Padé approximant

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1: 3.11 Approximation Techniques

… ►It is denoted by ${[p/q]}_f\left(z\right)$. Thus if $b_0\ne 0$, then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in $z^{p+q}$. … ►

3.11.24

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Symbols:

${[p/q]}_f$: Padé approximant

Permalink:

http://dlmf.nist.gov/3.11.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(iv), §3.11 and Ch.3

… ►Starting with the first column ${[n/0]}_f$, $n=0,1,2,\mathrm{\dots }$, and initializing the preceding column by ${[n/-1]}_f=\mathrm{\infty }$, $n=1,2,\mathrm{\dots }$, we can compute the lower triangular part of the table via (3.11.25). Similarly, the upper triangular part follows from the first row ${[0/n]}_f$, $n=0,1,2,\mathrm{\dots }$, by initializing ${[-1/n]}_f=0$, $n=1,2,\mathrm{\dots }$. …

2: 8.10 Inequalities

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Padé Approximants

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3: 13.31 Approximations

… ►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985). …

4: Bibliography W

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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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External Links:

ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(ii).

Encodings:

BibTeX

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5: 3.9 Acceleration of Convergence

… ►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)). …

6: Bibliography

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Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

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External Links:

ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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7: Bibliography B

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Padé Approximants. 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 59, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-45007-1, MathReview (Annie A. M. Cuyt), ZentralBlatt

Referenced by:

§3.11(iv), §3.11(iv).

Encodings:

BibTeX

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8: 18.40 Methods of Computation

… ►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. …