Padé approximant
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1: 3.11 Approximation Techniques
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►It is denoted by .
Thus if , then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in .
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3.11.24
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►Starting with the first column , , and initializing the preceding column by , , we can compute the lower triangular part of the table via (3.11.25).
Similarly, the upper triangular part follows from the first row , , by initializing , .
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2: 8.10 Inequalities
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Padé Approximants
…3: 13.31 Approximations
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►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985).
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4: Bibliography W
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Some explicit Padé approximants for the function and a related quadrature formula involving Bessel functions.
SIAM J. Math. Anal. 16 (4), pp. 887–895.
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5: 3.9 Acceleration of Convergence
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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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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8: 18.40 Methods of Computation
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►The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , 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 and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
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