Two methods for the verified inclusion of zeros of complex polynomials. (English) Zbl 0784.65043
Ullrich, Christian (ed.), Contributions to computer arithmetic and self- validating numerical methods. Papers from the first international conference held at the University of Basel, Switzerland, October 2-6, 1989. Basel: J. C. Baltzer AG, IMACS Ann. Comput. Appl. Math. 7, 229-244 (1990).
Summary: This paper presents two algorithms which compute verified inclusion of zeros of complex polynomials, i.e. the zeros are enclosed in narrow bounds by using the precise scalar product, complex interval arithmetic and complex circular arithmetic provided by the programming language PASCAL-SC.
In the first algorithm the problem to find the zeros of the complex polynomial is reduced to the eigenvalue problem of the companion matrix. Additionally the coefficients of the deflated polynomial are enclosed. Further it is described how to verify all simple zeros of the polynomial. A modification of this method allows to get inclusions of double zeros.
The second algorithm to be described allows to include zeros of arbitrary multiplicity. It is based on the Schur-Cohn algorithm which delivers the number of zeros of a polynomial lying in the interior of a closed disk of the complex plane.
For the entire collection see [Zbl 0778.00038].
In the first algorithm the problem to find the zeros of the complex polynomial is reduced to the eigenvalue problem of the companion matrix. Additionally the coefficients of the deflated polynomial are enclosed. Further it is described how to verify all simple zeros of the polynomial. A modification of this method allows to get inclusions of double zeros.
The second algorithm to be described allows to include zeros of arbitrary multiplicity. It is based on the Schur-Cohn algorithm which delivers the number of zeros of a polynomial lying in the interior of a closed disk of the complex plane.
For the entire collection see [Zbl 0778.00038].
MSC:
| 65H05 | Numerical computation of solutions to single equations |
| 65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |
| 65G30 | Interval and finite arithmetic |
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 12Y05 | Computational aspects of field theory and polynomials (MSC2010) |
