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A recurrence scheme for converting from one orthogonal expansion into another

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Herbert E. SalzerAuthors Info & Claims

Communications of the ACM, Volume 16, Issue 11

Pages 705 - 707

https://doi.org/10.1145/355611.362548

Published: 01 November 1973 Publication History

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Abstract

A generalization of a scheme of Hamming for converting a polynomial P_n(x) into a Chebyshev series is combined with a recurrence scheme of Clenshaw for summing any finite series whose terms satisfy a three-term recurrence formula. An application to any two orthogonal expansions P_n(x) = ∑ⁿ_m=0 a_mq_m(x) = ∑ⁿ_m=0 A_mQ_m(x) enables one to obtain A_m directly from a_m, m = 0(1)n, by a five-term recurrence scheme.

References

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Abramowitz, M., and Stegun, I. (Ed.) Handbook of Mathematical Functions. Nat. Bur. Stands., Appl. Math. Ser., no. 55, Dover, New York, 1965, p. 782.]]

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[2]

Clenshaw, C. W. A note on the summation of Chebyshev series. Mathematical Tables and Other Aids to Computation 9 (1955), 118-120.]]

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Clenshaw, C. W. Chebyshev series for mathematical functions. Nat. Phys. Lab., Math. Tables, Vol. 5, H.M. Stationery Office, London, 1962, p. 9.]]

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Fox, L., and Parker, I.B. Chebyshev Polynomials in Numerical Analysis. Oxford Univ. Press, London, 1968, p. 56.]]

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Hamming, R.W. Numerical Methods Jor Scientists and Engineers. McGraw-Hill, New York, 1962, pp. 255-257.]]

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Hamming, R.W. Introduction to Applied Numerical Analysis. McGraw-Hill, New York, 1971, pp. 305-306.]]

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Slevinsky R(2019)Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier seriesApplied and Computational Harmonic Analysis10.1016/j.acha.2017.11.00147:3(585-606)Online publication date: Nov-2019
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Published In

Communications of the ACM Volume 16, Issue 11

Nov. 1973

135 pages

ISSN:0001-0782

EISSN:1557-7317

DOI:10.1145/355611

Editor:
CORPORATE ACM TOMS Staff

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Published: 01 November 1973

Published in CACM Volume 16, Issue 11

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Galagusz RMcFee S(2018)An iterative domain decomposition, spectral finite element method on non-conforming meshes suitable for high frequency Helmholtz problemsJournal of Computational Physics10.1016/j.jcp.2018.11.016Online publication date: Nov-2018
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Skrzipek M(2004)Inversion of Vandermonde-Like MatricesBIT10.1023/B:BITN.0000039420.97768.4944:2(291-306)Online publication date: 1-May-2004
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Laurie Dde Villiers J(2004)Orthogonal polynomials and Gaussian quadrature for refinable weight functionsApplied and Computational Harmonic Analysis10.1016/j.acha.2004.06.00217:3(241-258)Online publication date: Nov-2004
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Laurie D(1997)Calculation of Gauss-Kronrod quadrature rulesMathematics of Computation10.1090/S0025-5718-97-00861-266:219(1133-1145)Online publication date: 1997
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Cox M(1993)Reliable determination of interpolating polynomialsNumerical Algorithms10.1007/BF022156775:3(133-154)Online publication date: Mar-1993
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Barnett S(1987)Change of Basis for Products of Orthogonal PolynomialsSIAM Journal on Algebraic and Discrete Methods10.1137/06080138:2(155-162)Online publication date: 1-Apr-1987
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An interpolation algorithm for orthogonal rational functions

Computing Zeros on a Real Interval through Chebyshev Expansion and Polynomial Rootfinding

Recurrence equations and their classical orthogonal polynomial solutions

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