The Chebyshev points of the first kind

Xu, Kuan

doi:10.1016/j.apnum.2015.12.002

Cited by 24 publications

(14 citation statements)

References 46 publications

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“…In other words, by setting n b = n we can recover Method 1. To subsample, we use the nested property of the Chebyshev nodes [41]. More precisely, let the Chebyshev nodes…”

Section: Methods 2: Randomized Interpolatory Tensor Decomposition Wit...mentioning

confidence: 99%

Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions

Saibaba¹,

Minster²,

Kilmer³

2021

Preprint

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In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We have detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes.

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“…In other words, by setting n b = n we can recover Method 1. To subsample, we use the nested property of the Chebyshev nodes [41]. More precisely, let the Chebyshev nodes…”

Section: Methods 2: Randomized Interpolatory Tensor Decomposition Wit...mentioning

confidence: 99%

Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions

Saibaba¹,

Minster²,

Kilmer³

2021

Preprint

View full text Add to dashboard Cite

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“…Therefore, together with the degrees of polynomials ranging from 0 to n − 1, an n × n discrete orthogonal matrix P i (x k ) can be constructed, where any two different row vectors are orthogonal. Discrete orthogonal matrices are commonly used in a number of orthogonal transformations over real intervals, such as the Chebyshev Polynomials [17], [18], the Legendre Polynomials [19], the Meixner polynomials [20], the Charlier polynomials [21], the Krawtchouk Polynomials [22], the Discrete Hartley Transform [23] and the well-known Discrete Cosine Transform [11]. A comprehensive overview of these orthogonal polynomials, along with the development of their discrete matrices, is also detailed in [24].…”

Section: Charliermentioning

confidence: 99%

A General Method for Generating Discrete Orthogonal Matrices

Chan

2021

IEEE Access

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Discrete orthogonal matrices have applications in information coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize continuous orthogonal functions that have been discovered. The need of such continuous functions is restrictive. Polynomials, as the simplest class of continuous functions, are widely studied for their orthogonalality, to serve the purpose of generating orthogonal matrices. However, beginning with continuous orthogonal polynomials still takes much work. To overcome this complexity while improving the efficiency and flexibility, we present a general method for generating orthogonal matrices directly through the construction of certain even and odd polynomials from a set of distinct positive values, bypassing the need of continuous orthogonal functions. We present a constructive proof by induction that not only asserts the existence of such polynomials, but also tells how to iteratively construct them. Besides the derivation of the method as simple as a few nested loops, we discuss two well-known discrete transforms, the Discrete Cosine Transform and the Discrete Tchebichef Transform, how they can be achieved using our method with the specific values, and show how to embed them into the transform module of video coding. By the same token, we also give the examples for generating new orthogonal matrices from arbitrarily chosen values. The demonstrative experiments indicate that our method is not only simpler to implement, but also more efficient and flexible. It can generate orthogonal matrices of larger sizes, compared with those existing methods.

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“…The Chebyshev zeros are defined as the roots of the Chebyshev polynomial of the first kind of degree M, see e.g. [27] for a recent review. In the interval (−1, 1), the nodes are given by the explicit formulas 1) or the equivalent expression…”

Section: B Chebyshev Zeros With One Additional Endpointmentioning

confidence: 99%

Numerical Bifurcation Analysis of Physiologically Structured Population Models via Pseudospectral Approximation

et al. 2020

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We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system is ten times more efficient than the one originally proposed in [Breda et al, SIAM Journal on Applied Dynamical Systems, 2016], as it avoids the numerical inversion of an algebraic equation.

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The Chebyshev points of the first kind

Cited by 24 publications

References 46 publications

Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions

Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions

A General Method for Generating Discrete Orthogonal Matrices

Numerical Bifurcation Analysis of Physiologically Structured Population Models via Pseudospectral Approximation

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