The nearest polynomial with a given zero, revisited

ABSTRACT

In his 1999 SIGSAM BULLETIN paper [7], H. J. Stetter gave an explicit formula for finding the nearest polynomial with a given zero. This present paper revisits the issue, correcting a minor omission from Stetter's formula and explicitly extending the results to different polynomial bases.Experiments with our implementation demonstrate that the formula may not after all, fully solve the problem, and we discuss some outstanding issues: first, that the nearest polynomial with the given zero may be identically zero (which might be surprising), and, second, that the problem of finding the nearest polynomial of the same degree with a given zero may not, in fact, have a solution. A third variant of the problem, namely to find the nearest monic polynomial (given a monic polynomial initially) with a given zero, a problem that makes sense in some polynomial bases but not others, can also be solved with Stetter's formula, and this may be more satisfactory in some circumstances. This last can be generalized to the case where some coefficients are intrinsic and not to be changed, whereas others are empiric and may safely be changed. Of course, this minor generalization is implicit in [7]; This paper simply makes it explicit.

REFERENCES

• 1: Robert M. Corless , Patrizia M. Gianni , Barry M. Trager , Stephen M. Watt, The singular value decomposition for polynomial systems, Proceedings of the 1995 international symposium on Symbolic and algebraic computation, p.195-207, July 10-12, 1995, Montreal, Quebec, Canada [doi>10.1145/220346.220371]
• 2: Markus A. Hitz , Erich Kaltofen , Y. N. Lakshman, Efficient algorithms for computing the nearest polynomial with a real root and related problems, Proceedings of the 1999 international symposium on Symbolic and algebraic computation, p.205-212, July 28-31, 1999, Vancouver, British Columbia, Canada [doi>10.1145/309831.309937]
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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.1 Expressions and Their Representation
          Subjects: Representations (general and polynomial)

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on polynomials

General Terms:
Algorithms