Fall Semester 2005

Numerical Methods I


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This course is the second year core for Computational Science. It is an integrated course with 3 hours of lectures and 1.5 hours of computational laboratory per week, and will run over two semesters.

The course covers the standard techniques of numerical computation from a theoretical as well as a practical perspective with a particular emphasis on large-scale high-performance computation, and provides the foundation for more specialized third year courses in computation and modeling.

It is assumed that participants have the mathematical background equivalent to two semesters of Engineering and Science Mathematics - attendance of the first year B track (Multivariable Calculus, ODE; Linear Algebra, Fourier Methods, Probability) is highly recommended but not formally required - or Analysis I/II and Linear Algebra I.

The course is appropriate as a home school elective for students of all majors with a particular interest in computation. It is recommended that students commit to this course for the full year. Students interested in a more compact introduction to methods of numerical computation are advised to take the one-semester Engineering and Science Mathematics 4A (Numerical Methods) instead.

Topics covered throughout the year are: computer arithmetic, condition of algorithms, systems of linear equations including iterative methods, computation of eigenvalues, interpolation and least square methods, numerical quadrature, numerical solution of ordinary differential equations (including stiff equations and boundary value problems); introduction to linear and nonlinear optimization. The last part of this course consists of special topics which may change from year to year.

Contact Information:
Instructor:Marcel Oliver
Office hours:  WF after class in Research I, 107

TA/grader:Stanislav Harizanov
Office hours:  TBA

Time and Place:
Lectures:  WF 9:45-11:00 in East Hall 4

Recommended Textbooks:

Additional Reading:

Homework and Projects:
The weekly homework and project sheets are handed out each Wednesday and are due Friday the following week. For computer projects, the following must be turned in:
  • Hardcopy printouts of the code.
  • A detailed description of the computer experiment including, if applicable, a description of the code, a discussion of the input data, a description of the output, a discussion of possible errors, efficiency, and implementation difficulties, and a verbose answer to the question posed.
  • If applicable, hardcopy printouts of the results.
  • The code should, moreover, be sent in a single email to to TBA.
You should be able to present and explain the code in the computer lab session following submission.

Collaborative project work in groups of two or three is permissible provided:

  • Each member of the group maintains a runnable version of the code (on their personal laptop or teaching lab account) and is able to demonstrate and explain the code without help from others.
  • Each member of the group submits a discussion, clearly distinct from the others', using their own words. The only exception to this rule are source code comments (which are encouraged!) and algorithmically generated output.
  • You state who you collaborated with (both in your written exposition and in the source cod)

You may consult books and internet resources provided you always quote the source.


Class Schedule (subject to change)

02/09/2005: Introduction
07/09/2005: Computer arithmetic, condition of algorithms
09/09/2005: Solving scalar nonlinear algebraic equations: bisection, Newton's method
14/09/2005: Solving scalar nonlinear algebraic equations: analysis of Newton's method, secant method
16/09/2005: Matrix norms and condition numbers
21/09/2005: Linear systems: Gauss elimination
23/09/2005: LU decomposition without pivoting
28/09/2005: LU decomposition with pivoting; error analysis
30/09/2005: QR decomposition; least square solutions to linear systems
05/10/2005: Iterative methods: Jacobi and Gauss-Seidel method
07/10/2005: Gradient method
12/10/2005: Conjugate Gradient method
14/10/2005: Review for midterm exam
19/10/2005: Midterm Exam
26/10/2005: Lagrange interpolation; estimates of the interpolation error; numerical differentiation
28/10/2005: Lagrange interpolation (continued)
02/11/2005: Splines: minimality properties and approximation error estimates
04/11/2005: Splines: Derivation of the coefficients for the natural spline; introduction to B-splines
09/11/2005: Trigonometric interpolation: basic setup, orthogonality properties of DFT
11/11/2005: Trigonometric interpolation: approximation error estimates; introduction to the FFT
16/11/2005: Numerical integration: Newton Cotes formulae
18/11/2005: Numerical integration: Gauss quadrature
23/11/2005: Numerical integration: extrapolation
25/11/2005: Adaptive integration
30/11/2005: Review of ordinary differential equations
02/12/2005: One step methods for ordinary differential equations
07/12/2005: Review for final exam
19/12/2005: Final Exam, 9:00-11:00 in East Hall 5

Last modified: 2005/11/30
This page: http://math.iu-bremen.de/oliver/teaching/iub/fall2005/cps211/index.html
Marcel Oliver (m.oliver@iu-bremen.de)