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An application of Gelfand's formula in approximating the roots of polynomials

Frank J. Hall, Rachid M. Marsli, Rachid M. Marsli

Abstract

The purpose of this paper is to show how Gelfand's formula and balancing can be used to improve the upper and lower bounds of the spectrum of a companion matrix associated with a given real or complex polynomial. Examples and other related ideas are provided.

An application of Gelfand's formula in approximating the roots of polynomials

Abstract

The purpose of this paper is to show how Gelfand's formula and balancing can be used to improve the upper and lower bounds of the spectrum of a companion matrix associated with a given real or complex polynomial. Examples and other related ideas are provided.
Paper Structure (7 sections, 4 theorems, 50 equations, 2 figures, 5 tables, 2 algorithms)

This paper contains 7 sections, 4 theorems, 50 equations, 2 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. localization of the roots of a real polynomial
  3. Refining the roots location through matrix balancing of $C_1$ and $C_2$
  4. Numerical Results for Larger Matrices
  5. Upper and lower bounds on the distance between $\lambda_1$ and $\lambda_n$
  6. Comparison with other bounds and localization regions
  7. Conclusion

Key Result

Theorem 2.2

Using the same notation as above, the roots of the polynomial $p(x)$ are contained in the annulus defined by

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (15)

  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Example 2.6
  • Example 3.2
  • Theorem 5.1
  • Theorem 5.2
  • proof
  • Remark 5.3
  • ...and 5 more