The central problem in Part II is the zero finding problem. Special attention goes to multiple zeros and approximate GCDs. Recent work of Zhonggang Zeng has yielded robust algorithms for this problem.

The topics discussed in this lecture are:

1. Accuracy of Multiple Zeros (proposition 5.1).
2. The Chebyshev Criterion gives us a recommendation for the number of decimal places in the working precision: html version of a Maple worksheet. The Chebyshev Criterion is discussed in section 28.5 in the book of R.W. Hamming: "Numerical Methods for Scientists and Engineers", Dover 1986 (unabridged republication of the second edition, McGraw-Hill, 1973).
3. The quotient ring and its dual space of the ideal generated by a univeriate polynomial.
4. Jordan Canonical Form of the Companion Matrix Example 5.5 is illustrated by the html version of a Maple worksheet.
5. Numerical Methods to Find Roots
   • For the method of Weierstrass (aka Durand-Kerner), see project two of MCS 471 in Fall 2003.
   • To illustrate the importance of the zero finding problem, see
     1. John Michael McNamee: "A bibliography on roots of polynomials". Journal of Computational and Applied Mathematics, Volume 47, Issue 3, 30 September 1993, Pages 391-394
     2. John Michael McNamee: "An updated supplementary bibliography on roots of polynomials." Journal of Computational and Applied Mathematics, Volume 110, Issue 2, 30 October 1999, Pages 305-306
     Available online at a bibliography on roots of polynomials.