Constrained multi-degree reduction with respect to Jacobi norms

Abstract

Please cite this article in press as: Ait-Haddou, R., Bartoň, M. Constrained multi-degree reduction with respect to Jacobi norms. Comput. Aided Geom. Des. (2015), http://dx.doi.org/10. 1016/j.cagd.2015.12.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights• We study the problem of constrained degree reduction with respect to Jacobi norms.• Our results generalize several previous findings on polynomial degree reduction.• We explore the space of Jacobi parameters on the reduced polynomial approximation.Constrained multi-degree reduction with respect to Jacobi norms AbstractWe show that a weighted least squares approximation of Bézier coefficients with factored Hahn weights provides the best constrained polynomial degree reduction with respect to the Jacobi L 2 -norm. This result affords generalizations to many previous findings in the field of polynomial degree reduction. A solution method to the constrained multi-degree reduction with respect to the Jacobi L 2 -norm is presented.

Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm

“…1.4649 × 10 −2 2.0349 × 10 −2 ω = 0, λ1 = 3, λ2 = 3, μ1 = 3; λ3 = 2, μ2 = 2 1.3253 × 10 −2 5.2874 × 10 −3 1.4618 × 10 −3 Example 3. For the degree reduction of traditional Bézier curves, scholars have done a lot of related researches; see [22][23][24][25][26][27][28][29][30]. Nevertheless, up to now, the related work about degree reduction of SG-Bézier curves has not been studied.…”

“…One method is to transform the problem of degree reduction into solving the optimization of objective function by using intelligent optimization algorithm; Ahn et al [29] showed that the constrained polynomial degree reduction in the L 2 -norm equals best weighted Euclidean approximation of Bézier coefficients. In 2016, Ait-Haddou and Bartoň [30] illustrated that a weighted least squares approximation of Bézier coefficients with factored Hahn weights provides the best constrained polynomial degree reduction with respect to the Jacobi L 2 -norm. Lu and Qin [31] proposed a method for the degree reduction of S-λ curves using a GSA algorithm.…”

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

“…Thus, for data conversation and transmission between various models, we properly investigated the degree reduction/elevation of curves. For the degree reduction of curves, the three methods were proposed based on the least square theory [22][23][24][25], the algebraic method [26][27][28][29][30], and the intelligent optimization algorithm based methods, in which the problem of degree reduction is formulated as an optimization one and is solved by incorporating intelligent optimization algorithms [31][32][33][34]. In 2019, based on the genetic simulated annealing algorithm, Lu and Qin [31] realized the multi-degree reduction approximation of the S-λ curve for the first time.…”

An index-based algorithm for fast on-line query processing of latent semantic analysis

“…QUIC-SVD [ 46 ] provides an algorithm which producing the approximation of the whole-matrix SVD based on a sampling mechanism called the cosine tree, and provides speedups of several orders of magnitude over exact SVD. [ 47 , 48 ] proposed an algorithm for accurately computation of SVD by inhering the high accuracy properties of the Jacobi algorithm [ 49 ]. [ 50 ] introduced a bi-iteration type subspace tracker for updating SVD approximation of the cross-correlation matrix of dimension N × M .…”