Embarrassingly parallel algorithms for algebraic number arithmetic — and some less trivial issues

Abstract

AbstractRepresenting algebraic numbers α and β by their defining polynomials is an alternative to the older representation in which the sum α + β would be represented by a list of character strings recursively involving root indices, +, −, x, and integer terms or radicands. Algorithms for sums, products, etc., in the defining polynomial representation, are based on the theory of symmetric functions and have relatively efficient implementations using polynomial resultants. Even better algorithms use power sums rather than the coefficients of the defining polynomials; on massively parallel systems α × β executes in constant time, and computation of α + β is linear (or logarithmic, if enough processors are available) in mn, where m and n are the degrees of the defining polynomials and mn is the degree of the result. However, given polynomials of degree m and n, these algorithms require their power sums to order mn. The best known power-sum algorithm is based on Newton's identity, which may be treated as a linear recurrence, whose solution, conventionally understood to be of complexity O(mn), will dominate the time of the multiplication algorithm and significantly increase the addition time. A parallel algorithm, reducing the recurrence solution to O((log2 n)(log2 m)), is discussed.Keywordsalgebraic numberspower sumslinear recurrences