Symbolic Computation of Padé Approximants

Geddes, K. O.

doi:10.1145/355826.355835

Cited by 23 publications

(10 citation statements)

References 8 publications

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“…Also, "d" is the first letter in "determinant".) The Chebyshev-Pad6 (1,5), (2,4), and (2, 5) agree with f(x) through less than the expected m + n + 1 terms; the approximant in position (1,4), which also corresponds to a singular Hankel matrix, does have agreement with f(x) through m + n + 1 terms. This triangular nature is typical, with the lower right triangle in a (finite) square block containing "degenerate" approximants.…”

mentioning

confidence: 82%

“…Also note that if the function f(x) is a rational function (as in this example), then there is an infinite block extending to the right and downward from the position containing the exact function. [4]. )…”

mentioning

confidence: 98%

“…Finally, when condition (3.13) fails to llold we have the case of nonuniqueness, which is discussed in 6. 4. The complete Chebyshev-Pad6 table.…”

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confidence: 99%

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Block Structure in the Chebyshev–Padé Table

Geddes¹

1981

SIAM J. Numer. Anal.

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The problem of obtaining Chebyshev-Pad6 approximants for the Chebyshev series ' c,Tt,(x) is directly related to the Pad6 approximation of the corresponding power series ' ct, w t'. The theory of the block structure in the Pad6 table is generalized for the Chebyshev-Pad6 table. An anomaly of "nonexistence" of Chebyshev-Pad6 approximants is clarified by relating it to the location of the poles of the corresponding Pad6 approximant.

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confidence: 82%

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confidence: 98%

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Block Structure in the Chebyshev–Padé Table

Geddes¹

1981

SIAM J. Numer. Anal.

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“…For practical reasons, we choose instead the algorithm of Cabay and Kao [5] which requires 66' operations (including additions and multiplications, and ignoring lower order terms) to compute the first 6 diagonal Pade fractions for a single power series. This algorithm is particularly suitable for this application because it is iterative on 6 and because it is faster than other methods using classical polynomial arithmetic for computing Padi fractions, such as those of Geddes [9] and Rissanen [20].…”

Section: Conversion To Rational Formmentioning

confidence: 99%

Systems of linear equations with dense univariate polynomial coefficients

Cabay

1987

J. ACM

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An algorithm for computing the power series solution of a system of linear equations with components that are dense univariate polynomials over a field is described and analyzed. A method for converting the power series solution to rational form is derived. Theoretical and experimental cost estimates are obtained and used to identify classes of problems for which the power series method outperforms modular methods. Finally, it is shown that the power series method also provides an effective mechanism for solving the problem in which the coefficients of the polynomials are from the ring of integers.

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“…B. Keiper, refers to Froberg (1985). The Reader may also refer to Geddes (1979), Czapor & Geddes (1984) for their seminal value.…”

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Computation of hyperngeometric functions for gravitationally radiating binary stars

Pierro¹,

Pinto²,

F.³

2002

Monthly Notices of the Royal Astronomical Society

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The exact solution for any eccentricity of the Peters–Mathews model of binary stars emitting gravitational waves leads to hypergeometric functions of Appell and Gauss type. Efficient (fast and accurate) numerical computations of such functions can be obtained by using the Padé approximation. Padé rational approximants, for equal numbers of coefficients, are more accurate than Taylor series representation. Unfortunately, the algorithm computing Padé coefficients numerically from Taylor series is known to be unstable. This numerical instability lowers the accuracy of the Padé approximant. In this paper, double precision accuracy Padé approximant coefficients are computed symbolically. Advantages of the symbolic computation with respect to the numerical treatment are shown, providing accurate representations of the Appell and Gauss hypergeometric functions. The symbolic approach can be extended to higher post‐Newtonian models.

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Symbolic Computation of Padé Approximants

Cited by 23 publications

References 8 publications

Block Structure in the Chebyshev–Padé Table

Block Structure in the Chebyshev–Padé Table

Systems of linear equations with dense univariate polynomial coefficients

Computation of hyperngeometric functions for gravitationally radiating binary stars

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