Fast structured Jacobi-Jacobi transforms

Shen, Jie; Wang, Yingwei; Xia, Jianlin

doi:10.1090/mcom/3377

Cited by 9 publications

(5 citation statements)

References 41 publications

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“…A transformation of the Jacobi polynomials from index pair (α 1 , β 1 ) to (α 2 , β 2 ) is also easy to perform [34].…”

Section: Jacobi Polynomialsmentioning

confidence: 99%

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Li,

Ma,

Zhao

2024

Axioms

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Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.

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“…A transformation of the Jacobi polynomials from index pair (α 1 , β 1 ) to (α 2 , β 2 ) is also easy to perform [34].…”

Section: Jacobi Polynomialsmentioning

confidence: 99%

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Li,

Ma,

Zhao

2024

Axioms

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“…This transform algorithm is based on the connection relations that exist between different families of Jacobi polynomials. We will follow a similar approach to [14,15,18] and adapt it to the Jones-Worland polynomials. The algorithm heavily relies on two recurrence relations for the Jacobi polynomials.…”

Section: Jones-worland Transform Algorithmmentioning

confidence: 99%

Accurate and efficient Jones-Worland spectral transforms for planetary applications

Marti

Jackson

2021

Proceedings of the Platform for Advanced Scientific Computing Conference

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Spectral transforms between physical space and spectral space are needed for fluid dynamical calculations in the whole sphere, representative of a planetary core. In order to construct a representation that is everywhere smooth, regular and differentiable, special polynomials called Jones-Worland polynomials, based on a type of Jacobi polynomial, are used for the radial expansion, coupled to spherical harmonics in angular variables. We present an exact, efficient transform that is partly based on the FFT and which remains accurate in finite precision. Application is to high-resolution solutions of the Navier-Stokes equation, possibly coupled to the heat transfer and induction equations. Expected implementations would be in simulations with 𝑃 3 degrees of freedom, where 𝑃 may be greater than 10 3 . Memory use remains modest at high spatial resolution, indeed typically 𝑃 times lower than competing algorithms based on quadrature. CCS CONCEPTS• Applied computing → Earth and atmospheric sciences; • Mathematics of computing → Mathematical software performance; • Computing methodologies → Massively parallel and high-performance simulations.

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“…18.6.1], it is sufficient to assume that β = δ in (5.4). 4 A different fast algorithm using off-diagonal low rank structure of the conversion matrix, which has a fast quasilinear complexity online cost and an O(N 2 ) precomputation is given in [34]. Another fast algorithm for computing Jacobi expansions coefficients of analytic functions is described in [40].…”

Section: Chebyshev-to-legendre Conversionmentioning

confidence: 99%

Fast polynomial transforms based on Toeplitz and Hankel matrices

2017

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Abstract. Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive O(N (log N ) 2 ) algorithms, based on the fast Fourier transform, for converting coefficients of a degree N polynomial in one polynomial basis to coefficients in another. Numerical results show that this approach is competitive with state-of-the-art techniques, requires no precomputational cost, can be implemented in a handful of lines of code, and is easily adapted to extended precision arithmetic.

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Fast structured Jacobi-Jacobi transforms

Cited by 9 publications

References 41 publications

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Accurate and efficient Jones-Worland spectral transforms for planetary applications

Fast polynomial transforms based on Toeplitz and Hankel matrices

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