Multivariate Polynomial Multiplication Research Articles

Multilinear polynomial systems: Root isolation and bit complexity

We exploit structure in polynomial system solving by considering polynomials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We enumerate all expressions of the multihomogeneous (or multigraded) resultant of such systems as a determinant of Sylvester-like matrices, aka generalized Sylvester matrices. We construct these matrices by means of Weyman homological complexes, which generalize the Cayley-Koszul complex.The computation of the determinant of the resultant matrix is the bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector multiplication, which corresponds to multivariate polynomial multiplication, by extending the seminal work on Macaulay matrices of Canny, Kaltofen, and Yagati Canny et al. (1989) to the multihomogeneous case.We compute a rational univariate representation of the roots, based on the primitive element method. In the case of 0-dimensional systems we present a Monte Carlo algorithm with probability of success 1−1/2ϱ, for a given ϱ≥1, and bit complexity O˜B(n2D4+ϵ(nN+1+τ)+nD2+ϵϱ(D+ϱ)) for any ϵ>0, where n is the number of variables, D equals the multilinear Bézout bound, N is the number of variable subsets, and τ is the maximum coefficient bitsize. We present an algorithmic variant to compute the isolated roots of overdetermined and positive-dimensional systems. Thus our algorithms and complexity analysis apply in general with no assumptions on the input.

Multivariate Polynomial Multiplication on GPU

Multivariate polynomial multiplication is a fundamental operation which is used in many scientific domains, for example in the optics code for particle accelerator design at CERN. We present a novel and efficient multivariate polynomial multiplication algorithm for GPUs using floating-point double precision coefficients implemented using the CUDA parallel programming platform. We obtain very good speedups over another multivariate polynomial multiplication library for GPUs (up to 548x), and over the implementation of our algorithm for multi-core machines using OpenMP (up to 7.46x).

Sparse polynomial multiplication and division in Maple 14

We demonstrate new routines for sparse multivariate polynomial multiplication and division over the integers that we have integrated into Maple 14 through the expand and divide commands. These routines are currently the fastest available, and the multiplication routine is parallelized with superlinear speedup. The performance of Maple is significantly improved. We describe our polynomial data structure and compare it with Maple's. Then we present benchmarks comparing Maple 14 with Maple 13, Magma, Mathematica, Singular, Pari, and Trip.

Robust Stability of Discrete Polynomials with Polynomic Structure of Coefficients

A method for Schur robust stability analysis of polynomials with polynomic structure of their coefficients is presented in this paper. The presented method tests the stability in the coefficient space and is based on the Modified Jury criterion. Coefficients of this table are multivariate polynomial functions, whose positivity is tested by Sign-decomposition. This is a method which makes it possible to test positivity of some set of multivariate functions on a convex set. Another approach consists in using linear fractional transformation of discrete polynomials into continuous polynomials and then in using the Modified Routh criterion or the Hurwitz criterion together with Sign-decomposition for robust stability analysis. To determine elements of any table a new procedure based on fast Fourier transform for multivariate polynomial multiplication is used. Both methods are resulting in a necessary and sufficient condition. They are demonstrated on an easy example and compared from computational point of view.

Simple Multivariate Polynomial Multiplication

We observe that polynomial evaluation and interpolation can be performed fast over a multidimensional grid (lattice), and we apply this observation in order to devise a simple algorithm for multivariate polynomial multiplication. Surprisingly, this simple idea enables us to improve the known algorithms for multivariate polynomial multiplication based on the forward and backward application of Kronecker's map; in particular, we decrease, by the factor log log N, the known upper bound on the arithmetic time-complexity of this computation (over any field of constants), provided that the degree d in each of the m variables is fixed, m grows to the infinity, and N = ( d + 1) m .