Clenshaw–Curtis formula (extended)

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1: 3.5 Quadrature

… ►If we add $-1$ and $1$ to this set of $x_k$, then the resulting closed formula is the frequently-used Clenshaw–Curtis formula, whose weights are positive and given by … ►For detailed comparisons of the Clenshaw–Curtis formula with Gauss quadrature (§3.5(v)), see Trefethen (2008, 2011). … ►A comparison of several methods, including an extension of the Clenshaw–Curtis formula (§3.5(iv)), is given in Evans and Webster (1999). …

2: Bibliography C

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A method for numerical integration on an automatic copmputer. Numer. Math. 2 (4), pp. 197–205.

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ISSN 0029-599X, Document, MathReview (A. H. Stroud), ZentralBlatt

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§3.5(iv).

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Generalized exponential and logarithmic functions. Comput. Math. Appl. Part B 12 (5-6), pp. 1091–1101.

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ISSN 0886-9561, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§4.12.

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Algorithms for special functions. I. Numer. Math. 4, pp. 403–419.

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ISSN 0029-599X, Document, MathReview (A. S. Householder), ZentralBlatt

Referenced by:

§5.24(ii).

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Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.1(iv).

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Beyond floating point. J. Assoc. Comput. Mach. 31 (2), pp. 319–328.

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Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.1(iv).

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3: Bibliography P

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An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

External Links:

Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

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Document, Code, ZentralBlatt

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1st item.

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Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.

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ISSN 0006-3835, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(vii).

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Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.

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Document

Referenced by:

§10.76(ii).

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Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.

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Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.4.

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4: Bibliography W

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Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.

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ISSN 0006-3835, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(iv).

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Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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Document

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(20.7.34), §20.7(ix).

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Rational Chebyshev approximations for the Bessel functions $J_0(x)$, $J_1(x)$, $Y_0(x)$, $Y_1(x)$ . Math. Comp. 39 (160), pp. 617–623.

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Notes:

Tables on microfiche

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ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

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9th item.

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Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.

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FullText, Document, MathReview (S. D. Bajpai), ZentralBlatt

Referenced by:

§16.3(ii).

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Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

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ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

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5: Bibliography T

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On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.

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Painlevé functions and asymptotic analysis (Japanese) (Kyoto, 1995)

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FullText, MathReview (Andrei A. Kapaev)

Referenced by:

§32.12(i).

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Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.

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ISSN 0025-5858, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.4(v), §24.6.

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Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.

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ISSN 0036-1445, Document, MathReview (Kai Diethelm), ZentralBlatt

Referenced by:

§3.5(iv).

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6: 29.20 Methods of Computation

… ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … ►§29.15(i) includes formulas for normalizing the eigenvectors. …

7: 28.34 Methods of Computation

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Representations for $w_{\text{I}}(\pi ;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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8: 36.15 Methods of Computation

… ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. … ►For details, see Connor and Curtis (1982) and Kirk et al. (2000). …

9: 33.23 Methods of Computation

… ►Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … ►A set of consistent second-order WKBJ formulas is given by Burgess (1963: in Eq. …

10: Bibliography O

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Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.8(v), §22.11, §4.27.

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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Bounds for the solutions of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.

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External Links:

MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§2.9(i).

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Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

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Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

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Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

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