Math 625: Spring 2006
Computational Mathematics and C Programming

Course information

Course Description

This is a hands-on computational mathematics course with emphasis on programming in C. The goal is quickly to impart the necessary algorithmic and programming skills to enable the students build moderate- to large-scale application programs and gain a mastery of the C programming language in the context of scientific computing.

Some programming knowledge is assumed. A substantial part of the course will consists of introduction to C with special emphasis on common idioms used in scientific computing. Through the semester the students will gradually build their own computing library and will successively apply it to solve increasingly complex computational projects.

Prerequisites: Math 221 (linear algebra), Math 251 (multivariable calculus)

Resources

• 2006-02-18: array.h: type-generic array allocation macros
• 2006-02-21: Assorted images for use with the image processing utility
• 2006-03-14: rank-vertices.c
• 2006-04-05: Files for the truss solver:
  • Makefile
  • truss-to-eps.c   truss-to-eps.h
  • cantilever.dat   bollman.dat
  • cantilever.solution produced by

    truss-demo < cantilever.dat > cantilever.solution
truss-demo: read 8 nodes, 15 links, DOF=12
Nelder-Mead search converged after 1221 function evaluations

  • bollman.solution produced by

    truss-demo < bollman.dat > bollman.solution
truss-demo: read 14 nodes, 39 links, DOF=25
Nelder-Mead search converged after 5958 function evaluations

• 2006-04-15: Files for the integration program:
  • Correction: In read-mesh-data.c, in read_node_data(), near the end, I have:
    fscanf(fp, "%d %lf %lf %*d", ....
    Please change that to:
    fscanf(fp, "%u %lf %lf %*d", ....
  • fem.h
  • twb-quad.h   twb-quad.c
  • integrate-demo.c

    Try the various combinations of test functions and domain geometries defined in this file. Verify that you are getting the expected answer in each case.

  • square.poly

    Recommended usage: Put square.poly in a directory of its own. Make a coarse triangulation (square.1.*) and a fine triangulation (square.2.*) by:
    triangle -Qa0.05q30 square.poly.
    triangle -Qa0.01q30 square.1.poly.

  • annulus.poly

    Recommended usage: Put annulus.poly in a directory of its own. Make a coarse triangulation (annulus.1.*) and a fine triangulation (annulus.2.*) by:
    triangle -Qa0.5q30 annulus.poly.
    triangle -Qa0.1q30 annulus.1.poly.

  • ell.poly

    An ell-shaped domain with a hole removed. Its area is 2.82.

• 2006-04-28: Files for the finite element solver:
  • poisson.h
  • maple-plot.h   maple-plot.c
    The function:

    void maple_plot(Mesh *mesh, double *u, const char *filename);

    receives pointers to the mesh structure mesh, the solution vector u, and the name of an output file. It writes the given data to the output file in a form that can be read into Maple to display the solution as a 3D surface. Instuctions for running Maple are given at the top of that output file.
• 2006-05-02: An UMFPACK demo:
  • A demo file for UMFPACK: umfpack-simple.c
  • This one works just like maple_plot() (see above) but produces output that can be viewed in Matlab: matlab-plot.h   matlab-plot.c

Syllabus

• Dynamic memory allocation for vectors and matrices
• Linear system and the LU decomposition algorithm
• Haar wavelets applied to the image processing and compression
• Minimizing functions of n variables with Nelder-Mead downhill simplex algorithm
• Analyzing statically indeterminate 2D trusses via minimization of energy
• Legendre polynomials and Gauss quadrature for numerical integration
• A finite element solver for elliptic partial differential equations on irregular 2D domains with mixed Dirichlet and Neumann boundary conditions
• Using the UMFPACK library for solving large systems with sparse coefficients

Textbook

There is no textbook which covers the selection of topics planned for this course. I will provide plenty of reading material from various sources throughout the semester. Students are expected to take detailed notes.

Required reading: The C Programming Language by Brian Kernighan and Dennis Ritchie.

It would be a good idea to go over the book's errata collected in the book's website and insert fixes in the book.

You will find much interesting information about the book at http://cm.bell-labs.com/cm/cs/cbook.

Grading method

I will collect and grade the weekly programming assignments. Course grade will be based on cumulative performance on these assignments.

Sample image filtered with Haar wavelets

Original image	...with 10% of the original information
...with 5% of the original information	...with 1% of the original information

The Official UMBC Honors Code