# Clenshaw algorithm

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## 1—10 of 12 matching pages

##### 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
• (d)

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 algorithm3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

• ##### 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 algorithm3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …
##### 4: 3.11 Approximation Techniques
###### Summation of Chebyshev Series: Clenshaw’s Algorithm
For error analysis and modifications of Clenshaw’s algorithm, see Oliver (1977). …
##### 5: Bibliography C
• C. W. Clenshaw, G. F. Miller, and M. Woodger (1962) Algorithms for special functions. I. Numer. Math. 4, pp. 403–419.
• ##### 6: Bibliography R
• A. Ralston (1965) Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.
• ##### 7: Bibliography P
• K. A. Paciorek (1970) Algorithm 385: Exponential integral $\mathrm{Ei}(x)$ . Comm. ACM 13 (7), pp. 446–447.
• R. Piessens and M. Branders (1983) Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.
• R. Piessens and M. Branders (1984) Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.
• G. P. M. Poppe and C. M. J. Wijers (1990) Algorithm 680: Evaluation of the complex error function. ACM Trans. Math. Software 16 (1), pp. 47.
• P. J. Prince (1975) Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.
• ##### 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
• J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
• F. W. J. Olver and D. J. Sookne (1972) Note on backward recurrence algorithms. Math. Comp. 26 (120), pp. 941–947.
• F. W. J. Olver (1964a) Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.
• F. W. J. Olver (1967b) Bounds for the solutions of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.
• H. Oser (1960) Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.
• ##### 10: Bibliography W
• J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
• T. Weider (1999) Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.
• E. J. Weniger (2003) A rational approximant for the digamma function. Numer. Algorithms 33 (1-4), pp. 499–507.
• H. S. Wilf and D. Zeilberger (1992a) An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities. Invent. Math. 108, pp. 575–633.
• M. E. Wojcicki (1961) Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.