Enclosing clusters of zeros of polynomials

Authors:
Arnold Neumaier
Affiliations:
Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
Venue:
Journal of Computational and Applied Mathematics
Year:
2003

Citing 9
Cited 5

Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions

Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
A bibliography on roots of polynomials

Journal of Computational and Applied Mathematics
A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem

SIAM Journal on Matrix Analysis and Applications
Detection and validation of clusters of polynomial zeros

Journal of Symbolic Computation - Special issue: validated numerical methods and computer algebra
Error Bounds for Zeros of a Polynomial Based Upon Gerschgorin's Theorems

Journal of the ACM (JACM)
Algorithm 493: Zeros of a Real Polynomial [C2]

ACM Transactions on Mathematical Software (TOMS)
Improvement of convergence of an iterative method for finding polynomial factors of analytic functions

Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
Ten methods to bound multiple roots of polynomials

Journal of Computational and Applied Mathematics
Rounding Errors in Algebraic Processes

Rounding Errors in Algebraic Processes

Ten methods to bound multiple roots of polynomials

Journal of Computational and Applied Mathematics
A verified method for bounding clusters of zeros of analytic functions

Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
An efficient higher order family of root finders

Journal of Computational and Applied Mathematics
A family of root-finding methods with accelerated convergence

Computers & Mathematics with Applications
Constraint propagation on quadratic constraints

Constraints

Quantified Score

Hi-index	7.29

Visualization

Abstract

Lagrange interpolation and partial fraction expansion can be used to derive a Gerschgorin-type theorem that gives simple and powerful a posteriori error bounds for the zeros of a polynomial if approximations to all zeros are available. Compared to bounds from a corresponding eigenvalue problem, a factor of at least two is gained.The accuracy of the bounds is analyzed, and special attention is given to ensure that the bounds work well not only for single zeros but also for multiple zeros and clusters of close zeros.A Rouché-type theorem is also given, that in many cases reduces the bound even further.