Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Root neighborhoods of a polynomial

Author: Ronald G. Mosier
Journal: Math. Comp. 47 (1986), 265-273
MSC: Primary 65G05; Secondary 12D10, 30C10, 30C15
MathSciNet review: 842134
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The root neighborhoods of $ p(z)$, a polynomial over the complex field, are the sets of complex numbers that are the roots of polynomials which are near to $ p(z)$. The term 'near' means that the coefficients of the polynomials are within some fixed magnitude of the coefficients of $ p(z)$. A necessary and sufficient condition for a complex number to be in the root neighborhoods is given and it is proved that each root neighborhood contains at least one root of $ p(z)$ and the same number of roots of each near polynomial. Finally, a necessary and sufficient condition is given for a root neighborhood to contain more than one root of $ p(z)$, and consequently more than one root of any of the near polynomials.

References [Enhancements On Off] (What's this?)

  • [1] W. Gautschi, "The condition of orthogonal polynomials," Math. Comp.., v. 26, 1972, pp. 923-924. MR 0313558 (47:2112)
  • [2] W. Gautschi, "On the condition of algebraic equations," Numer. Math., v. 21, 1973, pp. 405-424. MR 0339478 (49:4237)
  • [3] W. Gautschi, "The condition of polynomials in power form," Math. Comp., v. 33, 1979, pp. 343-352. MR 514830 (80f:65034)
  • [4] W. Gautschi, "Questions of numerical condition related to polynomials," MAA Studies in Numerical Analysis, Vol. 24 (Gene H. Golub, ed.), The Mathematical Association of America, 1984, pp. 140-177. MR 925213
  • [5] K. Kuratowski, Topology, Vol. II, Academic Press, New York and London, 1968. MR 0259835 (41:4467)
  • [6] M. Marden, Geometry of Polynomials, 2nd ed., Math. Surveys, No. 3, Amer. Math. Soc., Providence, R.I., 1966. MR 0225972 (37:1562)
  • [7] A. M. Ostrowski, Solutions of Equations and Systems of Equations, Academic Press, New York and London, 1960. MR 0216746 (35:7575)
  • [8] S. Saks & A. Zygmund, Analytic Functions, Elsevier, New York, 1971.
  • [9] J. H. Wilkinson, Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, N.J., 1963. MR 0161456 (28:4661)
  • [10] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965. MR 0184422 (32:1894)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65G05, 12D10, 30C10, 30C15

Retrieve articles in all journals with MSC: 65G05, 12D10, 30C10, 30C15

Additional Information

Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society