The problem is to calculate a simple zero of a non-linear function f n-1 by iteration. We exhibit a family of iterations of order 2 which use n evaluations of f and no derivative evaluations, as well as a second family of iterations of order 2 n ^ based on n-1 evaluations of f and one of f f . In particular, with four evaluations we construct an iteration of eighth order.The best previous result for four evaluations was fifth order.We prove that the optimal order of one general class of multipoint n 1 iterations is 2 and that an upper bound on the order of a multipoint iteration based on n evaluations of f (no derivatives) is 2 n .CONJECTURE. A multipoint iteration without memory based on n evaluations has optimal order 2 n

No abstract

No abstract

The Poisson equation occurs in many areas of science and engineering. Here we focus on its numerical solution for an equation in d dimensions. In particular we present a quantum algorithm and a scalable quantum circuit design which approximates the solution of the Poisson equation on a grid with error ε. We assume we are given a superposition of function evaluations of the right-hand side of the Poisson equation. The algorithm produces a quantum state encoding the solution. The number of quantum operations and the number of qubits used by the circuit is almost linear in d and polylog in ε −1 . We present quantum circuit modules together with performance guarantees which can also be used for other problems.

This paper presents an elementary treatment of the theory of subresultants, and examines the relationship of the subresultants of a given pair of polynomials to their polynomial remainder sequence as determined by Euclid's algorithm. Two important versions of Euclid's algorithm are discussed. The results are essentially the same as those of Collins, but the presentation is briefer, simpler, and somewhat more general.