Clenshaw%20algorithm

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1: Bibliography O

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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

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§3.11(ii).

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Note on backward recurrence algorithms. Math. Comp. 26 (120), pp. 941–947.

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FullText, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.74(iv), §3.6(v).

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Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.

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FullText, MathReview (J.C.P. Miller), ZentralBlatt

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§3.6(iii).

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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

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§2.9(i).

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Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

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Notes:

Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.

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ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

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2: Bibliography L

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

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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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ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

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Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

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Notes:

Includes Algol program, maximum accuracy 12D.

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ISSN 0001-0782, Document

Referenced by:

§7.25(iv).

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Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

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MathReview (F. Gotze), ZentralBlatt

Referenced by:

§13.31(iii).

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Algorithms for the Computation of Mathematical Functions. Academic Press, New York.

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MathReview (Gh. Adam)

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§16.26.

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3: Bibliography P

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Algorithm 385: Exponential integral $\mathrm{Ei}(x)$ . Comm. ACM 13 (7), pp. 446–447.

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Notes:

Includes Fortran code, maximum accuracy 20D. For certification and remarks see same journal v. 13 (1970), pp. 448–449 and p. 750; v. 15 (1972), p. 1074.

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ISSN 0001-0782, Document

Referenced by:

§6.21(ii).

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

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Notes:

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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Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.

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ISSN 0377-0427, Document, ZentralBlatt

Referenced by:

§10.77(ix).

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Algorithm 680: Evaluation of the complex error function. ACM Trans. Math. Software 16 (1), pp. 47.

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Notes:

Single-precision Fortran, maximum accuracy 14S.

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ISSN 0098-3500, Document, GAMS (module 680; package TOMS), MathReview, ZentralBlatt

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1st item.

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4: Bibliography R

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

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

Referenced by:

§3.11(i).

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

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Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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ISSN 0098-3500, Document

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1st item.

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Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

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Precision controlled trigonometric algorithms. Appl. Math. Comput. 2 (4), pp. 335–352.

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Document, MathReview (Luciano Biasini), ZentralBlatt

Referenced by:

§4.45(i).

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5: 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

Referenced by:

§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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External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.1(iv).

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6: 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 simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let $N$ be a positive integer and define … ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. Gautschi (2004, p. 119–120) has explored the $\epsilon \to 0^{+}$ limit via the Wynn $\epsilon$-algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in $w(x)$, depending smoothly on $x^{\prime }$, for $N\approx 4000$, for an example involving first numerator Legendre OP’s. …

7: 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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Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

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ISSN 0098-3500, Document, GAMS (module 794; package TOMS), ZentralBlatt

Referenced by:

7th item, 10th item.

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A rational approximant for the digamma function. Numer. Algorithms 33 (1-4), pp. 499–507.

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International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001)

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§3.9(vi), §5.23(i).

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An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities. Invent. Math. 108, pp. 575–633.

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ISSN 0020-9910, Document, MathReview (David R. Masson), ZentralBlatt

Referenced by:

§16.4(iii).

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Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.

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Notes:

Includes Algol code, maximum accuracy 8S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

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BibTeX

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