On the sign of a trigonometric expression

Abstract : We propose a set of simple and fast algorithms for evalu-ating and using trigonometric expressions in the form F = d k=0 f k cos k π n , f k ∈ Q, d < n for a fixed n ∈ Z>0: comput-ing the sign of such an expression, evaluating it numerically and computing its minimal polynomial in Q[x]. As critical byproducts, we propose simple and efficient algorithms for performing arithmetic operations (multiplication, division, gcd) on polynomials expressed in a Chebyshev basis (with the same bit-complexity than in the monomial basis) and for computing the minimal polynomial of 2 cos π n in O(n 2 0) bit operations with n0 < n is the odd squarefree part of n. Within such a framework, we can decide if F = 0 in O(d(τ +d)) bit operations , compute the sign of F in O(d 2 τ) bit operations and compute the minimal polynomial of F in O(n 3 τ) bit operations, where τ denotes the maximum bit-size of the f k 's.
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https://hal.inria.fr/hal-01108656
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Submitted on : Wednesday, January 28, 2015 - 11:29:20 AM
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  • HAL Id : hal-01108656, version 1

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Pierre-Vincent Koseleff, Fabrice Rouillier, Cuong Tran. On the sign of a trigonometric expression. 2015. ⟨hal-01108656⟩

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