6,427 research outputs found
On Polynomial Multiplication in Chebyshev Basis
In a recent paper Lima, Panario and Wang have provided a new method to
multiply polynomials in Chebyshev basis which aims at reducing the total number
of multiplication when polynomials have small degree. Their idea is to use
Karatsuba's multiplication scheme to improve upon the naive method but without
being able to get rid of its quadratic complexity. In this paper, we extend
their result by providing a reduction scheme which allows to multiply
polynomial in Chebyshev basis by using algorithms from the monomial basis case
and therefore get the same asymptotic complexity estimate. Our reduction allows
to use any of these algorithms without converting polynomials input to monomial
basis which therefore provide a more direct reduction scheme then the one using
conversions. We also demonstrate that our reduction is efficient in practice,
and even outperform the performance of the best known algorithm for Chebyshev
basis when polynomials have large degree. Finally, we demonstrate a linear time
equivalence between the polynomial multiplication problem under monomial basis
and under Chebyshev basis
Chebyshev model arithmetic for factorable functions
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
A fast and well-conditioned spectral method
A novel spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes operations, where is the number of Chebyshev points needed to resolve the coefficients of the differential operator and is the number of Chebyshev points needed to resolve the solution to the differential equation. We prove stability of the method by relating it to a diagonally preconditioned system which has a bounded condition number, in a suitable norm. For Dirichlet boundary conditions, this reduces to stability in the standard 2-norm
Chebyshev polynomials and the Frohman-Gelca formula
Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the
Kauffman bracket skein module of the torus. This basis is especially useful
because the Jones-Kauffman product can be described via a very simple
Product-to-Sum formula. Presented in this work is a diagrammatic proof of this
formula, which emphasizes and demystifies the role played by Chebyshev
polynomials.Comment: 13 page
Phases of N=1 Supersymmetric SO/Sp Gauge Theories via Matrix Model
We extend the results of Cachazo, Seiberg and Witten to N=1 supersymmetric
gauge theories with gauge groups SO(2N), SO(2N+1) and Sp(2N). By taking the
superpotential which is an arbitrary polynomial of adjoint matter \Phi as a
small perturbation of N=2 gauge theories, we examine the singular points
preserving N=1 supersymmetry in the moduli space where mutually local monopoles
become massless. We derive the matrix model complex curve for the whole range
of the degree of perturbed superpotential. Then we determine a generalized
Konishi anomaly equation implying the orientifold contribution. We turn to the
multiplication map and the confinement index K and describe both Coulomb branch
and confining branch. In particular, we construct a multiplication map from
SO(2N+1) to SO(2KN-K+2) where K is an even integer as well as a multiplication
map from SO(2N) to SO(2KN-2K+2) (K is a positive integer), a map from SO(2N+1)
to SO(2KN-K+2) (K is an odd integer) and a map from Sp(2N) to Sp(2KN+2K-2).
Finally we analyze some examples which show some duality: the same moduli space
has two different semiclassical limits corresponding to distinct gauge groups.Comment: 55pp; two paragraphs in page 19 added to clarify the relation between
confinement index and multiplication map index, refs added and to appear in
JHEP; Konishi anomaly equations corrected and some comments on the
degenerated cases for SO(7) and SO(8) adde
A low multiplicative complexity fast recursive DCT-2 algorithm
A fast Discrete Cosine Transform (DCT) algorithm is introduced that can be of
particular interest in image processing. The main features of the algorithm are
regularity of the graph and very low arithmetic complexity. The 16-point
version of the algorithm requires only 32 multiplications and 81 additions. The
computational core of the algorithm consists of only 17 nontrivial
multiplications, the rest 15 are scaling factors that can be compensated in the
post-processing. The derivation of the algorithm is based on the algebraic
signal processing theory (ASP).Comment: 4 pages, 2 figure
Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix
This paper studies how to sketch element-wise functions of low-rank matrices.
Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we
aim for finding an approximated low-rank representation of the (possibly
high-rank) matrix [f(Aij)]. To this end, we propose an efficient
sketching-based algorithm whose complexity is significantly lower than the
number of entries of A, i.e., it runs without accessing all entries of [f(Aij)]
explicitly. The main idea underlying our method is to combine a polynomial
approximation of f with the existing tensor sketch scheme for approximating
monomials of entries of A. To balance the errors of the two approximation
components in an optimal manner, we propose a novel regression formula to find
polynomial coefficients given A and f. In particular, we utilize a
coreset-based regression with a rigorous approximation guarantee. Finally, we
demonstrate the applicability and superiority of the proposed scheme under
various machine learning tasks
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