Clenshaw algorithm

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1: 29.20 Methods of Computation

… ►The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). …

2: 28.34 Methods of Computation

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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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3: 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. …

4: 3.11 Approximation Techniques

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Summation of Chebyshev Series: Clenshaw’s Algorithm

… ►For error analysis and modifications of Clenshaw’s algorithm, see Oliver (1977). …

5: Bibliography C

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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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6: 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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7: 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.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii).

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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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Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.

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ISSN 0098-3500, Document, GAMS (module 498; package TOMS), ZentralBlatt

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1st item, 4th item.

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8: 5.24 Software

… ►In this section we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter. … ►See also Borwein and Zucker (1992), Carmignani and Tortorici Macaluso (1985), Clenshaw et al. (1962), Cody (1991), Filho and Schwachheim (1967), and Temme (1994a). …

9: 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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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§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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External Links:

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

Referenced by:

§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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External Links:

MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§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.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

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10: 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

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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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Notes:

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).

Encodings:

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