Wynn%20epsilon%20algorithm

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1: 22.16 Related Functions

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§22.16(ii) Jacobi’s Epsilon Function

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Integral Representations

… ►See Figure 22.16.2. … ►

Quasi-Addition and Quasi-Periodic Formulas

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Relation to Theta Functions

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2: 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. … ►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 … ►It is now necessary to take the limit $\epsilon \to 0+$ of $F(x^{\prime }+\mathrm{i}\epsilon )$, and the imaginary part is the required Stieltjes–Perron inversion: …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. …

3: 33.24 Tables

§33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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Curtis (1964a) tabulates $P_{\mathrm{\ell }}(ϵ,r)$, $Q_{\mathrm{\ell }}(ϵ,r)$ (§33.1), and related functions for $\mathrm{\ell }=0,1,2$ and $ϵ=-2(.2)2$, with $x=0(.1)4$ for and $x=0(.1)10$ for $ϵ\ge 0$; 6D.

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4: 20 Theta Functions

Chapter 20 Theta Functions

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5: 3.9 Acceleration of Convergence

… ►The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: ►

${\epsilon }_{-1}^{(n)}=0,$

… ►Then $t_{n,2k}={\epsilon }_{2k}^{(n)}$. … ►If $s_n$ is the $n$th partial sum of a power series $f$, then $t_{n,2k}={\epsilon }_{2k}^{(n)}$ is the Padé approximant ${[(n+k)/k]}_f$ (§3.11(iv)). ►For further information on the epsilon algorithm see Brezinski and Redivo Zaglia (1991, pp. 78–95). …

6: 3.11 Approximation Techniques

… ►The Padé approximants can be computed by Wynn’s cross rule. Any five approximants arranged in the Padé table as … ►For the recursive computation of ${[n+k/k]}_f$ by Wynn’s epsilon algorithm, see (3.9.11) and the subsequent text. … ►is of fundamental importance in the FFT algorithm. …For further details and algorithms, see Van Loan (1992). …

7: Bibliography W

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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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External Links:

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

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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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.

External Links:

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

Referenced by:

7th item, 10th item.

Encodings:

BibTeX

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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)

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§3.9(vi), §5.23(i).

Encodings:

BibTeX

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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:

BibTeX

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Upon systems of recursions which obtain among the quotients of the Padé table. Numer. Math. 8 (3), pp. 264–269.

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External Links:

ISSN 0029-599X, Document, MathReview (F. L. Bauer), ZentralBlatt

Referenced by:

§3.11(iv).

Encodings:

BibTeX

8: 2.11 Remainder Terms; Stokes Phenomenon

… ►Similar improvements are achievable by Aitken’s ${\mathrm{\Delta }}^2$-process, Wynn’s $ϵ$-algorithm, and other acceleration transformations. … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …

9: 8 Incomplete Gamma and Related
Functions

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10: 28 Mathieu Functions and Hill’s Equation

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