Root-Refining for a Polynomial Equation

Abstract. Polynomial root-finders usually consist of two stages. At first a crude approximation to a root is slowly computed; then it is much faster refined by means of the same or distinct iteration. The efficiency of com-puting an initial approximation resists formal study, and the users rely on empirical data. In contrast, the efficiency of refinement is formally measured by the classical concept q1/d where q denotes the convergence order, whereas d denotes the number of function evaluations per iter-ation. In our case of a polynomial of a degree n we use 2n arithmetic operations per its evaluation of at a point. Noting this we extend the def-inition to cover iterations that are not reduced to function evaluations alone, including iterations that simultaneously refine n approximations to all n roots of a degree n polynomial. By employing two approaches to the latter task, both based on recursive polynomial factorization, we yield refinement with the efficiency 2d, d = cn / log2 n for a positive constant c. For large n this is a dramatic increase versus the record efficiency 2 of refining an approximation to a single root of a polynomial. The advance could motivate practical use of the proposed root-refiners

Pathological work-up of sentinel lymph nodes in breast cancer. Review of current data to be considered for the formulation of guidelines.

Controversies and inconsistencies regarding the pathological work-up of sentinel lymph nodes (SNs) led the European Working Group for Breast Screening Pathology (EWGBSP) to review published data and current evidence that can promote the formulation of European guidelines for the pathological work-up of SNs. After an evaluation of the accuracy of SN biopsy as a staging procedure, the yields of different sectioning methods and the immunohistochemical detection of metastatic cells are reviewed. Currently published data do not allow the significance of micrometastases or isolated tumour cells to be established, but it is suggested that approximately 18% of the cases may be associated with further nodal (non-SN) metastases, i.e. approximately 2% of all patients initially staged by SN biopsy. The methods for the intraoperative and molecular assessment of SNs are also surveyed