Gram Polynomials and the Kummer Function

Barnard, Roger W.; Dahlquist, Germund; Pearce, Kent; Reichel, Lothar; Richards, Kendall C.

doi:10.1006/jath.1998.3181

Cited by 26 publications

(17 citation statements)

References 19 publications

(12 reference statements)

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“…, that is, similarly to (3.4), they satisfy , we have that C (2) ) andq =q(y (2) ), then, similarly as in the proof of Theorem 1, we have that…”

Section: N ]mentioning

confidence: 59%

“…This code is essentially the same as the one in Figure 1 for roots of unity, except for the explicit computation of the barycentric weights for the grid of Chebyshev points and the use of the DCT Orthogonal polynomials with respect to ·, · y (2) have been recently derived by Eisinberg and Fedele [12]. They are…”

Section: N ]mentioning

confidence: 99%

“…It follows from Theorem 1 that a rational interpolant for y (2) can be constructed from these polynomials. The method, however, can be slightly modified in this case to make it simpler by using Chebyshev polynomials instead.…”

Section: N ]mentioning

confidence: 99%

“…For the implementation with the barycentric formula (2.8), the weights for y (2) are given by [44] (3.9)…”

Section: N ]mentioning

confidence: 99%

“…First when N is large, Gram polynomials of degrees close to N have exponentially large oscillations near the endpoints, making them unsuitable for approximation of functions [2]. This introduces large numerical errors in the computation of the product C * m+1:N ΦC 0:n in Theorem 1, affecting the accuracy of the coefficients vector β.…”

Section: Equidistant Pointsmentioning

confidence: 99%

See 4 more Smart Citations

Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

Pachon¹,

Gonnet²,

Deun³

2012

SIAM J. Numer. Anal.

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A new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented. When the interpolation nodes are roots of unity or Chebyshev points, the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The appearance of common factors in the numerator and denominator due to finite precision arithmetic is explained by the behaviour of the singular values of the linear system associated to rational interpolation problem. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koç. Short codes in Matlab and numerical experiments are included.

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“…, that is, similarly to (3.4), they satisfy , we have that C (2) ) andq =q(y (2) ), then, similarly as in the proof of Theorem 1, we have that…”

Section: N ]mentioning

confidence: 59%

Section: N ]mentioning

confidence: 99%

Section: N ]mentioning

confidence: 99%

“…For the implementation with the barycentric formula (2.8), the weights for y (2) are given by [44] (3.9)…”

Section: N ]mentioning

confidence: 99%

Section: Equidistant Pointsmentioning

confidence: 99%

See 3 more Smart Citations

Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

Pachon¹,

Gonnet²,

Deun³

2012

SIAM J. Numer. Anal.

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Modelling of thermo-chemical properties over the sub-solidus MgO–FeO binary, as a function of iron spin configuration, composition and temperature

et al. 2014

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Thermo-chemical properties and T–X phase relations diagram of the (Mg,Fe)O solid solution are modelled using mixing Helmholtz energy, ΔF(T,x)mixing, calculated by quantum mechanical and semi-empirical techniques. The sub-solidus MgO–FeO binary has been explored as a function of composition, with iron either in high-spin (HS) or low-spin (LS) configuration. Only the HS model provides physically sound results at room pressure, yielding a correct trend of cell edge versus composition, whereas LS’s issues are at variance with observations. Mixing Helmholtz energy has been parametrized by the following relationship: ΔF(T,x)mixing = x × y × [U0(T) + U1(T) × (x – y) + U2(T) × (x − y)²]−T × S(x,y)config, where y = 1−x and Uj(T) are polynomials in T of the second order. ΔF(T,x)mixing exhibits a quasi-symmetric behaviour and allows one to build the T–X phase relations diagram over the MgO–FeO join. The HS model including vibrational contribution to the Helmholtz energy predicts a solid solution’s critical temperature of some 950 K, remarkably larger than olivine’s and Mg–Fe garnet’s. All this points to a more difficult Mg–Fe mixing in periclase-like structure than olivine and garnet, which, in turn, provide more structure degrees of freedom for atomic relaxation. From ΔF(T,x)mixing, we have then derived ΔH(T,x)excess and ΔS(T,x)excess. The former, characterized by a quasi-regular behaviour, has been parametrized through W × x × (1−x), obtaining WH,Mg–Fe of 17.7(5) kJ/mol. ΔS(T,x)excess, in turn, increases as a function of temperature, showing absolute figures confined within 0.1 J/mol/K. Mixing Gibbs energy, calculated combining the present issues with earlier theoretical determinations of the magnesio-wüstite’s elastic properties, has shown that the HS configuration is stable and promote Mg–Fe solid solution up to ≈15 GPa

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A Domain Decomposition Fourier Continuation Method for Enhanced $$L_1$$ L 1 Regularization Using Sparsity of Edges in Reconstructing Fourier Data

Shi

Jung

2017

J Sci Comput

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Gram Polynomials and the Kummer Function

Cited by 26 publications

References 19 publications

Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

Modelling of thermo-chemical properties over the sub-solidus MgO–FeO binary, as a function of iron spin configuration, composition and temperature

A Domain Decomposition Fourier Continuation Method for Enhanced $$L_1$$ L 1 Regularization Using Sparsity of Edges in Reconstructing Fourier Data

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