Practical Divide-and-Conquer Algorithms for Polynomial Arithmetic

Abstract

Abstract. We investigate two practical divide-and-conquer style algorithms for univariate polynomial arithmetic. First we revisit an algorithm originally described by Brent and Kung for composition of power series, showing that it can be applied practically to composition of polynomials in Z [x] given in the standard monomial basis. We offer a complexity analysis, showing that it is asymptotically fast, avoiding coefficient explosion in Z[x]. Secondly we provide an improvement to Mulders' polynomial division algorithm. We show that it is particularly efficient compared with the multimodular algorithm. The algorithms are straightforward to implement and available in the open source FLINT C library. We offer a practical comparison of our implementations with various computer algebra systems.

Univariate real root isolation in an extension field and applications

“…Next, we evaluate h, that represents the common roots of f and g, at the endpoints of this refined interval. Each evaluation costs O B (dτ + ddτ ) = O B (d 2 τ ), using a divide and conquer approach [5, Lemma 6], see also [3,18]. The output bitsize of this evaluation is O(d 2 τ ).…”

On the Sign of a Trigonometric Expression

“…Bostan et al (see [4]) give a fast general conversion method from the monomial basis to orthogonal bases. The alternatives we propose are based on the general method of Hart & Novocin for polynomial composition (see [9]) which lead to simplier descriptions and complexity analysis. Given f = Suppose that we have to evaluate the sum S = d j=0 ujv j .…”

“…In order not to reinvent the wheel, we have considered these transformations as a particular case of composition method developed in [9] with additional refinements linked to the properties of Chebyshev polynomials.…”

“…In section 3, we show that passing from the monomial basis to the Chebyshev basis consists essentially to inject in the algorithm from [9] the fast operations we have introduced in Section 2. For the reverse operation, we make use of a the linear recurrence introduced in [4] and essentially keep the structure of the composition algorithm from [9], thanks to some remarkable properties of Chebyshev polynomials.…”

“…For the reverse operation, we make use of a the linear recurrence introduced in [4] and essentially keep the structure of the composition algorithm from [9], thanks to some remarkable properties of Chebyshev polynomials. We finally show that the two transformations we propose perform O(d) arithmetic operations in Z and O(d(τ + d)) bit operations if τ is the maximal bitsize of the f k 's.…”