English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The DLG root-squaring iterations, due to Dandelin 1826 and rediscovered byLobachevsky 1834 and Graeffe 1837, have been the main approach to root-findingfor a univariate polynomial p(x) in the 19th century and beyond, but not sonowadays because these iterations are prone to severe numerical stabilityproblems. Trying to avoid these problems we have found simple but novelreduction of the iterations applied for Newton's inverse ratio -p'(x)/p(x) toapproximation of the power sums of the zeros of p(x) and its reversepolynomial. The resulting polynomial root-finders can be devised and performedindependently of DLG iterations, based on Newton's identities or Cauchyintegrals. In the former case the computation involve a set of leading ortailing coefficients of an input polynomial. In the latter case we must scalethe variable and increase the arithmetic computational cost to ensure numericalstability. Nevertheless the cost is still manageable, at least for fastroot-refinement, and the algorithms can be applied to a black box polynomialp(x)--given by a black box for the evaluation of the ratio p'(x)/p(x) ratherthan by its coefficients. This enables important computational benefits,including efficient recursive as well as concurrent approximation of a set ofzeros of p(x) or even all of its zeros, acceleration where an input polynomialcan be evaluated fast, and extension to approximation of the eigenvalues of amatrix or a polynomial matrix, being efficient if the matrix can be invertedfast, e.g., is data sparse. We also recall our recent fast algorithms forapproximation of the root radii, that is, the distances to the roots from theorigin or any complex value to the zeros of p(x), and apply it for fast blackbox initialization of polynomial root-finding by means of functionaliterations, e.g., Newton's, Ehrlich's, and Weierstrass's.

New Progress in Classic Area: Polynomial Root-squaring and Root-finding