The spectral connection matrix for classical orthogonal polynomials of a single parameter

Bella, T.; Reis, Jenna

doi:10.1016/j.laa.2014.06.002

Cited by 6 publications

(12 citation statements)

References 31 publications

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“…In this paper, we continue the work in [24] by proving that the spectral connection matrix has quasiseparable rank structure when the target family is the large and very useful family of Jacobi polynomials, including Chebychev polynomials as special cases. We also prove the same result for any change of basis within the Hermite, Laguerre, and Jacobi types.…”

Section: Introductionmentioning

confidence: 82%

“…In [23] it was shown that the spectral connection matrix corresponding to a change of basis within the Laguerre or within the Jacobi families has semiseparable structure, which is a type of the quasiseparable structure considered here. In [24], the connection problem between different families chosen among Hermite, Laguerre, and Gegenbauer was solved. There, it was shown that the spectral connection matrix has quasiseparable rank structure, and explicit algorithms were given.…”

Section: Introductionmentioning

confidence: 99%

“…At the conclusion, we also address a change of basis into any of these types from the set of Bessel polynomials. Explicit formulas for the generators of these quasiseparable matrices are given, allowing a fast algorithm to be implemented as in [24].…”

Section: Introductionmentioning

confidence: 99%

“…In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix Mathematics 2015, 3 383 are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014). …”

mentioning

confidence: 99%

“…are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).…”

mentioning

confidence: 99%

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The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials

Bella¹,

Reis²

2015

Mathematics

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Abstract:The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change-of-basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change-of-basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix Mathematics 2015, 3 383 are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).

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Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).…”

mentioning

confidence: 99%

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The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials

Bella¹,

Reis²

2015

Mathematics

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Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

Jiao

Wang

Huang

2016

Journal of Computational Physics

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Abstract. The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order µ ∈ (0, 1) to compute that of any order k + µ with integer k ≥ 0, while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order µ ∈ (0, 1). Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at "interior" JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, are essential for our algorithm development.

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On connection sequence for equivalent polynomial sets

Chaggara

Mbarki

2016

Integral Transforms and Special Functions

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The spectral connection matrix for classical orthogonal polynomials of a single parameter

Cited by 6 publications

References 31 publications

The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials

The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

On connection sequence for equivalent polynomial sets

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