bisection method, Newton's method, Secant method. General iterative
method, fixed point theorem, order of convergence, error estimate,
Aitken's acceleration technique.

ASSESSMENTS:

A maximum of 20% will be allocated for assignments carry out during the semester, and a
two-hour examination will be conducted in April/May.

RECOMMENDED TEXTBOOK:

R. L. Burden and J. D. Faires, Numerical Analysis. 6th edition (1997).
Publisher: PWS-Kent.

OTHER REFERENCES:

W. H. Press, B. P. Flannery, S. A. Teukolsky \& W. T. Vetterling,
Numerical Recipes in C --- The Art of Scientific Computing (1994).
Publisher: Cambridge University Press.

K. E. Atkinson, Elementary Numerical Analysis. 2nd edition (1993).
Publisher: John Wiley.