Clenshaw–Curtis formula (extended)

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11: 18.38 Mathematical Applications

… ►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). … ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

… ►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. … ►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas. … ►These generalize the ladder operators, as reviewed and extended by Infeld and Hull (1951), and also called creation and annilhilation operators. …

12: 22.21 Tables

… ►Curtis (1964b) tabulates $\mathrm{sn}(mK/n,k)$, $\mathrm{cn}(mK/n,k)$, $\mathrm{dn}(mK/n,k)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. … ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.

13: Bibliography R

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Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.

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External Links:

ISSN 0029-599X, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§3.11(i).

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Notes:

Symbolic and extended-precision general purpose computer algebra system.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

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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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External Links:

ZentralBlatt

Referenced by:

§27.17.

Encodings:

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Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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External Links:

ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

Referenced by:

§34.6.

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14: 3.11 Approximation Techniques

… ►For details and examples of these methods, see Clenshaw (1957, 1962) and Miller (1966). … ►

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 $p_n(x)$ is replaced by a rational function $R_{k,\mathrm{\ell }}(x)$. …

15: 4.47 Approximations

… ►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. …

16: Preface

… ►Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook. … ►Thus the utilitarian value of the Handbook will be extended far beyond its original scope and the traditional limitations of printed media. …

17: 5.24 Software

… ►See also Borwein and Zucker (1992), Carmignani and Tortorici Macaluso (1985), Clenshaw et al. (1962), Cody (1991), Filho and Schwachheim (1967), and Temme (1994a). …

18: 33.1 Special Notation

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$P_{\mathrm{\ell }}(ϵ,r)=(2\mathrm{\ell }+1)!f(ϵ,\mathrm{\ell };r)/2^{\mathrm{\ell }+1}$, $Q_{\mathrm{\ell }}(ϵ,r)=-(2\mathrm{\ell }+1)!h(ϵ,\mathrm{\ell };r)/(2^{\mathrm{\ell }+1}A(ϵ,\mathrm{\ell }))$.

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19: Foreword

… ►In 1964 the National Institute of Standards and Technology published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. … ►The document you are now holding, or the Web page you are now reading, represents an effort to extend the legacy of A&S well into the 21st century. The new printed volume, the NIST Handbook of Mathematical Functions, serves a similar function as the original A&S, though it is heavily updated and extended. …

20: 4.12 Generalized Logarithms and Exponentials

… ►For further information, see Clenshaw et al. (1986). …