Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
The complexity of elementary algebra and geometry
Journal of Computer and System Sciences
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Parallel computation of polynomial GCD and some related parallel computations over abstract fields
Theoretical Computer Science
Graeffe's, Chebyshev-like, and Cardinal's processes for splitting a polynomial into factors
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
Sign determination in residue number systems
Theoretical Computer Science - Special issue on real numbers and computers
The Future Fast Fourier Transform?
SIAM Journal on Scientific Computing
On the Reduction of Number Range in the Use of the Graeffe Process
Journal of the ACM (JACM)
Variations on computing reciprocals of power series
Information Processing Letters - Special issue analytical theory of fuzzy control with applications
On the geometry of Graeffe Iteration
Journal of Complexity
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
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In this paper we present a suitable generalization of the classical Graeffe's iteration by showing that it provides effective algorithms for the fast evaluation of polynomials and residues of rational and meromorphic functions. In particular, if g(z) = q(z)/p(z) is a rational function with n finite poles &xgr;1,…,&xgr;n and &xgr; is an initial approximation of &xgr;k such that ¦&xgr; - &xgr;k¦ j¦ for any j ≠ k, 1 ⪇ j ⪇ n, then an extrapolation method is found for computing the residue of g(z) at &xgr;k by means of successive evaluations of the derivatives of q(z) and p(z) at the point z = &xgr;. In the case where q(z) = zp′(z) we thus obtain a set of iterations for the refinement of polynomial zeros. Moreover, by setting q(z) = r(z)p′(z), the same approach can also be used for the fast approximate evaluation of r(z) on the zeros of p(z).