Approximate square-free decomposition and root-finding of lll-conditioned algebraic equations
Journal of Information Processing
Mathematics for computer algebra
Mathematics for computer algebra
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
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)
Univariate polynomials: nearly optimal algorithms for factorization and rootfinding
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
A formula for separating small roots of a polynomial
ACM SIGSAM Bulletin
An iterated eigenvalue algorithm for approximating roots of univariate polynomials
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Algorithms for Quad-Double Precision Floating Point Arithmetic
ARITH '01 Proceedings of the 15th IEEE Symposium on Computer Arithmetic
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Given a univariate polynomial having well-separated clusters of close roots, we give a method of computing close roots in a cluster simultaneously, without computing other roots. We first determine the position and the size of the cluster, as well as the number of close roots contained. Then, we move the origin to a near center of the cluster and perform the scale transformation so that the cluster is enlarged to be of size O(1). These operations transform the polynomial to a very characteristic one. We modify Durand-Kerner's method so as to compute only the close roots in the cluster. The method is very efficient because we can discard most terms of the transformed polynomial. We also give a formula of quite tight error bound. We show high efficiency of our method by empirical experiments.