MCS 563: Analytic Symbolic Computation

The textbook for this course is Numerical Polynomial Algebra by Hans J. Stetter, SIAM 2004.

The goal of the course is study symbolic-numerical algorithms with their implementation and applications to science and engineering. There are three aspects: algorithms, software, and applications. We will follow the book "Numerical Polynomial Algebra", which could also have been called "Numerical Commutative Algebra".

From time to time, we will make an excursion into Numerical Algebraic Geometry, in particular:

1. Algorithms: homotopy continuation and polyhedral methods.
2. Software: the SNAP facilities in Maple and PHCpack.
3. Applications: mechanical design and linear systems control.
   

Supplements to the lectures:

• lecture three: See the html version of a Maple worksheet illustrating the computation of a basis and the relation with eigenvalues.
• lecture nine: See the html version of a Maple worksheet illustrating the computation of power series solutions using Newton's method.
• lecture nine: See the html version of a Maple worksheet with the example of Windsteiger showing the numerical instability of a pure lexicographic Groebner basis.
• lecture ten: Read the papers listed here.
• lecture eleven: Two Maple worksheets on the Chebyshev criterion and the Jordan Canonical Form, a pointer to the method of Weierstrass and a link to a bibliography of polynomial roots.
• lecture twelve: Pointers to mpsolve and eigensolve, two recent root finders, and a Maple experiment on the Wilkinson polynomial.
• lecture 14: some notes on Cayley-Bacharach for the well-posedness of multivariate polynomial interpolation.
• lecture 16: the html version of a Maple worksheet illustrates the computation of a certificate of a pseudodivisor.
• lecture 17: we looked at the paper of Zhonggang Zeng on "Computing Multiple Roots of Inexact Polynomials", Math. Comp. electronically posted on July 22, 2004. See the web site of Zhonggang Zeng for nice powerpoint slides.
• Review questions for Chapters 1 to 6
• Questions for the midterm exam (Friday 8 October 2004); with answers in html format of a Maple worksheet.
• lecture 21: the html version of a Maple worksheet on the importance of parametrizations and on the technique of Lagrange multipliers.
• lecture 27. We illustrate Example 8.23 on page 319 of the textbook using a Maple worksheet, showing the infeasibility of a set of monomials to be a normal set for a particular polynomial system. The worksheet is html format is here.
• lecture 29. We continue the lecture of last Friday on homotopies and polyhedral methods.

We did not cover Chapter 11 of the book. Instead, we made an excursion to polyhedral methods -- justified by the importance of the "BKK bound" in the numerical basis computation. Here is the full outline of the lectures on polyhedral methods:

• L-26 10/22/04: Homotopies and Polyhedral Methods
  the theorems of Bézout and Koushnirenko
• L-29 10/29/04: Koushnirenko's Theorem and Puiseux series
  polyhedral homotopies induced by regular triangulations
• L-34 11/10/04: Deficient Systems and Bernshtein's theorems
  statements of Bernshtein's Theorem A and Theorem B
• L-35 11/11/04: Polyhedral Methods, Theorem B
  Puiseux series and face systems
• L-36 11/12/04: Polyhedral Methods, Theorem A
  mixed-cell configurations and Puiseux series
• L-37 11/15/04: Polyhedral Methods to Compute Mixed Volumes
  the lift-and-prune algorithm