Clenshaw%E2%80%93Curtis%20formula%20%28extended%29
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11—20 of 373 matching pages
11: 4.47 Approximations
12: 3.11 Approximation Techniques
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►For details and examples of these methods, see Clenshaw (1957, 1962) and Miller (1966).
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Summation of Chebyshev Series: Clenshaw’s Algorithm
… ►For error analysis and modifications of Clenshaw’s algorithm, see Oliver (1977). … ►The theory of polynomial minimax approximation given in §3.11(i) can be extended to the case when is replaced by a rational function . …13: 5.24 Software
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►See also Borwein and Zucker (1992), Carmignani and Tortorici Macaluso (1985), Clenshaw et al. (1962), Cody (1991), Filho and Schwachheim (1967), and Temme (1994a).
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14: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions and , ,
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ACM Trans. Math. Software 3 (1), pp. 93–95.
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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument.
ACM Trans. Math. Software 16 (2), pp. 178–182.
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Special Functions.
Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.
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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.
J. Math. Anal. Appl. 416 (1), pp. 52–80.
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15: Bibliography R
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Rational Chebyshev approximation by Remes’ algorithms.
Numer. Math. 7 (4), pp. 322–330.
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Fourier analysis and signal processing by use of the Möbius inversion formula.
IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
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Mathieu functions of integral orders and real arguments.
IEEE Trans. Microwave Theory Tech. 28 (3), pp. 276–277.
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Another proof of the triple sum formula for Wigner -symbols.
J. Math. Phys. 40 (12), pp. 6689–6691.
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16: Bibliography L
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Elliptic Functions and Applications.
Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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Numerical evaluation of integrals containing a spherical Bessel function by product integration.
J. Math. Phys. 22 (7), pp. 1399–1413.
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Algorithm 567: Extended-range arithmetic and normalized Legendre polynomials [A1], [C1].
ACM Trans. Math. Software 7 (1), pp. 141–146.
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17: 4.12 Generalized Logarithms and Exponentials
18: 5.23 Approximations
19: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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20: 18.38 Mathematical Applications
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►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957).
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►If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding .
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►Light and Carrington Jr. (2000) review and extend the one-dimensional analysis to solution of multi-dimensional many-particle systems, where the sparse nature of the resulting matrices is highly advantageous.
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►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas.
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►These generalize the ladder operators, as reviewed and extended by Infeld and Hull (1951), and also called creation and annilhilation operators.
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