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11: 18.40 Methods of Computation

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

… ►See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain $w(x)$. … ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. …

12: 5.4 Special Values and Extrema

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5.4.7 $$\mathrm{\Gamma }\left(\frac{1}{3}\right)=\mathrm{2.67893\hspace{0.33em}85347\hspace{0.33em}07747\hspace{0.33em}63365}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.11 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073005; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

►

5.4.8 $$\mathrm{\Gamma }\left(\frac{2}{3}\right)=\mathrm{1.35411\hspace{0.33em}79394\hspace{0.33em}26400\hspace{0.33em}41694}\mathrm{\dots },$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.13 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073006; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

►

5.4.9 $$\mathrm{\Gamma }\left(\frac{1}{4}\right)=\mathrm{3.62560\hspace{0.33em}99082\hspace{0.33em}21908\hspace{0.33em}31193}\mathrm{\dots },$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.10 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A068466; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

►

5.4.10 $$\mathrm{\Gamma }\left(\frac{3}{4}\right)=\mathrm{1.22541\hspace{0.33em}67024\hspace{0.33em}65177\hspace{0.33em}64512}\mathrm{\dots }.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.14 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A068465; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

… ►As $n\to \mathrm{\infty }$, …

13: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►Temme (1997) shows how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions. … ►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. … ►Similar considerations apply to the spherical Bessel functions and Kelvin functions. … ►Newton’s rule (§3.8(i)) or Halley’s rule (§3.8(v)) can be used to compute to arbitrarily high accuracy the real or complex zeros of all the functions treated in this chapter. …

14: Bibliography G

… ►

Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.

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External Links:

ISSN 0008-414X, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(v).

Encodings:

BibTeX

… ►

Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

Encodings:

BibTeX

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How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.

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External Links:

ISSN 0006-3835, Document, MathReview (G. A. Evans), ZentralBlatt

Referenced by:

§18.40(ii), §3.5(v), §9.17(iii).

Encodings:

BibTeX

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

Encodings:

BibTeX

… ►

Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

Encodings:

BibTeX

…

15: 22.20 Methods of Computation

… ►To compute $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ to 10D when $x=0.8$, $k=0.65$. ►Four iterations of (22.20.1) lead to $c_4=\mathrm{6.5\times 10⁻¹²}$. … ►By application of the transformations given in §§22.7(i) and 22.7(ii), $k$ or $k^{\prime }$ can always be made sufficently small to enable the approximations given in §22.10(ii) to be applied. The rate of convergence is similar to that for the arithmetic-geometric mean. … ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute $\mathrm{am}(x,k)$. …

16: 3.6 Linear Difference Equations

… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►It therefore remains to apply a normalizing factor $\mathrm{\Lambda }$. … ►(This part of the process is equivalent to forward elimination.) … ►Similar considerations apply to the first-order equation …Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. …

17: 3.2 Linear Algebra

… ►To solve the system … ►During this reduction process we store the multipliers ${\mathrm{\ell }}_{jk}$ that are used in each column to eliminate other elements in that column. … ►The sensitivity of the solution vector $𝐱$ in (3.2.1) to small perturbations in the matrix $𝐀$ and the vector $𝐛$ is measured by the condition number … ►where $𝐱$ and $𝐲$ are the normalized right and left eigenvectors of $𝐀$ corresponding to the eigenvalue $\lambda$. … ►Lanczos’ method is related to Gauss quadrature considered in §3.5(v). …

18: 25.10 Zeros

… ►Calculations relating to the zeros on the critical line make use of the real-valued function …is chosen to make $Z(t)$ real, and $\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)$ assumes its principal value. … ►Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp \left(\mathrm{i}\vartheta (t)\right)$ to obtain the Riemann–Siegel formula: …where $R(t)=O\left(t^{-1/4}\right)$ as $t\to \mathrm{\infty }$. … ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

19: 1.13 Differential Equations

… ►Here dots denote differentiations with respect to $\zeta$, and $\{z,\zeta \}$ is the Schwarzian derivative: … ►For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984). … ►The functions $u(x)$ which correspond to these being eigenfunctions. … ►

Transformation to Liouville normal Form

►Equation (1.13.26) with $x\in [a,b]$ may be transformed to the Liouville normal form …

20: 19.36 Methods of Computation

… ►In the Appendix of the last reference it is shown how to compute $R_J$ without computing $R_C$ more than once. … ►As $n\to \mathrm{\infty }$, $c_n$, $a_n$, and $t_n$ converge quadratically to limits $0$, $M$, and $T$, respectively; hence … ►To (19.36.6) add … ►(19.22.20) reduces to $0=0$ if $p=x$ or $p=y$, and (19.22.19) reduces to $0=0$ if $z=x$ or $z=y$. … ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …