Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
The algebraic eigenvalue problem
The algebraic eigenvalue problem
A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Error Bounds for Zeros of a Polynomial Based Upon Gerschgorin's Theorems
Journal of the ACM (JACM)
Enclosing clusters of zeros of polynomials
Journal of Computational and Applied Mathematics
Enclosing clusters of zeros of polynomials
Journal of Computational and Applied Mathematics
A property of the nearly optimal root-bound
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
On the convergence condition of generalized root iterations for the inclusion of polynomial zeros
Mathematics and Computers in Simulation
A family of root-finding methods with accelerated convergence
Computers & Mathematics with Applications
On Newton-type methods for multiple roots with cubic convergence
Journal of Computational and Applied Mathematics
Compositional analysis of floating-point linear numerical filters
CAV'05 Proceedings of the 17th international conference on Computer Aided Verification
Arrangement computation for planar algebraic curves
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
| Hi-index | 7.31 |
Given a univariate polynomial P with a k-fold multiple root or a k-fold root cluster near some z, we discuss nine different methods to compute a disc near z which either contains exactly or contains at least k roots of P. Many of the presented methods are known, some are new. We are especially interested in the behaviour of methods when implemented in a rigorous way, that is when taking into account all possible effects of rounding errors. In other words every result shall be mathematically correct. We display extensive test sets comparing the methods under different circumstances. Based on the results we present a tenth, a hybrid method combining five of the previous methods which, for given z, (i) detects the number k of roots near z and (ii) computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster. Therefore, the resulting discs are numerically nearly optimal.