Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
On condition numbers and the distance to the nearest III-posted problem
Numerische Mathematik
LAPACK's user's guide
A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Polynomial roots from companion matrix eigenvalues
Mathematics of Computation
Matrix computations (3rd ed.)
Applied numerical linear algebra
Applied numerical linear algebra
Error Bounds for Zeros of a Polynomial Based Upon Gerschgorin's Theorems
Journal of the ACM (JACM)
Efficient Collision Detection for Curved Solid Objects
Proceedings of the seventh ACM symposium on Solid modeling and applications
An iterated eigenvalue algorithm for approximating roots of univariate polynomials
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Relations between roots and coefficients, interpolation and application to system solving
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Efficient isolation of polynomial's real roots
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Parametrization of approximate algebraic curves by lines
Theoretical Computer Science - Algebraic and numerical algorithm
Parametrization of approximate algebraic surfaces by lines
Computer Aided Geometric Design
Improved algorithms for computing determinants and resultants
Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
Can the TPRI structure help us to solve the algebraic eigenproblem?
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Distance bounds of ε-points on hypersurfaces
Theoretical Computer Science
Structured matrix methods for polynomial root-finding
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Parametrization of approximate algebraic surfaces by lines
Computer Aided Geometric Design
Method for finding multiple roots of polynomials
Computers & Mathematics with Applications
Approximately singular systems and ill-conditioned polynomial systems
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Polynomial root-finding via structured matrices
ACM Communications in Computer Algebra
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We analyze an iterative algorithm that approximates all roots of a univariate polynomial. The algorithm is based on (hardware) floating-point eigenvalue computation of a generalized companion matrix. With some assumptions, we show that it approximates the roots to floating-point accuracy within about log&rgr;/e X(P) iterations, where ∈ is the relative error of floating-point arithmetic, &rgr; is the relative separation of the roots, and X(P) is the condition number of the polynomial. Each iteration requires an n × n floating-point eigenvalue computation, n the polynomial degree, and evaluation of the polynomial to floating-point accuracy at n points. On some hard examples of ill-conditioned polynomials, e.g. high-degree Wilkinson polynomials, the algorithm is an order of magnitude faster than the best alternative.