Computing Floating-Point Square Roots via Bivariate Polynomial Evaluation

Accurate Polynomial Evaluation in Floating Point Arithmetic

One of the three main processes associated with polynomials is evaluation; the two other ones being interpolation and root finding. Higham [1, chap. 5] devotes an entire chapter to polynomials and more especially to polynomial evaluation. The small backward error the Horner scheme introduce when evaluated in floating point arithmetic justifies its practical interest. It is well known that the c...

Roots of Bivariate Polynomial Systems via Determinantal Representations

Fast Floating Point Square Root

Hain and Freire have proposed different floating point square root algorithms that can be efficiently implemented in hardware. The algorithms are compared and evaluated on both performance and precision.

The complexity of class polynomial computation via floating point approximations

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton ...

On the stability of computing polynomial roots via confederate linearizations

A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems t...