Division algorithms for Bernstein polynomials

Robust arithmetic for multivariate Bernstein-form polynomials

As a zero set of some multivariate splines, the piecewise algebraic variety is a kind of generalization of the classical algebraic variety. This paper presents an algorithm for isolating real roots of the zero-dimensional piecewise algebraic variety which is based on interval evaluation and the interval zeros of univariate interval polynomials in Bernstein form. An example is provided to show the proposed algorithm is effective.

Polynomial and spline quasi-interpolants (QIs) are practical and effective approximation operators. Among their remarkable properties, let us cite for example: good shape properties, easy computation and evaluation (no linear system to solve), uniform boundedness independently of the degree (polynomials) or of the partition (splines), good approximation order. We shall emphasize new results on various types of univariate and multivariate polynomial or spline QIs, depending on the nature of coefficient functionals, which can be differential, discrete or integral. We shall also present some applications of QIs to numerical methods.

Among the applications of the Bernstein polynomials in one variable is their use in solving problems associated with stochastic computing. Taking as a starting point the notion of stochastic logic in the sense of Qian-Riedel-Rosenberg, the aim of this paper is to investigate some necessary and sufficient conditions for guaranteeing whether polynomial operations can be implemented with stochastic logic based on multivariate Bernstein polynomials with coefficients in the unit interval.

In this paper we develop a new approach for learning decision trees and multivariate polynomials via interpolation of multivariate polynomials. This new approach yields simple learning algorithms for multivariate polynomials and decision trees over finite fields under any constant bounded product distribution. The output hypothesis is a (single) multivariate polynomial that is an /spl epsiv/-approximation of the target under any constant bounded product distribution. The new approach demonstrates the learnability of many classes under any constant bounded product distribution and using membership queries, such as j-disjoint DNF and multivariate polynomial with bounded degree over any field. The technique shows how to interpolate multivariate polynomials with bounded term size from membership queries only. This in particular gives a learning algorithm for O(log n)-depth decision tree from membership queries only and a new learning algorithm of any multivariate polynomial over sufficiently large fields from membership queries only. We show that our results for learning from membership queries only are the best possible.

We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the Bernstein-Bézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of L-bases, a class of bases that include the Bernstein-Bézier, multinomial, and a rich subclass of Lagrange and Newton bases. All of the known evaluation algorithms can be generated either by considering up recursive evaluation algorithms for L-bases or by examining change of basis algorithms for L-bases. For polynomials of degree n n in s s variables, the class of up recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity O ( n s + 1 ) O(n^{s+1}) , a nested multiplication algorithm with computational complexity O ( n s log ⁡ n ) O(n^s \log n) and a ladder recurrence algorithm with computational complexity O ( n s ) O(n^s) . These algorithms also generate a new generalization of the Aitken-Neville algorithm for evaluation of multivariate polynomials expressed in terms of Lagrange L-bases. The second class of algorithms, based on certain change of basis algorithms between L-bases, include a nested multiplication algorithm with computational complexity O ( n s ) O(n^s) , a divided difference algorithm, a forward difference algorithm, and a Lagrange evaluation algorithm with computational complexities O ( n s ) O(n^s) , O ( n s ) O(n^s) and O ( n ) O(n) per point respectively for the evaluation of multivariate polynomials at several points.

We derive low-complexity matrix-free finite element algorithms for simplicial Bernstein polynomials on simplices. Our techniques, based on a sparse representation of differentiation and special block structure in the matrices evaluating B-form polynomials at warped Gauss points, apply to variable coefficient problems as well as constant coefficient ones, thus extending our results in Kirby (Numer Math, 2011, in press).

The LLL basis reduction algorithm was the first polynomial-time algorithm to compute a reduced basis of a given lattice, and hence also a short vector in the lattice. It approximates an NP-hard problem where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm has applications in number theory, computer algebra and cryptography. In this paper, we provide an implementation of the LLL algorithm. Both its soundness and its polynomial running-time have been verified using Isabelle/HOL. Our implementation is nearly as fast as an implementation in a commercial computer algebra system, and its efficiency can be further increased by connecting it with fast untrusted lattice reduction algorithms and certifying their output. We additionally integrate one application of LLL, namely a verified factorization algorithm for univariate integer polynomials which runs in polynomial time.

This paper presents a modular algorithm for computing the greatest common right divisor (gcrd) of two univariate Ore polynomials over Z[t]. The subresultants of Ore polynomials are used to compute the evaluation homomorphic images of the gcrd. Rational number and rational function reconstructions are used to recover coefficients. The experimental results illustrate that the present algorithm is markedly superior to the Euclidean algorithm and the subresultant algorithm for Ore polynomials.

This paper presents a GPU-based real-time raycasting algorithm for piecewise algebraic surfaces in terms of tensor product B-splines. 3DDDA and depth peeling algorithms are employed to traverse the piecewise surface patches along each ray. The intersection between the ray and the patch is reduced to the root-finding problem of the univariate Bernstein polynomial. The polynomial is obtained via Chebyshev sampling points interpolation. An iterative and unconditionally convergent algorithm called Bezier point insertion is proposed to find the roots of the univariate polynomials. The Bezier point insertion is robust and suitable for the SIMD architecture of GPU. Experimental results show that the proposed root-finding algorithm performs better than other root-finding algorithms, such as Bezier clipping and B-spline knot insertion. Our rendering algorithm can display thousands of piecewise algebraic patches of degrees 6–9 in real time and can achieve the semi-transparent rendering interactively.

Chapter 12 - Multiplying with Synapses and Neurons

An identity for multivariate Bernstein polynomials

The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer coefficients. The code is mainly written in CilkPlus [11] targeting multicore processors. The current distribution focuses on dense polynomials and the sparse case is work in progress. A strong emphasis is to put on adaptive algorithms as the library aims at supporting a wide variety of situations in terms of problem sizes and available computing resources. One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers. The BPAS library is publicly available in source at www.bpaslib.org.

Polynomial Inequalities on Measurable Sets and Their Applications II. Weighted Measures

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y − f 1(X))…(Y − f n (X)) + g(X), with f, f 1, ⋅, f n , g univariate polynomials over an arbitrary field.