Math. 655 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
Bibliography for the course in PDF format

Topic no.  Description Explanation / References / Projects
1. Overview of the course and a review of interval arithmetic
  • Kreinovich's web page
  • Kearfott's Euromath Bulletin article
  • Sun's interval compiler introduction
  • Kearfott's preprint page and 
  • Sun's Interval Arithmetic Programming Reference
  • 2. A review of interval Newton methods
  • preprint of Kearfott's Encyclopedia of Optimization article and 
  • §1.5 of Rigorous Global Search: Continuous Problems (email for a copy.)
  • 3. A review of necessary and sufficient conditions for global optimization
  • §3.3 and §3.4 of Gill / Murray / Wright, 
  • §5.2.5 of Rigorous Global Search: Continuous Problems (email for a copy.)
  • 4. Hansen's technique for substituting points for some intervals in the derivative matrices
  • §1.3 of Rigorous Global Search: Continuous Problems (email for a copy.), also 
  • slope-based existence and uniqueness in §1.5 of Rigorous Global Search: Continuous Problems (email for a copy.)
  • 5. Interval Newton methods applied to the Fritz / John equations
  • Examples of interval Newton methods with equality, inequality, and bound constraints; discussion of the Lagrange multipliers; 
  • E. Hansen and W. Walster, "Bounds for Lagrange Multipliers and Optimal Points," Comput. Math. Appl.25 (10), pp. 59 ff
  • Project: Investigate practical behavior of estimates for Lagrange multipliers in local computational existence / uniqueness.
  • 6 Interval Newton Methods and non-Smooth problems
  • Chapter 6 of Rigorous Global Search: Continuous Problems, and an excerpt from 
  • I. Zang, "A Smoothing-Out Technique for Min-Max Optimization, Mathematical Programming 19 (1), pp. 61--67 (1980) (email for a copy.)
  • Project: Compare the method of representing breakpoints in Zang to that in "Rigorous Global Search."  Which method gives smaller intervals? How would we compute interval slopes for the Zang representation?
  • Project: Compare verification processes obtained from adding constraints to verification processes associated with handling the non-smooth problems directly.
  • 7. Generalizations of interval Newton Methods
  • Project: Investigate the "higher order" interval Newton method described in M. Berz and J. Hoefkens, "Verified High-Order Inversion of Functional Dependencies and

  • Interval Newton Methods, Reliable Computing 7 (5), pp. 379--398, 2001. (email for a copy.)
    8. Extended interval arithmetic and optimization
  • G. W. Walster, J. D. Pryce, and E. R. Hansen," Exception-Free Arithmetic on the Extended Reals," seminar notes from the Royal Military College of Science, Cranfield, U.K. (email for a copy.)
  • We will be doing some experiments with interval Newton methods and extended interval arithmetic, in particular, looking at the relationship to  necessary conditions for optima.
  • Project: In Sun's extended interval arithmetic, does 0 need to be an element of grad(f)(x) for x to contain a critical point? (There are issues of domain of definition and of smoothness.)
  • 9. Singular systems and the topological degree
  • §1.6 of Rigorous Global Search: Continuous Problems (email for a copy.), 
  • "Existence Verification for Singular Zeros of Complex Nonlinear Systems, SIAM J. Numer. Anal. 38 (2), pp. 360--379 (2000).  (Copyright  SIAM.)
  • "Existence Verification for Higher Degree Singular Zeros of Complex Nonlinear Systems," (joint with J. Dian), submitted to the SIAM J. Numer. Anal. 
  • "Verifying Topological Indices for Higher-Order Rank Deficiencies," submitted for the proceedings of the 2000 AMS-IMS-SIAM summer  research conference on Algorithms and their Complexity for Nonlinear Problems, July 16 to July 20, 2000, Mt. Holyoke College, edited by Christopher Sikorski. 
  • Projects:
  • Implement computation of the topological index for higher-order rank deficiencies (worth at least a publication)
  • Investigate degree computation in the non-smooth case.
  • 10. Interval Newton methods and least squares: the augmented system
  • M. Arioli, I. S. Duff, and P. P. M. de Rijk, "On the Augmented System Approach to Sparse Least-Squares Problems," Numer. Math. 55 (1989), pp. 667--684.
  • Notes on implementing the augmented system (email for a copy.)
  • 11. Least squares, a possible new paradigm
  • Uncertainties in the data can be explicitly taken into account with interval techniques

  • References: 
  • A. Neumaier, "Solving Linear Least Squares Problems by Interval Arithmetic," Freburger Intervall-Berichte 87, 6, pp. 37--42.
  • A. Neumaier, "Linear Interval Equations," in Interval Mathematics 1985 (Lecture Notes in Computer Science no. 212), ed. K. Nickel, Springer Verlag, 1985
  • Project: Implement the idea in the Neumaier reference and see how it does with some realistic data sets.
  • 12. Workshop on good programming style
  • W. S. Brainerd, C. H. Goldberg and J. C. Adams, Programmer's Guide to Fortran 90, third edition, Springer Verlag, New York, 1995.
  • We will cover the following:
    1. Global versus local data (Proper use of modules, locality of reference, ease and readability of subroutine calls; various examples will be given.)
    2. Indentation and other readability considerations (loop indentation, if-then-else and case, choosing variable and label names, avoidance of "jumps")
    3. Modularity (making subroutines into logical units, internal versus external subroutines)
    4. Machine efficiency versus readability and maintainability (human efficiency)
    5. A discussion of user-defined data types
    13. Workshop on mathematical writing
  • "Plain Writing ... What's Needed and Why," Address given by I. E. Block, SIAM Managing Editor, July 18, 1989 (Copies will be handed out in class.) Also, Handbook of Writing for the Mathematical Sciences by Nicholas J. Higham, SIAM, Philadelphia, 1993 (in Dupré Library).
  • 12.