| 释义 |
Padé ApproximantApproximants derived by expanding a function as a ratio of two Power Series and determining both the Numeratorand Denominator Coefficients. Padé approximations are usually superior to TaylorExpansions when functions contain Poles, because the use of RationalFunctions allows them to be well-represented.
The Padé approximant  corresponds to the Maclaurin Series. When it exists, the  Padéapproximant to any Power Series
 | (1) |
 is a Transcendental Function, then the terms are given by the Taylor Series about 
 | (2) |
 | (3) |
 can be multiplied by an arbitrary constant which will rescale theother Coefficients, so an addition constraint can be applied. The conventional normalization is
 | (4) |
These give the set of equations
where  for  and  for  . Solving these directly gives
 | (13) |
where sums are replaced by a zero if the lower index exceeds the upper. Alternate forms are for
and  .
The first few Padé approximants for  are
Two-term identities include
 | (16) |
 | (17) |
 | (18) |
 | (19) |
 | (20) |
 | (21) |
where  is the C-Determinant. Three-term identities can be derived using theFrobenius Triangle Identities (Baker 1975, p. 32).
A five-term identity is
 | (22) |
 | (23) |
 | (24) |
 | (25) |
 | (26) |
 | (27) |
References
 Padé ApproximantsBaker, G. A. Jr. ``The Theory and Application of The Pade Approximant Method.'' In Advances in Theoretical Physics, Vol. 1 (Ed. K. A. Brueckner). New York: Academic Press, pp. 1-58, 1965. Baker, G. A. Jr. Essentials of Padé Approximants in Theoretical Physics. New York: Academic Press, pp. 27-38, 1975. Baker, G. A. Jr. and Graves-Morris, P. Padé Approximants. New York: Cambridge University Press, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Padé Approximants.'' §5.12 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 194-197, 1992.
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