On Polynomial Multiplication in Chebyshev Basis

Giorgi, Pascal

doi:10.1109/tc.2011.110

Cited by 13 publications

(15 citation statements)

References 24 publications

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“…Another area of improvement would concern the binary product operation, whose complexity currently scales exponentially in the number of variables and the expansion order. Several studies have explored approaches to speeding up the multiplication of univariate and bivariate polynomials in Chebyshev basis (e.g., [2,5,22,45]). Ways to use some of these developments for multivariate Chebyshev polynomials could be explored as part of future work.…”

Section: Discussionmentioning

confidence: 99%

Chebyshev model arithmetic for factorable functions

et al. 2016

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This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.

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Section: Discussionmentioning

confidence: 99%

Chebyshev model arithmetic for factorable functions

et al. 2016

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“…The 3C-algorithm is a special case of a more general class of algorithms dealing with the product of polynomials. We note that according to reference (Giorgi 2012) an even faster algorithm (although moderate) might be implemented in a future version of Angpow if necessary. We note finally that this general method can be applied to use cases beyond the power spectrum computation in other fields of interest.…”

Section: Appendix A: Clenshaw-curtis-chebyshev Algorithm (3c-algorithm)mentioning

confidence: 99%

Angpow: a software for the fast computation of accurate tomographic power spectra

Campagne

Neveu

Plaszczynski

2017

A&A

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Aims. The statistical distribution of galaxies is a powerful probe to constrain cosmological models and gravity. In particular, the matter power spectrum P(k) provides information about the cosmological distance evolution and the galaxy clustering. However the building of P(k) from galaxy catalogs requires a cosmological model to convert angles on the sky and redshifts into distances, which leads to difficulties when comparing data with predicted P(k) from other cosmological models, and for photometric surveys like the Large Synoptic Survey Telescope (LSST). The angular power spectrum C (z 1 , z 2 ) between two bins located at redshift z 1 and z 2 contains the same information as the matter power spectrum, and is free from any cosmological assumption, but the prediction of C (z 1 , z 2 ) from P(k) is a costly computation when performed precisely. Methods. The Angpow software aims at quickly and accurately computing the auto (z 1 = z 2 ) and cross (z 1 z 2 ) angular power spectra between redshift bins. We describe the developed algorithm based on developments on the Chebyshev polynomial basis and on the Clenshaw-Curtis quadrature method. We validate the results with other codes, and benchmark the performance. Results. Angpow is flexible and can handle any user-defined power spectra, transfer functions, and redshift selection windows. The code is fast enough to be embedded inside programs exploring large cosmological parameter spaces through the C (z 1 , z 2 ) comparison with data. We emphasize that the Limber's approximation, often used to speed up the computation, gives incorrect C values for cross-correlations.

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“…, we obtain first that τX (Un+1) ≤ τX (Tn), using Formulas (5) and (7). The sum of the absolute values of the coefficients of Tn is k n n−k…”

Section: Chebyshev Polynomialsmentioning

confidence: 99%

“…Note that this very simple algorithm might be replaced by the one from Giorgi ( [7]) which claims the equivalent of two multiplications between polynomials of degree d with coefficients of bitsize in O(τ ).…”

Section: Chebyshev Formsmentioning

confidence: 99%

“…This simple transformation is also used in Section 2 to set simple and fast algorithms for multiplying, dividing, computing greatest common divisors for polynomials expressed in the Chebyshev basis. In particular, we provide, for the multiplication, a very simple alternative to [7] with an equivalent bit complexity. More generally, the complexity of these operations is the same as for polynomials in the monomial basis (see [16]).…”

Section: Introductionmentioning

confidence: 99%

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On the Sign of a Trigonometric Expression

Koseleff

Rouillier

Tran

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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We propose a set of simple and fast algorithms for evaluating and using trigonometric expressions in the form, f k ∈ Q, d < n for a fixed n ∈ Z>0: computing 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

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On Polynomial Multiplication in Chebyshev Basis

Cited by 13 publications

References 24 publications

Chebyshev model arithmetic for factorable functions

Chebyshev model arithmetic for factorable functions

Angpow: a software for the fast computation of accurate tomographic power spectra

On the Sign of a Trigonometric Expression

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