Math. 656 Course Outline

This outline is a tentative guide, and will be updated as the course progresses.  Time will be allocated as appropriate for student presentations and discussion.

Note: Some of the material referenced below is copyrighted, and hence is not generally available by clicking on it. Please email me,, and I will email you either a Postscript or a PDF copy of the appropriate excerpts, according to the ``fair use" provision of the copyright law.

Home page for the course

Topic no.  Description Explanation / References / Projects
1. Review of the first semester and outlook for this semseter.
  •  Nonlinear systems of equations versus operator equations
  • "Computational Functional Analysis"
  • Examples of operator equations, contrasted with finite-dimensional nonlinear systems
  • Underlying manipulative techniques for solving operator equations
  • 2. Review of (or introduction to) underlying functional analysis concepts (This topic may take several class periods.) Parts of chapters 1 through 12 of R. E. Moore, An Introduction to Numerical Functional Analysis, Halsted Press (1985)
    3. Contraction mappings and fixed point iterations in function spaces Chapter 15 of R. E. Moore
    4. Frechét derivatives Chapter 16 of R. E. Moore
    5. Newton's method in Banach spaces Chapter 17 of R. E. Moore
    6 Example of a simple iteration method in an infinite-dimensional space Chapter 8 of Ole Stauning's Ph.D. dissertation
    7. Introduction to Green's functions See the web page at
    8.. A review of more sophisticated methods for elliptic boundary value problems M. Plum, "Inclusion Methods for Elliptic Boundary Value Problems," in Topics in Validated Computations, ed. J. Herzberger, North-Holland, 1994.
    (Copies to be supplied.)
    9. More on the solution of integral equations L. B. Rall, "Application of Interval Integration to the Solution of Integral Equations," J. Integral Equations 6, pp. 127-141 (1984) (copies will be supplied)
    10. Miscellaneous additional applications Note: Some of these are finite-dimensional, and may be presented before topics 1-8 above.
    1. Ray tracing: W. Barth, R. Lieger, M. Schindler, "Ray Tracing General

    2. Parametric Surfaces Using Interval Arithmetic", The Visual
      Computer, International Journal of Computer Graphics, 10  (7), pp. 363-371 (1984).