Clenshaw%E2%80%93Curtis%20quadrature%20formula

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1: 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

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§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

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§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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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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2: 29.20 Methods of Computation

… ►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►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. …

3: 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). …

4: 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

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Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Double-precision Fortran, maximum accuracy 20S.

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ISSN 0010-4655, Document, Code

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6th 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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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

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§33.4.

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Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.

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Document, ZentralBlatt

Referenced by:

§8.24(ii).

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5: 18.40 Methods of Computation

… ►For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. For further information see Clenshaw (1955), Gautschi (2004, §§2.1, 8.1), and Mason and Handscomb (2003, §2.4). … ►

A numerical approach to the recursion coefficients and quadrature abscissas and weights

… ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►The quadrature points and weights can be put to a more direct and efficient use. …

6: 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)

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§32.12(i).

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Single-precision Fortran.

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ISSN 0010-4655, Document, Code

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9th item.

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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:

Algol procedures, maximum accuracy 14S.

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FullText, ZentralBlatt

Referenced by:

§12.21(ii).

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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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Six myths of polynomial interpolation and quadrature. Math. Today (Southend-on-Sea) 47 (4), pp. 184–188.

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FullText

Referenced by:

§3.5(iv), §3.5(iv), Ch.3.

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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: 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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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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Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.

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ISSN 0923-2958, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§25.18(i).

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9: 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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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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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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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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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(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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10: 6.20 Approximations

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Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\mathrm{Ei}\left(x\right)$, with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$, with accuracies up to 20S.

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Clenshaw (1962) gives Chebyshev coefficients for $-E_1\left(x\right)-\ln |x|$ for $-4\le x\le 4$ and ${\mathrm{e}}^x{E}_1\left(x\right)$ for $x\ge 4$ (20D).

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