Improved Newton iteration for integral roots

Quadratic Newton Iteration for Systems with Multiplicity

Newton’s iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton’s iterator is still an open problem: How to design an efficient iterator wi...

The end-game for Newton iteration

We consider the angular orientations of convergent iterates generated by Newton’s method in multiple space dimensions. We show that the limiting behavior of the iterates can be described by a tensor eigenproblem. We give an extensive computational analysis of this tensor eigenproblem in two dimensions. In a large fraction of cases, the tensor eigenproblem has a discrete number of solutions to w...

Newton Iteration and Integer Division

An Improved Newton Iteration for the Generalized Inverse of a Matrix, with Applications

Accelerated Newton Iteration: Roots of Black Box Polynomials and Matrix Eigenvalues

We study the problem of computing the largest root of a real rooted polynomial p(x) to within error ε given only black box access to it, i.e., for any x ∈ ’, the algorithm can query an oracle for the value of p(x), but the algorithm is not allowed access to the coefficients of p(x). A folklore result for this problem is that the largest root of a polynomial can be computed in O (n log(1/ε)) pol...